Defect stability in phase-ﬁeld crystal models: Stacking ...provatas/papers/Joel_PRB.pdf ·...

PHYSICAL REVIEW B 86, 224112 (2012)

Defect stability in phase-field crystal models: Stacking faults and partial dislocations

Joel Berry,1,3 Nikolas Provatas,1,2 Jörg Rottler,3 and Chad W. Sinclair41Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7

2Physics Department, McGill University, 3600 rue University, Montréal, Québec, Canada H3A 2T83Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,

British Columbia, Canada V6T 1Z14Department of Materials Engineering, University of British Columbia, 309-6350 Stores Road, Vancouver, British Columbia, Canada V6T 1Z4

(Received 26 September 2012; published 27 December 2012)

The primary factors controlling defect stability in phase-field crystal (PFC) models are examined, withillustrative examples involving several existing variations of the model. Guidelines are presented for constructingmodels with stable defect structures that maintain high numerical efficiency. The general framework combinesboth long-range elastic fields and basic features of atomic-level core structures, with defect dynamics operableover diffusive time scales. Fundamental elements of the resulting defect physics are characterized for the case offcc crystals. Stacking faults and split Shockley partial dislocations are stabilized for the first time within the PFCformalism, and various properties of associated defect structures are characterized. These include the dissociationwidth of perfect edge and screw dislocations, the effect of applied stresses on dissociation, Peierls strains forglide, and dynamic contraction of gliding pairs of partials. Our results in general are shown to compare favorablywith continuum elastic theories and experimental findings.

DOI: 10.1103/PhysRevB.86.224112 PACS number(s): 61.72.Bb, 61.72.Lk, 61.72.Nn, 62.20.F−

I. INTRODUCTION

Structural kinetics in crystalline solids are driven hetero-geneously at the atomic level by localized defects, which inturn drive mesoscopic and macroscopic phenomena such asstructural phase transformations, fracture, and other formsof plastic flow. A complete description of such processestherefore requires a multiscale approach. Existing modelingmethods typically operate either exclusively on atomic scalesor on meso- and macroscopic scales. Phase-field crystal(PFC) models on the other hand provide a framework thatcombines atomic length scales and mesoscopic/diffusive timescales,1–3 with the potential to reach mesoscopic lengthsthrough systematic multiscale expansion methods.4–8

The PFC approach naturally incorporates elasticity, plas-ticity, and effects of local crystal orientation into a relativelysimple atomic-level continuum theory.1,2 The literature pub-lished to date demonstrates that such a formulation, with onlya few minimal ingredients, gives rise to a broad range ofphysics associated with diffusive nonequilibrium processesin liquid and solid systems. One of its strengths in terms ofdescribing periodic systems is the wealth of inherent defectstructures that automatically emerge from the basic free energyfunctional. This permits the study of crystalline defects withina description that captures both long-range elastic fields andbasic features of atomic-level core structures. In addition tofundamental, local defect properties, the role of these defects indynamic materials phenomena that operate over long, diffusivetime scales can be examined. Therefore, defect stability, thecore structure of stable defect configurations, and the dynamicsand interactions of various defect structures within the PFCdescription all become central issues as the method is advancedfurther into solid-state phenomena. However, in light of theapproximations inherent to the approach, its limitations needto be adequately realized and understood as well.

The basic properties of perfect PFC crystals are relativelywell understood, but the various classes and types of defects

relevant to each given lattice symmetry require further studyif material-specific applications are to be pursued. Perfectdislocations and simple grain boundaries in 2D triangular and3D bcc crystals have been examined and are known to betopologically stable under typical conditions of small localstrain and low thermodynamic driving force.2,3,9–19 Certainother defect structures, notably stacking faults in close-packed3D crystals, have inherently lower topological stability, andin such cases the proper balance between crystal stabilityand defect stability in the model formulation becomes morerestrictive and difficult to achieve. It is also not necessarilyclear which of the now many versions of PFC are best suitedfor describing defect-mediated processes.

The aims of the first part of this article are to characterizethe nature of defect stability in PFC models in terms of afew general model features, and to examine how variousversions compare in this respect. We will show that fine controlof crystal structure often leads to reduced defect stability.A balance must therefore be found that sufficiently favorscrystalline order yet does not destabilize relatively unprotecteddefect structures such as planar faults. A few ways to achievethis balance are identified and compared critically. The mostefficient of these approaches is then, in the second part ofthis article, applied to the case of fcc crystals, in whichthe stability of stacking faults plays a central role. A rangeof defect properties and behaviors consistent with theoryand/or experiment are shown to naturally emerge from whatis still a very simple set of equations. Potential implicationsfor atomistic studies of plastic deformation involving slow,diffusive processes are discussed, and initial results concerninga few processes of interest are described. More detailedfindings relative to these will be outlined further in a futurepublication.

PFC models can be viewed as simplified versionsof classical density functional theory (CDFT).3,20,21 TheRamakrishnan-Yussouff (RY) CDFT22 provides a useful

224112-11098-0121/2012/86(22)/224112(12) ©2012 American Physical Society

http://dx.doi.org/10.1103/PhysRevB.86.224112

BERRY, PROVATAS, ROTTLER, AND SINCLAIR PHYSICAL REVIEW B 86, 224112 (2012)

reference point, with a free energy functional given by anexpansion around the liquid state correlation functions,

F

kBT=

∫d�r[ρ(�r) ln(ρ(�r)/ρ�) − δρ(�r)]

− 12

∫∫d�rd�r2δρ(�r)C2(�r,�r2)δρ(�r2) + · · · , (1)

where ρ(�r) is the atomic number density field, ρ� is a constantreference density, δρ(�r) = ρ(�r) − ρ�, and C2(�r,�r2) = C2(|�r −�r2|) is the two-point direct correlation function of the fluid,assumed isotropic.

A general PFC-type functional can be derived from Eq. (1),truncated beyond C2(|�r − �r2|), as described in Ref. 3. In termsof the rescaled atomic density field n(�r) = ρ(�r)/ρ� − 1, anexpansion of the logarithm in Eq. (1) generates a functionalF̃ = F/(kBTρ�) that can be written

F̃ =∫

d�r[

1

2n2(�r) − w

6n3(�r) + u

12n4(�r)

]

− 12

∫∫d�rd�r2n(�r)C2(|�r − �r2|)n(�r2). (2)

The expansion coefficients w and u are treated as freeparameters to provide additional model flexibility, and n(�r)is also typically allowed to assume nonzero average values n0.Such an expansion is justified for kernels C2 that producelow-amplitude density profiles, which in general requiressuppression of large wave number two-body correlations. Thecoefficient w will be set to zero throughout this article, asthe un4(�r)/12 term automatically gives rise to the effectiveterms un0n3(�r)/3 and un20n2(�r)/2.23 It will also be impliedthat u = 3 throughout this study.

Three dynamic equations for n(�r) will be considered here.The first is a purely diffusive model B form,

∂n(�r)∂t

= ∇2 δF̃δn(�r) , (3)

where t is dimensionless time. The second equation of motionintroduces a faster inertial, quasiphonon dynamic componentin addition to diffusive dynamics,10

∂2n(�r)∂t2

+ β ∂n(�r)∂t

= α2∇2 δF̃δn(�r) , (4)

where α and β are constants related to sound speed anddamping rate, respectively. The final equation of motion,applicable to Eq. (1), is

∂ ln [n(�r) + 1]∂t

= − δF̃δn(�r) . (5)

With imposed density conservation this equation provides anaccelerated path to local energy minima.24 All simulationsin this study were performed in 3D using pseudospectralalgorithms and periodic boundary conditions. Those thatemployed Eqs. (3) or (4) used semi-implicit time stepping,while those that employed Eq. (5) used explicit time stepping.

II. ORDER AND DEFECTS IN PFC MODELS

In this section, the primary factors controlling defectstability in PFC and CDFT models are first outlined through

an analysis of stacking faults applicable to both model types.Three potential solutions to the problem of defect instabilityare examined and shown to be sufficient for stabilizationof stacking faults in fcc crystals. These approaches centeron multipeaked correlation functions, few-peaked correla-tion functions with broad effective envelopes, and entropy-driven formulations, respectively. Section III expands onthese concepts with a more detailed examination of specificissues concerning defect structure geometry, interactions, anddynamics.

A. Large wave number or multipeaked models

The efficiency and tractability of PFC models relativeto CDFT are direct consequences of the central PFC ap-proximation: truncation of two-body correlations beyond thefirst few primary correlation peaks in Fourier space [whichpermits the truncated expansion of the logarithm in Eq. (1)].All structural information is consolidated into the wavenumber range around the first few primary peaks in thestructure factor. This still permits control of basic structuralsymmetries, but produces local density peaks with broad,sine-wave-like profiles rather than the sharply peaked Gaussianprofiles that emerge from CDFT functionals with large wavenumber correlations. Though this small wave number approx-imation reduces model complexity and increases efficiencyby orders of magnitude, it can in some cases also reducedefect stability, as will be shown in the present subsection.Methods by which defects can be stabilized while stillretaining the small wave number approximation are discussedin Sec. II B.

1. Many-peaked XPFC model

The effect of large wave number correlations on defectstability can be demonstrated through an examination of thePFC model formulation of Greenwood et al.25–27 (XPFC).This approach allows one to stabilize many different crystalsymmetries by constructing a correlation kernel with peaks,typically Gaussians, located at the first few primary reflectionsof a given lattice structure. The Fourier transformed XPFCkernel can be written

Ĉ2(k)i = −r + He−(k−ki )2/(2α2i )e−σ 2k2i /(2ρiβi ), (6)where i denotes a family of lattice planes at wave numberki , and σ is a temperature parameter. The constants αi ,ρi , and βi are the Gaussian width (which sets the elasticconstants), atomic density, and number of planes, respectively,associated with the ith family of lattice planes. We haveintroduced the additional constants r and H to permit Ĉ2(k)to assume negative values between peak maxima, whichimproves numerical stability when used with Eq. (1). Theenvelope of all selected Gaussians i shifted by the constant −rthen composes the final Ĉ2(k). For example, two such peakswith ratio k2/k1 =

√4/3 produce an fcc structure in which the

first peak corresponds to {111} and the second to {200}, thetwo primary fcc reflections.

The issue of defect stability and its reliance on thecorrelation kernel can be illustrated with an analysis of thefcc intrinsic stacking fault energy (γISF or SFE) and stabilityas a function of the number of fcc reflections considered

224112-2

DEFECT STABILITY IN PHASE-FIELD CRYSTAL . . . PHYSICAL REVIEW B 86, 224112 (2012)

(a) (b) (c)

x = [1̄12]y = [110]

z = [11̄1]

x

z

x

x

z

ABCACABC

FIG. 1. (Color online) (a) An intrinsic NL = 35 fcc stacking faultin a periodic simulation cell. (b) Close-up and full xz views of thefault. (c) Full xz view of a sheared/unfaulted crystal in the same cell.Yellow lines are guides to the eye. These images were generated usingEqs. (2) and (9) with parameter values n0 = −0.48, r = −0.63, andBx = 1.

in Ĉ2(k). The full CDFT of Eq. (1) with Ĉ2(k) given byEq. (6) was employed, as discussed below. Faulted crystalswere evaluated within a periodic simulation cell with (�x,�y,�z)axes in ([1̄12],[110],[11̄1]) directions, and a single intrinsicstacking fault was created by removing one close-packed layerand shifting the layers above down by a/

√3, where a =

2√

3π/k1 = 1.8537 is the equilibrium fcc lattice constant.The final number of close-packed layers NL must satisfyNL = 3n − 1, where n > 1 is an integer (NL = 35 was usedin most cases here). Initial states were relaxed using Eq. (5). Inthis geometry, the stacking fault may be stable, metastable,or unstable, with instability unfolding through a shearingoperation τzx or τzy that removes the stacking fault and createsa uniformly sheared perfect crystal with total shear strain� = (√2NL)−1, as shown in Fig. 1.

Some results are displayed in Fig. 2 along with specificparameter values employed. Stacking faults were found tobe highly unstable in the two-peaked model for virtually allmeaningful values of αi (� 2). With five peaks, metastablefaults were observed, while ten peaks produced fully stablefaults for which the energy of the faulted crystal was veryslightly lower than that of the sheared state (δF̃ = F̃SF −F̃Shear � 0). A 15-peak model was also found to be fully stableagainst shear, with a larger negative δF̃ . The relevant structurefactors for the 15-peak model are shown in Fig. 2. These resultsindicate that the stability of stacking faults to shear increasesas larger-k correlations are considered. The reason for thisbehavior is discussed later in this section.

Significantly, the computational efficiency of these modelsdecreases rapidly with increasing kmax. The amplitude of thedensity peaks must be allowed to grow roughly as kemax if thelarger-k modes are to contribute significantly to the free energyof the system. This necessitates use of the full logarithmicone-body term in the free energy, without expansion, sincethe truncated expansion of Eq. (2) limits stable amplitudes torelatively small values, O(1). The logarithm greatly reducesthe maximum allowable time step, while the higher-k modes

-2

-1

0

1

C2(

k)

XPFCPY

-2-1 0 1

5 10 15 20 25 30

SF

E,

F

kmax

SFEF

102103104105106

S(k

)

Faulted fccSheared fcc

0

1

2

3

5 10 15 20 25 30

S(k

) x1

05

k

Faulted fccSheared fcc

ˆ˜

˜

FIG. 2. (Color online) Effect of large-k modes on stacking faultstability. From top to bottom: Cross-sectional normalized n(�r) mapsof the XPFC fcc crystal with 2, 5, 10, and 15 fcc reflections fromleft to right; Ĉ2(k) of the PY hard-sphere model at ρ� = 0.9445 andof the XPFC model with 2, 5, 10, and 15 fcc peaks; log plot of thestructure factor S(k) = 〈|n̂(k)|2〉 for faulted and sheared fcc crystalsin the 15 peak XPFC model; linear S(k) plot for faulted and shearedfcc crystals in the PY hard-sphere model; XPFC γISF and NL = 35driving force for instability, δF̃ = F̃SF − F̃Shear vs kmax. Symbols inthe S(k) plots are Bragg reflection maxima of perfect fcc crystals ineither model. XPFC parameter values: ρ� = 1.0488, αi = 1, σ = 0,H = 3, and r = 2. H and r were varied from their default values,H = 1 and r = 0, to ensure numerical stability of the logarithmicterm in Eq. (1).

reduce the density peak width (as shown in Fig. 2, top) andtherefore the maximum allowable grid spacing x.

2. Many-peaked CDFT model

A similar analysis was attempted using the full RY CDFTof Eq. (1), with Ĉ2(k) given by the Percus-Yevick (PY)expression for hard spheres28,29 and dynamics given by Eq. (5).Our findings indicate that hcp crystals in this model actuallytend to have a slightly lower free energy than fcc crystals,a fact that apparently has not been appreciated before. Thisleads to a negative γISF and therefore stacking faults that areinherently stable to the sheared state. Nonetheless, a perfectedge dislocation in a metastable fcc crystal was found tosplit into two well-separated Shockley partial dislocationsconnected by a stacking fault, as expected for low γISFmaterials (Fig. 3). The reaction is

�b = a2

[110] → a6

[121] + γISF + a6

[211̄]. (7)

224112-3


SF

bθ

b1

θ1

b2

θ2

x = [1̄12]

y = [110]

z = [11̄1]

StackingFault

ShockleyPartials

b2

b1

FIG. 3. (Color online) Split fcc edge dislocation in the ρ� =0.9445 PY hard-sphere CDFT of Eq. (1). A cutaway view of thedensity field ρ(�r) is shown, with the color scale truncated aboveρ(�r) = 1, though ρ(�r) � 7800 at the peak maxima. Left: Diagram-matic representation of the dissociation of a perfect dislocation �b intoShockley partials �b1 and �b2.

Though both γISF and δF̃ in this system are always negativeor very small and positive, making instability to shear virtuallyimpossible, we did find that δF̃ increases as kmax is reduced.This was determined by modifying the PY function as Ĉ2(k) =ĈPY2 (k)e

−(k−kmax)/2 for k � kmax, while holding x constant.The observed trend in δF̃ indicates that the underlying linkbetween stability and kmax found in the XPFC model is alsopresent here.

3. Analysis and generalization

This link can be understood by examining the spec-tra of the two relevant states, the faulted crystal and thesheared/unfaulted crystal (see Fig. 2). An infinite stackingfault can be expressed mathematically as a 1D step functionin the 3D displacement field u(�r) of the crystal, with thestep direction normal to the fault. The structure factor of afaulted crystal therefore contains the same 1/k2 line shapescharacteristic of scattering from a surface. In this case eachlattice reflection k(i), except the set of {111} planes parallelto the stacking fault, will possess its own set of 1/(kz − k(i)z )2modes, and these modes will only extend in the reciprocalspace direction perpendicular to the plane of the fault, kz.The spherically averaged structure factor S(k) thus exhibitsa sequence of broadened fcc reflections, each with effective1/(k − k(i))2 line shapes of roughly the same width.30

The structure factor of a sheared perfect crystal also exhibitsa form of peak broadening, but in this case the degree ofbroadening is proportional to k. Depending on the orientationof shear strain �, or deformation in general, the spacing dbetween certain lattice planes within any given family changesto some value d ′. A simple analysis shows that the resultingshift in k space under shear is

δk = 2π(

1

d ′− 1

d

)= 1 − cos ω

cos ωk, (8)

where ω = arctan �. Thus the primary low-k reflections remainsharp and/or slightly split, while the degree of shift or effectivebroadening increases in proportion to k due to this growingdecoherence effect across high-index, low-d planes.

The driving force for stacking fault instability is primarilya function of δF̃ = F̃SF − F̃Shear. When a low-mode kernelis used, F̃Shear will tend to be smaller than F̃SF because theslightly shifted low-k modes of a sheared crystal induce asmall energy cost relative to that of the long-range line-shapetails of a faulted crystal, provided that the shear strain is nottoo large. As kmax increases, this situation eventually reverses.The high-order reflections from the sheared lattice becomebroader or less coherent than those of the faulted crystalsuch that if enough high-k modes are considered, stackingfault stability can eventually be attained when F̃Shear > F̃SF.This effect can be seen in the data of Fig. 2. We emphasizethat the high-k modes do not necessarily decrease γISF orF̃SF, but that the increase in F̃Shear alone leads to a dramaticincrease in stacking fault stability. In the geometry of oursimulations, the critical wave number at which F̃SF = F̃Shearincreases with NL since � = (

√2NL)−1, but for fixed NL

stability increases with kmax. The exact wave number at whichstability is attained (kmax � 18 in this case) also dependson model parameters, but the basic trends should generallyhold.

Similar arguments can be constructed for other typesof defects by reconsidering for each case the two genericparameters controlling stability; driving force and energybarrier. The driving force for instability is controlled by boththe energy of the defected state and the energies of the statesresulting from defect removal. In the case just examined, thisforce was quantified by δF̃ , the energy difference between thefaulted state and the competing sheared state. The magnitudeof this force is therefore deeply tied to the precise form ofĈ2(k) and the energy cost that it imposes on defect line shapes,a few examples of which are shown in Fig. 4. These spectrawere obtained using the modulated version of the original

10-710-610-510-410-310-210-1100101102103104105

0 2 4 6 8 10 12 14 16 18

S(k

)

k

Perfect fcc, H0=0.00

Intrinsic Stacking Fault, H0=0.00

Edge Dislocation Pair, H0=0.08

Screw Dislocation Quadrupole, H0=0.08

Grain Boundary Pair, H0=0.00

FIG. 4. (Color online) Structure factors of fcc crystals withvarious types of defects. Note that the most compact structure, theplanar stacking fault, produces the overall broadest line shapes. Thesespectra were generated using Eqs. (2) and (12) with parameter valuesn0 = −0.48, r = −0.63, Bx = 1, α0 = 1/2, and k0 = 6.2653.

224112-4


PFC model presented in Sec. III A, though the basic featuresdo not depend on the model used. The four crystals shownhave roughly equivalent defect densities and their structurefactors exhibit similar line shapes, with the stacking faultproducing the greatest degree of broadening. It is apparentthat in general, broad correlation kernels will reduce theenergies of all of these defect structures and therefore thedriving force for their instability, as demonstrated explicitlyin the next subsection. Alternatively, stacking faults werestabilized in the previous example by instead increasingthe energy of the post-defect-removal sheared state throughhigh-k correlation effects. This simply illustrates that thedriving force can be controlled by varying the energy ofthe initial state, the final state, or both through the formof Ĉ2(k).

The second parameter relevant to stability, the energy bar-rier for defect removal or topological protection, is controlledby the nature of the process by which a given defect can beremoved. The shearing operation that removes a stacking faultrequires only small local translations of the density peaks.No peaks are created or removed, thus no long-range masstransport is involved, and the resulting energy barrier is small.Stability in such a case therefore requires that the drivingforce be minimized, since the energy barrier is in generalmore difficult to manipulate. Removal of dislocations andgrain boundaries on the other hand requires more elaboratetransformations that often involve climb, with redistributionof atoms by long-range diffusive mass transport (adding orremoving density peaks in the case of a PFC model). Thesedefects thus have greater inherent stability than stacking faultsdue to both their somewhat narrower line shapes and theirgenerally larger energy barriers for removal.31

The results presented in this section demonstrate that certaindefect structures can be stabilized in PFC models by buildingstructural information back into the model with additionalcorrelation peaks. The resulting multipeaked models requirethe full logarithmic free energy functional and thereforesuffer from marked inefficiencies and greater complexityrelative to existing PFC models studied in the literature.However, the sources of defect instability uncovered herewill prove useful in analyzing and constructing few-peakedPFC models that both retain efficiency and stabilize relevantdefects.

B. Small wave number or low-mode models

The inefficiencies associated with large wave numbermodels can be avoided by maintaining the small wave numberPFC approximation and giving closer treatment to the detailsof the one or few correlation peaks employed. We havesuccessfully stabilized stacking faults using four variationsof PFC with the small wave number approximation, though ingeneral this requires some degree of parameter tuning and/orslight modification of Ĉ2(k).

Perhaps the simplest way to improve defect stability in thisapproximation is to use a broad Ĉ2(k) in the region of thefirst few primary lattice reflections. This in general lowers theenergy associated with defect line shapes while having littleto no effect on the energy of, for example, uniformly shearedstates or undeformed crystals. A sufficiently broad envelope

can thus stabilize the 1/k2 stacking fault modes relative tothe first few sheared state δk shifts. The main drawback isthat the stability of the equilibrium crystal structure tends todecrease as Ĉ2(k) is broadened. For example, a two-peakedXPFC fcc model can prefer hcp, bcc, rod, or lamellar structuresover fcc for sufficiently large αi . Thus defect stability is notgiven freely; it comes inversely bound to crystal stability.This trade-off must be managed by ensuring that the primaryreflections of the desired crystal structure continue to bepreferred over those of competing symmetries that may existwithin a broad Ĉ2(k) envelope. Specific parameter valuesrelated to the width of Ĉ2(k) (elastic moduli) therefore becomerestricted, but a workable balance between defect stabilityand crystal stability is still often attainable in the small wavenumber PFC approximation, as demonstrated in the followingsubsections.

1. Original PFC model

We begin with the original PFC model of Elder et al.,1,2

for which the correlation kernel and the primary fcc and hcpreflections are shown in Fig. 5. This PFC or Brazovskii32

functional specifies a kernel

Ĉ2(k) = −r + 1 − Bx(1 − k̃2)2, (9)where Bx is a constant proportional to the solid-phase elasticmoduli and k̃ = k/(2π ). This Ĉ2(k) produces equilibrium fccstructures within a certain parameter range,33,34 but the energyof the hcp crystal is necessarily always very close to that offcc, and is sometimes lower. This means that the fcc γISFis always vanishingly small in the original PFC model, whichnaturally stabilizes stacking faults relative to shear. Perfect dis-locations, as with the CDFT hard-sphere model, split into two

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

C2(

k)

k

Original PFCWu et al.

XPFC*fcc

hcp

5 6 7

ˆ

FIG. 5. (Color online) Sample PFC kernels that produce stablestacking faults, with relevant fcc and hcp reflections also indicated.Inset: Close-up view of the peak maxima. Original PFC parameters:n0 = −0.48, r = −0.63, and Bx = 1. XPFC∗ parameters, Eq. (10):n0 = −3/10, r = −9/40, w2 = 1/50, w4 = 49/50, αi � 1, σ = 0,and H � 1.0625. Wu et al. parameters: n0 = −3/10, r = −9/40,R1 = 1/20, and Bx = 1. Note that for display purposes all kernelshave been shifted by −3n20 to maintain consistency with Eq. (1), andthat k has been rescaled in the XPFC∗ and Wu et al. kernels to matchthe equilibrium fcc reflections of the original PFC kernel.

224112-5


SFb2

b1

x = [1̄12]

y = [110]

z = [11̄1]

Edge

H0 = 0.01

b2

b1

SFb2

b1

y = [110]

x = [1̄12]

z = [11̄1]

Screw

H0 = 0.005

b2b1

FIG. 6. (Color online) Dissociated �b = a/2[110] edge and screwdislocations in the (modulated) original PFC model of Eqs. (2) and(12). The lower images show cutaway views of n(�r), illustrating themixed Shockley partials connected by a single intrinsic stacking fault.The upper images show atomic representations of the peak positionsin n(�r): Red atoms have local hcp coordination (stacking fault),gray atoms have local noncrystalline coordination (defect cores),and atoms with fcc coordination are not shown. Parameter valuesn0 = −0.48, r = −0.63, Bx = 1, α0 = 1/2, and k0 = 6.2653 wereused.

well-separated Shockley partials and a stacking fault, as shownin Fig. 6. Note that, for display purposes, the defects shownin Fig. 6 were generated using a modified version of Eq. (9),discussed in Sec. III A. The small value of γISF produced byEq. (9) leads to an equilibrium splitting distance ∼32a for �b =a/2[110] edge dislocations, which is significantly larger thanvalues �10a observed in typical single-component fcc metals.

The broad Ĉ2(k) function of the original PFC modeltherefore naturally stabilizes defects, but is relatively limited interms of flexibility in controlling crystal symmetry and elasticproperties. One is restricted to the deep quench region of fccstability, where small changes to the elasticity parameter Bx

tend to destabilize fcc, and the inherent γISF is extremely small.It is possible to tune γISF within a small range by varyingthe envelope width Bx , but a more effective approach is toexplicitly enhance or suppress specific hcp and fcc modes, asdemonstrated in Sec. III.

2. Low-mode XPFC model

XPFC models with only a few low-k peaks can also betuned to support less protected defect structures, though somemodifications seem to be necessary in the case of fcc stackingfaults. As αi is increased above ∼1 or 2, Gaussian kernel peaksgenerally become too broad to maintain a consistent crystalstructure before they are broad enough near their maxima tosufficiently support the 1/k2 stacking fault modes. We haveobtained better results in this regard using a modified Gaussian

shape function,

Ĉ2(k)i = −r + H exp[−w2(k − ki)

2 + w4(k − ki)42α2i

]

× exp[− σ

2k2i

2ρiβi

], (10)

where the coefficients w2 + w4 = 1 set the relative weight ofk2 versus k4 shape functions. This formulation still permitscontrol of the phase diagram in the original spirit of the modelas well as the magnitude and anisotropy of elastic constants,though the range of parameter values sufficient for stabilityremains restricted. Values in the vicinity of w2 = 1/50, w4 =49/50, αi = 1, σ = 0, H = 1, r = −9/40, and n0 = −3/10have been found sufficient to stabilize stacking faults in a three-peaked fcc model with two fcc reflections at k2 =

√4/3k1

and one commensurate hcp reflection at k3 =√

41/36k1 (seeFig. 5). Similar results have been obtained using only the twofcc reflections with their maxima connected by any nearlylinear bridge function.

These findings highlight the conflict between crystal sta-bility and defect stability in small wave number models. Verynarrow crystal-stabilizing XPFC kernels destabilize defects ofall but the most protected types, somewhat broader kernels stilltend to destabilize susceptible planar fault structures, whilevery broad fully defect-stabilizing XPFC kernels ultimatelyresemble the simpler kernels of the original PFC model.These inherent trade-offs suggest a limitation to the rangeof atomic-level crystal and defect structures that low-mode,two-body PFC models can capture, though the ultimate limitapparently lies at some level of complexity beyond that of theprimary defect structures in fcc crystals.

3. Wu et al. fcc PFC model

The two-mode fcc model introduced by Wu et al.35

also produces stable stacking faults within certain parameterranges. To facilitate tuning of γISF and stability, we find ithelpful to retain a bulk modulus coefficient Bx analogous tothat of the original PFC kernel,

Ĉ2(k) = −r + 1 − Bx(1 − k̃2)2[(

Q21 − k̃2)2 + R1

Bx

],

(11)

where Q1 =√

4/3 is the wave number of the second modeand R1 sets its height. We find that quenches of intermediatedepth tend to produce fcc crystals with moderate γISF, andthat γISF decreases with increasing |n0|. This is because theenergy of hcp approaches and may become lower than thatof fcc as the quench becomes deeper. As an example, theparameter set r = −9/40, R1 = 1/20, Bx = 1, and n0 =−1/4 produces metastable stacking faults for NL = 35 andperfect dislocations that do not split into partials. As n0 islowered, NL = 35 stacking faults eventually become fullystable, with n0 = −3/10 producing a very low γISF and widelysplit partial dislocations (see Fig. 5). γISF also varies rapidlywith Bx , again permitting only a small degree of elastic modulituning.

The source of defect stability in this PFC model is apparentfrom examination of the kernel around the primary fcc modes.

224112-6


Elasticity parameters Bx � 1 produce broad profiles withvery shallow wells between the two modes, while Bx � 1produces narrowing profiles with increasingly deeper wells.Using the parameter values above at n0 = −3/10 and Bx = 2,the stacking fault line shapes are already unfavorable enoughto fully destabilize NL = 35 faulted crystals. This two-modemodel comes with similar restrictions in terms of acceptableparameter ranges to those of the original and XPFC models, butdoes provide greater control of fcc stability than the originalPFC model, allowing use of shallower quenches closer to thelinear elastic regime.

4. Vacancy PFC model

The vacancy PFC or VPFC model typically employs thesame Ĉ2(k) as the original PFC model, but adds a stronglocal nonlinear cutoff term that destroys the inherent one-mode nature of inhomogeneous density states and producesrelatively autonomous individual density peaks separated byregions of zero density.36,37 This can lead to stabilizationof individual lattice vacancies and in general produces aBrownian hard-sphere-like system in which fcc and hcpstructures have lowest free energy, with a slight preferencetoward fcc. When vacancies are stabilized in this way, packingeffects become more important and the natural similaritiesbetween fcc and hcp in this regard should lead to low-γISFsystems. We find that a low γISF, comparable to that ofthe original PFC kernel, is indeed obtained with the VPFCdescription, though in this case we attribute the low valueprimarily to entropic or packing effects. In this sense, theVPFC model is likely the most defect friendly of existing PFCmodels, but one must reinterpret the time scales accessible tothis description since the explicit vacancy diffusion mechanismhas been partially reintroduced. Nonetheless, stacking faultsare quite stable in this formulation and perfect dislocationssplit properly into partials joined by a stacking fault.

III. SURVEY OF FCC DEFECT PROPERTIES IN A SMALLWAVE NUMBER PFC MODEL

The findings presented in Sec. II suggest that many typesof crystalline defects observed in real materials, even thosewith relatively low inherent stability, can be stabilized withinexisting PFC models. The general requirement for stabilityunder the small wave number approximation involves asufficiently broad correlation kernel maximum, which leads torestrictions on the parameter ranges that provide both defectand crystal stability. Nonetheless, these restrictions appearto be manageable in the case of close-packed crystals, andcloser examinations of fcc systems reveal that a wide varietyof realistic defect properties naturally emerge within theseparameter ranges. Some of these properties are discussed inthe present section.

Emphasis is placed on fcc crystals and tuning of thestacking fault energy to generate either undissociated per-fect dislocations (high γISF) or dissociated Shockley partialdislocations bound by stacking faults (low γISF). Variousfundamental properties of both edge and screw dislocationsin these configurations are examined.

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8 9

C2(

k)

k

k111 k200k1101

k0001

k1100

Modulated Original PFCOriginal PFC

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12

0.00 0.02 0.04 0.06 0.08

ISF

FHCP-FFCC

H0

1

2

ˆ

˜ ˜

FIG. 7. (Color online) Modulated original PFC kernel of Eq. (12)where n0 = −0.48, r = −0.63, Bx = 1, α0 = 1/2, and k0 = 6.2653.H0 = 0, 0.04, and 0.08 are shown. Inset: γISF obtained with the givenmodulated kernel vs F̃hcp − F̃fcc and H0. γ1 corresponds to NL =53 and γ2 to NL = 35. As in Fig. 5, all kernels have been shiftedby −3n20.

A. Modulation of the original PFC model

As discussed in Sec. II B1, the dissociation width of aperfect dislocation in the original PFC model is larger thanthat of most single component fcc metals, due to the smallvalue of γISF relative to the fcc elastic moduli. This behaviorcan be tuned by various means. Since γISF is strongly linked tothe difference in free energy between commensurate fcc andhcp states, one method to control γISF is to selectively promoteor suppress the primary hcp reflections that do not coincidewith those of the fcc crystal. For example, the commensuratehcp reflection at k0 = 2π

√41/12/a can be suppressed by

subtracting a small XPFC-type Gaussian centered at this wavenumber from the Ĉ2(k) of Eq. (9), such that the two fccreflections are not affected,

Ĉ2(k) = −r + 1 − Bx(1 − k̃2)2 − H0e−(k−k0)2/(2α20 ). (12)The constants H0 and α0 are analogous to those of Eq. (6).

The original PFC kernel modified in this way is shownin Fig. 7. The result is greater stabilization of fcc due tothe increase in F̃hcp, and a corresponding increase in γISFas the height of the subtracted Gaussian H0 becomes larger.This Ĉ2(k) is similar to that of Wu et al. with variable Bx ,though some subtle differences are apparent. Only γISF ofthe equilibrium fcc state changes as H0 is varied, producingan essentially fixed reference system with tunable γISF. Thestandard phase diagram is of course modified for nonzeroH0, proportionally expanding the region of fcc stability.The dependence of γISF on H0 is also shown in Fig. 7.NL = 35 faults are stable or metastable for H0 � 0.05, andvery little system size dependence is observed in γISF forNL � 35.

B. Dissociation width vs γISF

This method of controlling γISF can be used to examinedefect properties that depend on stacking fault energy, and

224112-7


to compare functional dependencies with those predicted bycontinuum elastic theories. For example, the elastic strainenergy of a fcc crystal containing a perfect dislocation withb = a/√2 is lowered by dissociation into two Shockleypartials with bp = a/

√6 and a stacking fault, as described by

Eq. (7). This is confirmed by the Frank criterion, b2 = a2/2 >2b2p = a2/3. Continuum elastic theories provide predictionsfor the equilibrium separation deq between the two resultantShockley partials. The long-range elastic energies of thepartials produce a repulsive force per length of dislocationline proportional to 1/d, while the energy cost of thefault increases with d, producing an attractive force perlength proportional to γISF. The separation that balancesthese two forces for an elastically isotropic material isgiven by38

deq = 2 − ν1 − ν

(1 − 2ν cos 2β

2 − ν)

μb2p

8πγISF, (13)

where ν is the Poisson ratio, β is the angle between the Burgersvector of the perfect dislocation and its line direction, and μ isthe isotropic shear modulus of the crystal. Similar though morecomplicated results can be derived using anisotropic elasticityor various approximations thereof.38

We have measured deq for both edge and screw dislocationsin the modulated PFC model, as a function of γISF. Baseparameter values n0 = −0.48, r = −0.63, and Bx = 1 wereused, giving an fcc lattice constant a = 1.8537. The crystalorientation of Sec. II was employed (see Fig. 6), though inthis case with periodicity broken in the z direction by a thinlayer of liquid. A single perfect �b = a/2[110] edge dislocationwith line direction [1̄12] was initialized at (Ly/2,Lz/2)using system dimensions (Lx,Ly,Lz) = (16,3986,1820) and

x = a√3/512 (203 668 atoms). Perfect �b = a/2[110]screw dislocation dipoles with 〈110〉 line directions wereinitialized similarly at (Lx/4,Lz/2) and (3Lx/4,Lz/2) with(Lx,Ly,Lz) = (4278,10,1820) and x = a/

√200 (107 692

atoms). The symmetry of the screw dislocation displacementfield in the x direction necessitates a dipole configuration. Inboth cases the diffusive dynamics of Eq. (3) were used. Resultsare shown in Fig. 8, including comparisons with Eq. (13) andthe available anisotropic theories.

The expected linear trend is clearly observed for bothdislocation types, and the agreement with linear elasticpredictions is excellent. The fully isotropic theory veryslightly overestimates deq, while the isotropic theories withapproximate anisotropic moduli of Voigt and Reuss38 andthe fully anisotropic theory (which is tractable only for thescrew dislocation) match the data extremely well.39 We notethat it was necessary to simulate relatively large systems toeliminate significant boundary and image effects. The periodicimage forces nonetheless induce a small bias toward largerd that grows with d and appears to become appreciable ford � 25a at this system size. Otherwise, the largest proportionaldisagreement is seen at small values of d where the linearelastic predictions are likely somewhat inaccurate due tocore overlap effects. We believe that these results providean important validation of fundamental defect energetics andinteractions within the PFC description.

0

5

10

15

20

25

30

35

0 1 2 3 4 5

Spl

ittin

g D

ista

nce,

d [a

]

Inverse Stacking Fault Energy, 10 -4/ ISF

Edge Dislocation: DataIsotropic Theory

Anisotropic Moduli (Voigt)Anisotropic Moduli (Reuss)

Screw Dislocation: DataIsotropic Theory

Anisotropic Moduli (Voigt)Anisotropic Moduli (Reuss)

Anisotropic Theory

FIG. 8. (Color online) Equilibrium splitting distance betweenShockley partials within dissociated perfect �b = a/2[110] edge andscrew dislocations, shown as a function of γ −1ISF . Lines represent thepredictions of isotropic and anisotropic continuum elastic theories,with no adjustable parameters. Model parameters are the same asthose of Fig. 7.

C. Dissociation width vs applied (Escaig) strain

When the glide-inducing shear strain �zy is applied to eitherof the dissociated dislocation lines shown in Fig. 6, the twopartials will glide together with a nearly constant separation(except at large velocities as discussed below). This is becausethe components of their respective Burgers vectors in the sheardirection [110] have the same sense. This is not the case forshear strain �zx . The two partials have components in thisshear direction [1̄12] with opposite sense, which producesglide forces in opposite directions that extend the faulted regionin between. This particular strain orientation is sometimesreferred to as the Escaig strain. The new equilibrium separationfor fixed stress τzx can again be calculated within isotropicelasticity theory, and for a general dislocation one obtains40

d = μb2pf (θ1,θ2)

π (2γISF − τzxbp| sin θ2 − sin θ1|) , (14)

where f (θ1,θ2) = cos θ1 cos θ2 + sin θ1 sin θ2/(1 − ν), and θ1,θ2 are the angles between the overall dislocation line directionand the Burgers vector directions of the leading and trailingpartials, respectively. For the case of a screw dislocation, whereθ1 = −30◦ and θ2 = 30◦, Eq. (14) reduces to

d = 2 − 3ν8π (1 − ν)

μb2p

γISF − τzxbp/2 . (15)

Shear strain �zx was applied to the screw dipole configurationsdescribed in the previous subsection by adding a penaltyfunction to the first few crystalline surface layers near theliquid boundaries and translating the penalty field at someconstant rate �̇zx to drive the external deformation. An affineshear deformation was also applied to the entire crystal ateach time step, and the inertial equation of motion, Eq. (4),with α = 1 and β = 1/100 was used. These features togethershould ensure that the resolved shear strain at the dislocationremains as close to the applied shear strain as possible. Selected

224112-8


0

5

10

15

20

25

30

35

40

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Spl

ittin

g D

ista

nce,

d [a

]

Applied Shear Strain, zy

H0=0.00H0=0.01H0=0.02H0=0.03H0=0.04H0=0.08

FIG. 9. Divergence of the equilibrium splitting distance betweenShockley partials within dissociated perfect �b = a/2[110] screwdislocations, shown as a function of �zx , the Escaig shear strain.Lines represent the predictions of isotropic continuum elastic theory,Eq. (15), as discussed in the text. Model parameters are the same asthose of Fig. 7.

fixed values of �zx were held periodically to allow the system tofully relax to a steady configuration. Though a reliable methodof directly quantifying stress in our simulations is currentlylacking, the assumption of linear elasticity will produceaccurate stress-strain conversions whenever plastic flow / stressdissipation is negligible and strain is not exceedingly large.We can confidently apply this assumption here (τzx = Czx�zx)to obtain results in terms of stress, which can be directlycompared with Eq. (15). Expected errors are


stress-strain relations to report Peierls stresses rather thanstrains (τP � μ�P). The unsplit screw Peierls stress approachesa value within the expected range τP � 6 × 10−4μ, while theaverage Peierls stress for the split screw is slightly lower,τP � 3.5 × 10−4μ. The measured values for the leading andtrailing screw partials differ by a small amount. This may bea consequence of weakly asymmetric periodic image forcesgenerated by the simulation boundaries, or of an initial dvalue that is incommensurate with the periodicity of the Peierlspotential, as discussed in Ref. 45.

The widely split edge dislocation exhibits much lessdeviation between leading and trailing τP values, thoughthe apparent limiting value is not smaller than that ofthe perfect edge dislocation to within expected error. Bothconfigurations approach a value that appears to extrapolate toτP � 3 × 10−5μ, also within the expected range for typical fccedge dislocations. It was observed that edge dislocations tendto glide with a relatively uniform, continuous progression,while screws exhibit a more pronounced stick-slip hoppingmotion with each unit of translation, indicative of a largerPeierls barrier. These results together demonstrate that thefundamental features of fcc Peierls stresses are captured quitereasonably by this relatively simple PFC model.

The apparent relation �P ∼√

�̇ for larger shear rates canbe obtained as follows. Based on our results, there appears tobe an intrinsic time scale τ associated with relaxation of anentire defect structure, including the core region and the long-range displacement fields. If shear is applied slowly relativeto τ , the core region and the long-range displacements canevolve cooperatively such that the inherent Peierls barrier ofthe given dislocated system, δF̃P, is realized. If �̇−1 is largerelative to τ , then the short-range and long-range componentscannot relax with optimal synchronization. This introducesnonequilibrium effects and alternate relaxation processes thatlead to measurement of some larger effective barrier δF̃Eff =δF̃P + δF̃NEQ.

In the PFC formulation, slow relaxations near equilibriumare diffusive whether one uses Eq. (3) or Eq. (4). Thusthe intrinsic defect time scale τ can be approximated as anexponential relaxation time, and neglecting other relaxations,F̃NEQ should be, to a first approximation, an exponentialfunction of the dimensionless parameter τ �̇. We thereforemay write δF̃NEQ ∼ eτ �̇ − 1 or δF̃NEQ ∼ τ �̇ in the small τ �̇limit. Combining this relation with a linear elastic response,δF̃Eff ∼ �2, we obtain �P ∼

√δF̃Eff ∼

√δF̃P + cτ �̇ where c

is a constant. This form and the corresponding unexpandedform both fit our results well, indicating that the competitionbetween rate of applied shear and rate of diffusive strainfield relaxation leads to an increased effective Peierls strainwith relevant control parameter τ �̇. The observed system sizedependence of �P is also consistent with this expression.

E. Dynamic contraction of dissociated dislocations

When the glide-inducing stress τzy is applied to a dis-sociated dislocation, the Peierls barrier is overcome quitequickly and glide is initiated within the slip system ofthe defect. The structure of the extended defect may thenbe altered by dynamic effects associated with navigation ofthe Peierls potential as well as drag or damping forces of

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.5

Spl

ittin

g D

ista

nce,

d [a

]

Dislocation Velocity, v [a/t]

Drafting Theory

RelativisticContraction

Steady-States9.50 x 10-7

2.85 x 10-6

5.70 x 10-6

8.55 x 10-6

1.19 x 10-5

1.52 x 10-5

FIG. 11. (Color online) Measured dissociation width of a glidingedge dislocation vs dislocation velocity for various applied shearstrain rates. Steady-state widths and velocities at each shear rate areseparately indicated. The solid line is a fit to the functional form ofthe drafting theory presented in the text, and the dashed line is anillustrative example of the behavior expected from semirelativisticeffects (d ∼ √1 − v2/α2). Model parameters are the same as thoseof Fig. 7.

various origins. Employing the dynamics of Eq. (4), we appliedfixed shear strain rates to the dissociated edge and screwdislocation configurations already described, and allowed eachto approach a steady-state glide velocity v, while monitoringthe separation between paired partials d. Similar results wereobtained for both dislocation types, and those for the H0 = 0edge dislocation are shown in Fig. 11. For relatively lowsteady-state velocities (v � 0.1α), d was found to exhibitregular oscillations in time. These oscillations, which canbe seen in the lowest shear rate data displayed in Fig. 11,are not unlike the so-called breathing modes observed in MDsimulations,46,47 though their period appears to be significantlylonger and more regular in these PFC simulations.

At higher velocities a dramatic decrease in d was observed,followed by a return approximately to the initial d upon releaseof strain. Since the shear strain �zy does not significantly alterγISF or the repulsive force between static paired partials, thiscontraction appears to be a fully dynamic effect. We believethat the primary mechanism at work is a difference in thefrictional drag force experienced by each partial at high PFCvelocities or, similarly, a reduction of the drag force exerted onthe extended defect structure as a whole. Either scenario mayresult from the difference in screw-edge character betweenthe two partials38 or may be driven by a cooperative drag-reduction process analogous to the phenomenon of drafting orslipstreaming in fluid dynamics.

At low velocities, where the effect of drag is small, vis determined primarily by the height of successive barriersin the Peierls potential. Thus any forces that might changethe dissociation width are small, though they may becomelarge enough to generate low-amplitude breathing modes. Asv increases, the Peierls potential becomes less relevant andthe effect of frictional drag becomes large. In this regime, adissociated defect may experience a larger total drag forcethan an undissociated one due to its greater extension. The

224112-10


total drag force may be reduced, thus increasing v at a givenstrain, by reducing d such that the trailing partial assumesan optimized lower-drag position within the leading wake orslipstream. At large enough velocities, the reduction in dragforce experienced by the full defect under contraction maycompensate for the increase in local defect energy. The energyof the entire system is reduced by the excess strain relief thatis a consequence of faster dislocation glide.

We can modify the analysis given in Ref. 38 to incorporatesuch an effect into the predicted steady-state value of d. Thesteady-state force balance between gliding paired partials canbe written38

γISF + B = D1 + Ad

,

(16)

γISF + D2 = C + Ad

,

where A is an abbreviation of the terms multiplying γ −1ISF inEq. (13), B and C are the forces per unit length on �b1 and�b2, respectively, due to the applied shear, and D1 and D2 arethe damping forces per unit length on the moving partials.First we assume that both partials experience the same forcedue to the applied shear strain (B = C) and that the nominaldrag force on each PFC dislocation obeys a Stokes’ law form(Di � Cdv), as shown in Ref. 9. The drag coefficient shouldroughly be given by Cd � μ/M , where M = v/� � 3 is thepartial dislocation mobility. Finally, since we assume that D2decreases with d [D2 = Cdvf (d)] a plausible functional formfor f (d) must be proposed. We will use f (d) = δd/(δd + cvv)where δd = d + d0, and d0, cv are constants.

A fit to the resulting equation for d vs v is shown in Fig. 11.The only adjustable parameters in the fit are d0 and cv , andthe agreement is very good. Semirelativistic contraction ofstrain fields in the glide direction has also been predictedfor dislocations moving at velocities near the sound speed αof the crystal.38 A curve illustrating this effect is shown inFig. 11, where it is assumed that both partials experiencethe same contraction effect and reduce d accordingly. Thismechanism does not have a large effect until the sound speedis approached and its form is not consistent with our results.Furthermore, we observe similar behavior to that shown inFig. 11 when dynamics are given by Eq. (3), which does notintroduce a sound speed.

The PFC description is generally best suited for examiningbehavior under relatively low strain rates or driving forces,where dislocation velocities tend not to closely approachthe sound speed. Nonetheless, a better understanding of thenature of the interaction between moving dislocations and thequasiphonons described by Eq. (4) would be useful in terms ofconfirming that the artificially low sound speeds employedin PFC simulations do not qualitatively alter the low andintermediate velocity dislocation dynamics. Details of theseissues are deferred to a future publication.

IV. DISCUSSION AND CONCLUSIONS

The primary factors controlling defect stability in PFCmodels have been examined, and it has been demonstratedthat broad correlation kernels or elastically soft crystalsproduce the greatest defect stability. Maximally broadened

kernels appear to be necessary for stabilization of certaindefects such as planar faults in the small wave numberPFC approximation. Higher wave number correlations withnarrower peaks were also shown to improve stability in somecases, but this feature leads to greatly reduced model efficiency.The inherent conflict between crystal stability and defectstability in broad kernel models suggests potential limitationsto the complexity of structures that can be described by suchmodels. Any such limitations were shown to be nonfactors inthe case of the primary defect structures in close-packed fcccrystals.

Stacking faults with low inherent stability were stabilizedin four PFC variants, indicating that considerable defectstability can be obtained without losing crystal stability. Thecentral defect structure in fcc plasticity, dissociated partialdislocations with stacking faults, has been examined in somedetail and shown to be well described by PFC methods.The dependence of the equilibrium dissociation width onstacking fault energy has been shown to agree with continuumelastic predictions for both edge and screw dislocations,under zero and nonzero applied external stresses. Peierlsstresses for glide of both dislocation types have also beenmeasured and shown to fall within the typical ranges andrelative magnitudes determined from experiments and MDsimulations. Contraction of gliding pairs of partials has beenobserved at high velocities and argued to be a consequence oflarge frictional drag forces on widely split partials.

The findings presented in this article are intended to lay thegroundwork for larger-scale PFC studies of plasticity in fcccrystals, with the potential to examine experimentally relevantstrain rates, nonconservative/climb-driven defect evolution,and processes in which diffusing solute atoms interact withmobile and/or immobile defect structures. These, in practice,are inaccessible to conventional MD simulations. Conversely,PFC models cannot reliably access the rapid time scales of MDand likely cannot predict some details of more complex defectatomic core structures as accurately as MD. Microscopic phasefield (MPF) models of defects48 are related to PFC in thesense that both employ phase field methodologies, but MPFmodels utilize a top-down approach, explicitly building thedefect physics in by hand. PFC models employ a far simpler,atomic-level free energy functional, from which all defectphysics automatically emerge with fewer imposed constraintsand with straightforward dynamical extensions. Nonetheless,by directly incorporating ab initio data such as γ -surfaceenergies,49,50 MPF models have proven highly effective interms of predicting static core-level defect structures andenergies with greater quantitative accuracy than current PFCdescriptions. At a more coarse-grained level, both discretedislocation dynamics models51 and coarse-grained phase fielddislocation dynamics models48 readily describe length scalesinaccessible to MD, MPF, and PFC. Similarly large lengthscales are in principle also accessible to coarse-grainedcomplex amplitude representations of PFC models,4–8 whichstill retain atomistic resolution.

Targeted PFC studies, examining more complex problemsin defect physics that involve atomic length scales anddiffusive time scales, will be the subject of an upcomingpublication. Topics that have been analyzed and will beaddressed include climb fundamentals, the structure of jogged

224112-11


dislocation lines, jog constriction, jog pair annihilation, jogdrag, screw dislocation cross-slip, formation and collapse ofstacking fault tetrahedra from triangular Frank vacancy loops,Lomer-Cottrell or stair-rod dislocations, the interaction ofdissociated edge and screw dislocations with stacking faulttetrahedra, and dislocation creation mechanisms.

ACKNOWLEDGMENTS

This work has been supported by the Natural Scienceand Engineering Research Council of Canada (NSERC), andaccess to supercomputing resources has been provided byCLUMEQ/Compute Canada.

1K. R. Elder, M. Katakowski, M. Haataja, and M. Grant, Phys. Rev.Lett. 88, 245701 (2002).

2K. R. Elder and M. Grant, Phys. Rev. E 70, 051605 (2004).3K. R. Elder, N. Provatas, J. Berry, P. Stefanovic, and M. Grant,Phys. Rev. B 75, 064107 (2007).

4N. Goldenfeld, B. P. Athreya, and J. A. Dantzig, Phys. Rev. E 72,020601 (2005).

5B. P. Athreya, N. Goldenfeld, J. A. Dantzig, M. Greenwood, andN. Provatas, Phys. Rev. E 76, 056706 (2007).

6D. Yeon, Z. Huang, K. R. Elder, and K. Thornton, Philos. Mag. 90,237 (2010).

7Z.-F. Huang, K. R. Elder, and N. Provatas, Phys. Rev. E 82, 021605(2010).

8R. Spatschek and A. Karma, Phys. Rev. B 81, 214201 (2010).9J. Berry, M. Grant, and K. R. Elder, Phys. Rev. E 73, 031609 (2006).

10P. Stefanovic, M. Haataja, and N. Provatas, Phys. Rev. Lett. 96,225504 (2006).

11P. Stefanovic, M. Haataja, and N. Provatas, Phys. Rev. E 80, 046107(2009).

12T. Hirouchi, T. Takaki, and Y. Tomita, Comput. Mater. Sci. 44, 1192(2009).

13P. Y. Chan, G. Tsekenis, J. Dantzig, K. A. Dahmen, andN. Goldenfeld, Phys. Rev. Lett. 105, 015502 (2010).

14A. Jaatinen, C. V. Achim, K. R. Elder, and T. Ala-Nissila, Phys.Rev. E 80, 031602 (2009).

15R. Backofen, A. Voigt, and T. Witkowski, Phys. Rev. E 81,025701(R) (2010).

16A. Jaatinen, C. V. Achim, K. R. Elder, and T. Ala-Nissila,Technische Mechanik 30, 169 (2010).

17J. Berry, K. R. Elder, and M. Grant, Phys. Rev. B 77, 224114 (2008).18J. Mellenthin, A. Karma, and M. Plapp, Phys. Rev. B 78, 184110

(2008).19D. L. Olmsted, D. Buta, A. Adland, S. M. Foiles, M. Asta, and

A. Karma, Phys. Rev. Lett. 106, 046101 (2011).20Y. Singh, Phys. Rep. 207, 351 (1991).21D. Oxtoby, Annu. Rev. Mater. Res. 32, 39 (2002).22T. V. Ramakrishnan and M. Yussouff, Phys. Rev. B 19, 2775 (1979).23The latter implies that the Fourier transformed kernel Ĉ2(k) is

effectively shifted by a constant un20 relative to the kernel defined inEq. (1). To maintain consistency with existing notation we willneglect this constant, and it will be implied that un20 must besubtracted from all kernels Ĉ2(k) associated with Eq. (2) if onewishes to recover the kernel that is fully consistent with Eq. (1).

24C. Dasgupta, Europhys. Lett. 20, 131 (1992).25M. Greenwood, N. Provatas, and J. Rottler, Phys. Rev. Lett. 105,

045702 (2010).26M. Greenwood, J. Rottler, and N. Provatas, Phys. Rev. E 83, 031601

(2011).27M. Greenwood, N. Ofori-Opoku, J. Rottler, and N. Provatas, Phys.

Rev. B 84, 064104 (2011).

28J. K. Percus and G. J. Yevick, Phys. Rev. 110, 1 (1958).29M. S. Wertheim, Phys. Rev. Lett. 10, 321 (1963).30The {111} reflection will be broadened as well because the three

additional {111} planes that are not parallel to the fault will contain1/(kz − k(i)z )2 modes.

31Depending on local defect configurations some dislocation andgrain boundary structures can annihilate by glide alone, leadingto much lower energy barriers for removal. This effect is naturallycaptured by CDFT and PFC descriptions.

32S. A. Brazovskii, Sov. Phys. JETP 41, 85 (1975).33A. Jaatinen and T. Ala-Nissila, J. Phys.: Condens. Matter 22, 205402

(2010).34G. I. Tóth, G. Tegze, T. Pusztai, G. Tóth, and L. Gránásy, J. Phys.:

Condens. Matter 22, 364101 (2010).35K.-A. Wu, A. Adland, and A. Karma, Phys. Rev. E 81, 061601

(2010).36P. Y. Chan, N. Goldenfeld, and J. Dantzig, Phys. Rev. E 79,

035701(R) (2009).37J. Berry and M. Grant, Phys. Rev. Lett. 106, 175702

(2011).38J. P. Hirth and J. Lothe, Theory of Dislocations (John Wiley & Sons,

Inc., New York, 1982), 2nd ed.39The independent elastic constants were measured numerically

by using Eq. (3) and applying the appropriate strains, givingC11 = 0.0967, C12 = 0.0388, and C44 = μ = 0.032. An isotropicapproximation of Poisson’s ratio is therefore ν = C12/(C11 +C12) = 0.2863, and the anisotropy ratio is 2C44/(C11 − C12) =1.11. Thus there is a small degree of elastic anisotropy in thissystem, as evidenced by the slightly lower slopes for the anisotropicpredictions in Fig. 8. The expressions for the Voigt and Reussaverage anisotropic moduli can be found in Ref. 38.

40T. Byun, Acta Mater. 51, 3063 (2003).41D. L. Olmsted, K. Y. Hardikar, and R. Phillips, Modell. Simul.

Mater. Sci. Eng. 9, 215 (2001).42M. S. Duesbery, Modell. Simul. Mater. Sci. Eng. 6, 35

(1998).43Z. S. Basinski, Philos. Mag. 4, 393 (1959).44M. Bujard, G. Gremaud, and W. Benoit, J. Appl. Phys. 62, 3173

(1987).45F. R. N. Nabarro, Philos. Mag. A 75, 703 (1997).46D. Mordehai, Y. Ashkenazy, I. Kelson, and G. Makov, Phys. Rev.

B 67, 024112 (2003).47D. Mordehai, I. Kelson, and G. Makov, Phys. Rev. B 74, 184115

(2006).48Y. Wang and J. Li, Acta Mater. 58, 1212 (2010).49C. Shen and Y. Wang, Acta Mater. 52, 683 (2004).50A. Hunter, I. J. Beyerlein, T. C. Germann, and M. Koslowski, Phys.

Rev. B 84, 144108 (2011).51S. Groh and H. M. Zbib, J. Eng. Mater. Technol. 131, 041209

(2009).

224112-12

http://dx.doi.org/10.1103/PhysRevLett.88.245701http://dx.doi.org/10.1103/PhysRevLett.88.245701http://dx.doi.org/10.1103/PhysRevE.70.051605http://dx.doi.org/10.1103/PhysRevB.75.064107http://dx.doi.org/10.1103/PhysRevE.72.020601http://dx.doi.org/10.1103/PhysRevE.72.020601http://dx.doi.org/10.1103/PhysRevE.76.056706http://dx.doi.org/10.1080/14786430903164572http://dx.doi.org/10.1080/14786430903164572http://dx.doi.org/10.1103/PhysRevE.82.021605http://dx.doi.org/10.1103/PhysRevE.82.021605http://dx.doi.org/10.1103/PhysRevB.81.214201http://dx.doi.org/10.1103/PhysRevE.73.031609http://dx.doi.org/10.1103/PhysRevLett.96.225504http://dx.doi.org/10.1103/PhysRevLett.96.225504http://dx.doi.org/10.1103/PhysRevE.80.046107http://dx.doi.org/10.1103/PhysRevE.80.046107http://dx.doi.org/10.1016/j.commatsci.2008.08.001http://dx.doi.org/10.1016/j.commatsci.2008.08.001http://dx.doi.org/10.1103/PhysRevLett.105.015502http://dx.doi.org/10.1103/PhysRevE.80.031602http://dx.doi.org/10.1103/PhysRevE.80.031602http://dx.doi.org/10.1103/PhysRevE.81.025701http://dx.doi.org/10.1103/PhysRevE.81.025701http://dx.doi.org/10.1103/PhysRevB.77.224114http://dx.doi.org/10.1103/PhysRevB.78.184110http://dx.doi.org/10.1103/PhysRevB.78.184110http://dx.doi.org/10.1103/PhysRevLett.106.046101http://dx.doi.org/10.1016/0370-1573(91)90097-6http://dx.doi.org/10.1146/annurev.matsci.32.100401.103425http://dx.doi.org/10.1103/PhysRevB.19.2775http://dx.doi.org/10.1209/0295-5075/20/2/007http://dx.doi.org/10.1103/PhysRevLett.105.045702http://dx.doi.org/10.1103/PhysRevLett.105.045702http://dx.doi.org/10.1103/PhysRevE.83.031601http://dx.doi.org/10.1103/PhysRevE.83.031601http://dx.doi.org/10.1103/PhysRevB.84.064104http://dx.doi.org/10.1103/PhysRevB.84.064104http://dx.doi.org/10.1103/PhysRev.110.1http://dx.doi.org/10.1103/PhysRevLett.10.321http://dx.doi.org/10.1088/0953-8984/22/20/205402http://dx.doi.org/10.1088/0953-8984/22/20/205402http://dx.doi.org/10.1088/0953-8984/22/36/364101http://dx.doi.org/10.1088/0953-8984/22/36/364101http://dx.doi.org/10.1103/PhysRevE.81.061601http://dx.doi.org/10.1103/PhysRevE.81.061601http://dx.doi.org/10.1103/PhysRevE.79.035701http://dx.doi.org/10.1103/PhysRevE.79.035701http://dx.doi.org/10.1103/PhysRevLett.106.175702http://dx.doi.org/10.1103/PhysRevLett.106.175702http://dx.doi.org/10.1016/S1359-6454(03)00117-4http://dx.doi.org/10.1088/0965-0393/9/3/308http://dx.doi.org/10.1088/0965-0393/9/3/308http://dx.doi.org/10.1088/0965-0393/6/1/005http://dx.doi.org/10.1088/0965-0393/6/1/005http://dx.doi.org/10.1080/14786435908233412http://dx.doi.org/10.1063/1.339342http://dx.doi.org/10.1063/1.339342http://dx.doi.org/10.1080/01418619708207197http://dx.doi.org/10.1103/PhysRevB.67.024112http://dx.doi.org/10.1103/PhysRevB.67.024112http://dx.doi.org/10.1103/PhysRevB.74.184115http://dx.doi.org/10.1103/PhysRevB.74.184115http://dx.doi.org/10.1016/j.actamat.2009.10.041http://dx.doi.org/10.1016/j.actamat.2003.10.014http://dx.doi.org/10.1103/PhysRevB.84.144108http://dx.doi.org/10.1103/PhysRevB.84.144108http://dx.doi.org/10.1115/1.3183783http://dx.doi.org/10.1115/1.3183783

Defect stability in phase-ﬁeld crystal models: Stacking ...provatas/papers/Joel_PRB.pdf ·...

Documents

Transcript of Defect stability in phase-ﬁeld crystal models: Stacking ...provatas/papers/Joel_PRB.pdf ·...