PHYSICAL REVIEW E 98, 043002 (2018)

Statistical model for diffusion-mediated recovery of dislocation and point-defect microstructure

I. Rovelli,1,2,* S. L. Dudarev,2 and A. P. Sutton1

1Department of Physics, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom2Culham Centre for Fusion Energy, UK Atomic Energy Authority, Abingdon, Oxfordshire OX14 3DB, United Kingdom

(Received 12 February 2018; published 15 October 2018)

The evolution of the defect microstructure in materials at high temperature is dominated by diffusion-mediatedinteractions between dislocations, cavities, and surfaces. This gives rise to complex nonlinear couplings betweeninterstitial and vacancy-type dislocation loops, cavities, and the field of diffusing vacancies that adiabaticallyfollows the evolution of microstructure. In our previous work, we developed a nonlocal model for the climb ofcurved dislocations and the morphological evolution of cavities during postirradiation annealing of structuralcomponents in nuclear reactors. We now expand the formalism to include the treatment of population ofvery small defects and dislocation loops that are below the experimental detection limit. These are taken intoaccount through a mean field approach coupled with an explicit real-space treatment of larger-scale discretedefect clusters. We find that randomly distributed small defects screen diffusive interactions between largerdiscrete clusters, renormalizing the free diffusion Green’s functions and transforming them into Yukawa-typepropagators. The evolution of the coupled system is modelled self-consistently, showing how the defectmicrostructure evolves through a nonmonotonic variation of the distribution of sizes of dislocation loops andcavities, treated as discrete real-space objects.

DOI: 10.1103/PhysRevE.98.043002

I. INTRODUCTION

In a metal at sufficiently high temperature, the evolu-tion of its microstructure following irradiation is effected bydiffusion. Point defects, predominantly vacancies, propagatebetween dislocations, cavities, grain boundaries, and surfaces,producing changes in their morphologies. The evolution rateof each microstructural defect is governed by the imbalancebetween the local chemical potential of vacancies in the atmo-sphere surrounding it, and the chemical potential of vacanciesin the bulk. Each defect, acting as a sink or source of vacan-cies, introduces a perturbation of the vacancy concentrationin the medium, varying as the inverse distance away from thedefect. Therefore dislocations, vacancy clusters, and surfacesevolve through coupled diffusive interactions carried by thevacancy field, and the evolution of each one depends on theglobal defect microstructure.

A model for the evolution of an arbitrary configurationof vacancy clusters and dislocations at elevated temperaturesis needed to predict changes of engineering properties ofmaterials in nuclear fusion and fission reactors. High energyneutrons produced by nuclear reactions in fusion plasma, oremitted by fissile nuclear fuel, impact atoms in the structuralcomponents, generating vacancies and self-interstitials thataggregate to form prismatic vacancy and interstitial disloca-tion loops and cavities [1–5]. The accumulation of such de-fects degrades mechanical and thermal properties of materials.To design annealing protocols for the recovery of neutronirradiation damage [2], it is desirable to be able to predict the

*Corresponding author: ir1014@ic.ac.uk

evolution of arbitrary populations of such defects in real spaceas a function of time and temperature postirradiation.

Molecular dynamics (MD) simulations have been instru-mental in understanding the initial stage of defect productionduring irradiation due to short-lived (∼0.1–1 ps) collision cas-cades [6–8] and in characterizing the structure and mobilityof small defect clusters [9]. However, the diffusion-drivenevolution of interacting defect clusters involves time scales(μs to years) not accessible to MD simulations.

Two coarse-grained modeling frameworks have risen toprominence for the treatment of defect cluster evolution inirradiation conditions and during postirradiation annealing:mean field rate theory (MFRT) and kinetic Monte Carlo(KMC) [10,11].

In MFRT spatial distributions of defect clusters are rep-resented by spatially independent or slowly spatially vary-ing continuous size distribution functions (SDFs). The timeevolution of SDFs are governed by coupled master equationsinvolving defect generation rates (when considering processesduring irradiation) and fluxes, in the defect size space, propor-tional to the rates of point-defect absorption and emission byvarious defect cluster types. Reaction rates are assumed to de-pend on the available populations of point defects, which arealso dynamical quantities of the theory, evolving according tothe average effective sink strength of defect clusters and othermicrostructural features [12]. While it is possible to introducea degree of spatial information, for instance by consideringmultiple subsystems with different but co-evolving clusterdensities, MFRT models fundamentally employ spatially av-eraged descriptions of defect cluster distributions and hencethey are unable to treat real-space diffusion-driven processeslike the correlated diffusion-mediated evolution of dislocationloops.

2470-0045/2018/98(4)/043002(21) 043002-1 ©2018 American Physical Society

I. ROVELLI, S. L. DUDAREV, AND A. P. SUTTON PHYSICAL REVIEW E 98, 043002 (2018)

KMC models treat defect clusters and individual point de-fects as discrete objects with explicit positions in the simula-tion cell, while their time evolution is governed stochasticallyinstead of deterministically. A set of possible transition eventsis specified for the various classes of objects, such as migra-tion, emission or aggregation of point defects from or to clus-ters, and vacancy-interstitial annihilation. At each time step,an event is randomly chosen with probability proportional tothe Arrhenius factor with the relevant energy barrier for thereaction [13]. A particular object able to undergo the reactionis then randomly selected, the state of the system is updatedaccordingly, and time is advanced by an amount inverselyproportional to the total reaction rate. The computational costof KMC is generally considerably higher than MFRT, there-fore simulations are limited to smaller cell sizes, typicallyof the order of a few hundred nm, and relatively low [<1displacements per atom (dpa)] doses [14]. The main limitationof KMC models is that the method requires choosing theobjects and reaction rules at the start of the simulation. It ispossible to model the evolution of populations of dislocationloops and vacancy clusters as long as the defect structuresformed as a result of reactions between the existing objectsbelong to the same class of objects, for which the rules ofevolution are defined. For example, if the objects defined in aKMC simulation are circular prismatic dislocation loops, thena simulation is unable to predict the formation of dislocationloops with noncircular shapes or the formation of a dislocationnetwork, such as those often observed in experiment. Thenumber of individual defects that a KMC simulation canhandle is also limited, as the total reaction rate increases inproportion to the total number of objects in a simulation cell.This reduces the simulation time step and makes it difficultto treat evolution in the high temperature limit where theconcentration of mobile vacancies is high.

In an earlier paper [15] we presented a formalism thatextends the nonlocal dislocation climb model of Gu et al. [16]to simulate the evolution of discrete dislocation loops and va-cancy clusters in a finite medium, at high temperatures and inreal space. One of the motivations for developing a real-spacemodel, in contrast to a spatially averaged MFRT approach,is that the sizes of clusters of vacancies or self-interstitialsproduced by neutron irradiation appear to obey a power-lawprobability distribution [6,17] of the form f (n) ∼ n−s , withn the number of self-interstitials or vacancies in the cluster.For s < 2, corresponding to experimental observations [17],the average size of a defect cluster is not defined, and soa representative cluster size of a rate theory model remainsundefined. A real-space model can also treat local variations inthe number density of clusters, for instance those arising fromdepleted zones [18] at grain boundaries and free surfaces.

The nonhomogeneous spatial arrangement of defects anddislocation segments in irradiated samples is clearly evidentin transmission electron microscopy (TEM) images of ionirradiated tungsten foil shown in Fig. 1. Furthermore, thein situ TEM observations of dynamics of postirradiation an-nealing have highlighted a strong dependence of evolution ofindividual defect clusters on the local defect distribution intheir local environment [2].

In principle, KMC models might allow, subject to thelimitations noted above, the simulation of arbitrary spatial

FIG. 1. TEM snapshot of in situ annealing of ion-irradiated tung-sten (2-MeV W+ ions, 500 ◦C, 1014 W+/cm2) at 700 ◦C. Adaptedwith permission from [2].

distributions of defect clusters. However, the computationalcost of our real-space model is many orders of magnitudelower, as individual point defects are not explicitly taken intoaccount as stochastic discrete entities. As a result, we areable to treat larger simulation cells, higher densities of defectclusters, and considerably longer time scales. In addition, ourmodel does not preclude the treatment of arbitrary dislocationstructures outside of circular loops, and in principle can beintegrated in discrete dislocation dynamics (DDD) models.

It is likely that the experimentally determined number ofthe smallest defect clusters is always underestimated owingto detection limits of the instrumentation. The sizes of ex-perimentally observed defect clusters is usually assumed toobey a Gaussian-like distribution. Such a distribution can beunderstood as a product of the true distribution with a sigmoidfunction representing the instrumental sensitivity, as sketchedin Fig. 2. Therefore, experimental data on radiation induceddefect clusters is generally incomplete and the number densityof the smallest “invisible” defect clusters can be much greaterthan the observed population, as recently noted by Liu et al.[19].

It would be useful, therefore, to include a statistical de-scription of the unknown small-size cluster population, intro-ducing an effective mean field which evolves self-consistentlywith the observable cluster population. This approach wouldprovide an efficient way to investigate the effect of invisibleclusters on the distribution of sizes of visible clusters. It turnsout that the mean field is governed by only a few low-ordermoments of the size distribution of visible clusters, which canbe determined by the experimentally measured evolution ofthe observable population. In this paper we present a hybridmodel that couples the evolution of an experimentally visible,discrete population of defect clusters in real space with a meanfield, representing the experimentally invisible clusters.

In Sec. II we estimate the expected time scales for the evap-oration of nanometric cavities as a function of temperaturein W, Fe, and Be. In Sec. III we introduce the mean fieldformulation by averaging the positions and sizes of the small

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0

1

0 1 2 3 4

Observed distribution

Real distributionSensitivity function

R/b0.5 3.51.5 2.5

0.8

0.6

0.4

0.2

f(R)

[arb

. un

its]

FIG. 2. Qualitative explanation of the observed Gaussian-likesize distribution of defect clusters in neutron irradiated tungsten. Thedashed black line represents the true, power-law like, size distri-bution. The dotted blue line represents the experimental sensitivityfunction plotted as a function of cluster size. The solid black line isthe observed distribution, given by the product of the true distributionand the sensitivity function. The red shaded area is proportional tothe number density of the observed clusters, while the yellow shadedarea is proportional to the number density of mostly “invisible”clusters.

clusters, using a technique developed in scattering theory byEdwards [20]. In this way we obtain a general expression,Eq. (32), for the coarse-grained vacancy concentration in thepresence of an arbitrary distribution of small clusters.

In Appendix B we present a perturbative approach thatexpands Eq. (32) in terms of an increasing number of scat-tering events, deriving an analog of the self-energy expansionfor condensed matter systems. In the case of homogeneousand uncorrelated distributions, we provide closed form ex-pressions for the self-energy up to the third order terms inEq. (B11).

By considering the first order approximation to the effec-tive self-energy, we show that the effective Green’s functiongoverning the interaction between the observable clusterstakes the form of a Yukawa propagator, Eq. (40). Thus,we show that diffusive interactions mediated by vacanciesbetween larger clusters are screened by the mean field ofsmall clusters. The formalism has similarities to the diffusionscreening theory, originally due to Marqusee and Ross [21]and further developed by other authors [22–26], which ad-dresses the issue of divergences in Ostwald ripening theoriesfor finite volume fractions of particles. In diffusion screeningtheory the usual Laplacian diffusive interactions are screenedas a result of coarse-graining second phase particles, whichis analogous to our mean field treatment of invisible defectclusters. This procedure enables us in Eq. (43) to providea closed form expression for the vacancy field where onlyobservable clusters are treated explicitly.

The rates of growth of cavities can be calculated self-consistently in the same manner as in our previous model [15]by evaluating the vacancy field at the cluster positions withdefined boundary conditions. We couple the evolution of themean field with large clusters via Eqs. (46) and (49).

The theory is developed for an infinite medium. However,the majority of experimental data on radiation-induced defectclusters has so far been obtained using thin film samples. InSec. IV we present an extension of the theory to a thin filminfinitely extended in the lateral direction, making use of avariant of Ewald summation developed for Yukawa potentials[27]. In Sec. V we briefly present a mean field treatment ofall defect clusters. Finally, in Sec. VI we present numericalsimulations to illustrate applications of the theory to theevolution of distributions of cluster sizes in irradiated thinfilms of W, Be, and Fe.

II. PRELIMINARY ESTIMATE OF GOVERNINGTIME SCALES

An estimate of the time scale required for the completeevaporation of a nanometric defect cluster can be obtained byconsidering the isolated cluster in a vacancy atmosphere inlocal thermodynamic equilibrium. In general, the characteris-tic time scales for vacancy equilibration are much shorter thanthe time scales for significant morphological changes of defectclusters. As a consequence, the usual approach in model-ing dislocation-climb and diffusion-mediated microstructuralevolution consists in assuming instantaneous equilibration ofthe vacancy field with respect to the defect cluster morphology[28–32]. We have discussed the validity of this assumptionquantitatively in our previous work [15]. In particular, in thecase of nanometric dislocation loops or cavities, transienteffects due to vacancy equilibration become important only ona spatial scale roughly 100 times larger than the defect clustersize. At such distances, however, perturbations of the vacancyfield arising from the cluster evolution are already negligibledue to the 1/r dependence of the diffusion propagator.

Consider an isolated spherical cavity of radius R. Assum-ing an adiabatic evolution of the vacancy concentration withrespect to the cavity, we have

dR

dt= Dv

R[c0 − c� (R)], (1)

where c� is the local vacancy concentration at the cavitysurface and c0 is the equilibrium vacancy concentration peratomic site, given by c0 = exp(−Ev/kBT ) where Ev is thevacancy formation energy. c� (R) is determined by the condi-tion of local equilibrium, i.e., there is no change in free energyif vacancies attach to or detach from the cavity, which leads tothe Gibbs-Thomson expression:

c� (R) = c0 exp

(2γ�

RkBT

), (2)

where γ is the cavity surface energy per unit area and � is theatomic volume. Therefore, if we consider a system of cavitiesof average radius R evolving adiabatically with a vacancyfield in local thermodynamic equilibrium we can estimate thecharacteristic time scale for cavity evaporation as follows:

τe = R2

Dv[c0 − c� (R)]= R2

D0v

exp

(Ev + Em

kBT

)

×[

exp

(2γ�

RkBT

)− 1

]−1

, (3)

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TABLE I. Material parameters used in the present work. Parameters for Be are averages of basal and nonbasal values. 1 eV/nm2 ∼0.16 J/m2.

Material γ (eV/nm2) μ (GPa) ν b (nm) � (nm3) Ev (eV) Em (eV) D0v (nm2/s)]

W 20.4a 161b 0.28b 0.27b 0.016b 3.56c 1.78c 4.0 × 1012d

Fe 15.3a 82b 0.29b 0.29b 0.012b 2.07c 0.65c 2.8 × 1014e

Be 12.5a 132b 0.03b 0.18b 0.008b 0.95c 0.81c 5.7 × 1013f

aReference [34].bReference [35].cReference [36].dReference [37].eReference [38].fReference [39].

where Ev and Em are respectively the vacancy formation andmigration energies, and D0

v is the pre-exponential factor of thediffusion coefficient.

In a similar way, we can estimate the characteristic timescale of dislocation loop evaporation by considering the rateof change of the radius of an isolated circular prismatic loopdue to nonconservative dislocation climb [16]:

dR

dt= ± 2πDv

be ln(8R/rd )[c0 − c�(R)], (4)

where the plus or minus sign denotes respectively the caseof a vacancy or interstitial loop, be is the edge (out of plane)component of the dislocation Burgers vector, rd is the radiusof the dislocation core, and c� is the vacancy concentrationinfinitesimally close to the dislocation line. The assumptionof local equilibrium between the dislocation loop and thevacancy field surrounding it leads to the following relation:

c�(R) = c0 exp

[−fcl(R)�

bekBT

], (5)

where, for a circular prismatic loop [33],

fcl(R) = ∓ μb2e

4π (1 − ν)R

[ln

(8R

rd

)− 1

](6)

is the climb force per unit length of the dislocation,1 ν is Pois-son’s ratio and μ is the shear modulus, and the minus or plussigns distinguish between vacancy type and interstitial type,respectively. We can therefore estimate the characteristic timescale for the evaporation of a dislocation loop of radius R as

τe = ±beR ln(8R/rd )

2πD0v

exp

(Ev + Em

kBT

)

×{

exp

[± μ�be

4π (1−ν)kBT

(ln

(8R

rd

)−1

)]−1

}−1

.

(7)

1In this treatment we neglect the additional contributions to theclimb force due to stresses imposed on the system as a whole orarising from other loops. This is a good approximation because weare considering very small loops (a few nanometers wide) for whichthe self-interaction stress dominates. We have defined the climb forceas the projection of the Peach-Koehler force in the direction of thecross product between the dislocation line direction and the Burgersvector. Therefore, the climb force acting on a vacancy loop is thenegative of that acting on an interstitial loop.

The parameters used for the investigated materials throughoutthe present work are given in Table I.

We applied Eq. (3) to bcc iron, tungsten, and beryllium,which are candidate materials for nuclear fusion engineeringapplications, with R = 1 nm, which is within the range ofexperimentally observed sizes of radiation-induced cavitiesand dislocation loops. The computed τe are plotted in Fig. 3as a function of the homologous temperature T/Tm, where Tm

is the melting point. It is evident that τe can vary over manyorders of magnitude: from milliseconds to years, dependingon temperature and material properties.

We note that although W shares the same crystal structureas Fe in Fig. 3, there is a systematic difference of at leastone order of magnitude in the time scale τe for cavities anddislocation loops, even after scaling the temperature to therelevant melting point.

In Fig. 4 we plot the dependence of the relative changeof predicted time scales �τe/τe for cavity evaporation onrelative changes in the activation energy for diffusion byvacancy mechanism �Ea/Ea , where Ea = Ev + Em, andrelative changes in the surface energy �γ /γ , of up to ±10%,at T = 0.4 Tm. In Fig. 4(a) we see that an overestimation of

10-6

10-4

10-2

100

102

104

106

108

1010

0.25 0.3 0.35 0.4 0.45 0.5 0.55

τ e [

s]

T/Tm

W - cavitiesFe - cavitiesBe - cavitiesW - vac. loopsFe - vac. loopsBe - vac. loopsW - int. loopsFe - int. loopsBe - int. loops

FIG. 3. Estimated time scales τe for the evaporation and anni-hilation of ∼1 nm cavities (solid lines), vacancy dislocation loops(dashed lines), and interstitial dislocation loops (solid-dashed lines)in W (red, circles), Fe (yellow, triangles), and Be (blue, diamonds),as functions of homologous temperature T/Tm.

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-100 %

-10 %

-1 %

1 %

10 %

100 %

1000 %

10000 %

-10 % -5 % 0 % 5 % 10 %

∆τ e

/τe

∆Ea/Ea

TungstenIronBeryllium

-40 %

-20 %

0 %

20 %

40 %

60 %

80 %

-10 % -5 % 0 % 5 % 10 %

∆τ e

/τe

∆γ/γ

TungstenIronBeryllium

(a) (b)

FIG. 4. Relative change of the estimated time scale τe for cavity evaporation at T/Tm = 0.4 with respect to small deviations of activationenergy for diffusion by vacancy mechanism Ea = Ev + Em (a) and surface energy γ (b). In plot (a) the scale of the y axis is linear between−1% and 1% and logarithmic elsewhere.

Ea by just ∼2% leads to a relative error of at least 100%for τe in all three materials. This extreme sensitivity, arisingfrom the exponential dependence of τe on the formation andmigration energies of vacancies should be kept in mind whenattempting to compare experimentally observed time scaleswith theoretical estimates. Activation energies for diffusioncan be affected by a number of factors which may not beunder experimental control, such as the presence of boundimpurity-vacancy complexes, or a dependence of the entropyof activation on temperature, which may become significantat higher temperatures [40]. The sensitivity to errors in theassumed surface energy is less extreme, but also a source ofuncertainty because the surface energy per unit area of clusterswith radius as small as 1 nm or less may differ significantlyfrom the surface energies per unit area of larger clusters.

III. DEFINITION OF THE MEAN FIELD FORMALISM

A. Adiabatic vacancy field in the presenceof a small defect cluster

Consider an infinite crystal containing a circular prismaticdislocation loop of radius R, with its center at the originand lying in the x-y plane. The loop may be of vacancy orinterstitial character. Since the loop lies in the x-y plane,the component be of the Burgers vector normal to the loop(i.e., along the z axis) remains constant at all points aroundthe loop. Let c(x) be the vacancy concentration per atomicsite, i.e., c(x) is the (dimensionless) probability of finding avacancy at a lattice site at x, and let us assume that vacanciesare the only mobile point defects. If we further assume thatthe evolution of the vacancy field is adiabatic with respectto changes in the loop configuration,2 we can write [16] thefollowing equation:

c(x) = − be

4πDv

∮�

vcl(x′)|x − x′|dl′ + c∞, (8)

2This assumption implies that local equilibrium exists betweeneach dislocation segment and the surrounding vacancy field. Forquantitative bounds on this assumption see [15].

where � denotes the dislocation line, vcl(x) is the dislocationclimb velocity, and c∞ is the vacancy concentration at infinity.We define vcl(x) as the projection of the velocity of a pointon � along the vector defined by the cross product betweenthe tangent vector to the dislocation line at x and the Burgersvector.

Throughout this paper we adhere to the definition of theBurgers vector used by Hirth and Lothe [33], i.e., given acircular path C, drawn around a dislocation line accordingto the right hand rule with respect to the dislocation linedirection, the Burgers vector b is defined as

b =∮

C

∂u∂l

dl, (9)

where u is the displacement field arising from the dislocation.This implies that the cross product between the tangent vectorto the dislocation line and the Burgers vector points inwardtoward the center of a prismatic interstitial loop, and outwardaway from the center of a prismatic vacancy loop. We warnthe reader that the opposite convention is used by Landau[41], Trinkaus [42], and others, where the direction of theBurgers vector is reversed with respect to the direction ofthe dislocation line. We also define the normal direction n tothe surface enclosed by a dislocation loop according to theright hand rule with respect to the dislocation line direction,so that b and n are parallel for a prismatic vacancy loop andantiparallel for a prismatic interstitial loop.

We point out that Eq. (8) is the scalar form of a generalvector equation (see Appendix A for details):

c(x) = − 1

4πDv

∮�

v(x′)|x − x′| · (dl′ × b) + c∞

=∮

�

G(|x − x′|)v(x′) · (dl′ × b) + c∞, (10)

where G(|x|) = −1/4πDv|x| is the free space Green’s func-tion of the steady-state diffusion equation, Dv∇2G(|x|) =δ(x), v(x′) is the vector velocity of a point x′ ∈ �, b is theBurgers vector of the dislocation loop, and the differential dl′is tangential to the dislocation line at x′.

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By evaluating Eq. (8) on � and using Eq. (5) we can self-consistently find an analytical solution for vcl(x), leading toa closed-form expression for the vacancy concentration in themedium:

c(x)=c∞− 1

2 ln(8R/rd )

∮�

1

|x − x′| [c∞ − c�(R)]dl′. (11)

Let r denote the distance between x and the center of the dislo-cation loop and let φ = cos−1(z/r ), where z is the componentof x out of the x-y plane. In the limit of r R we can expandthe above integrand in powers of R/r to obtain

c(r, φ) = c∞ − [c∞ − c�(R)]π

ln (8R/rd )

(R

r

)

×[

1+1

2

(R

r

)2(3 sin2 φ

2−1

)+O

(R

r

)3], (12)

where φ = cos−1(z/r ), and z is the component of x out of thex-y plane. The above expression shows that to second orderin R/r , a dislocation loop, or in fact any defect cluster, can betreated as a spherically symmetric vacancy source or sink.

On the other hand, the vacancy field sufficiently far froman isolated spherical cavity at the origin can be exactly in-terpreted as originating from a pointlike source. Indeed, atall r > R, Newton’s shell theorem shows that the vacancyconcentration is a function of the distance to the center of thecavity r , and is given by [15,43]

c(r ) = c∞ − [c∞ − c� (R)]

(R

r

). (13)

We note that the above expressions are examples of a moregeneral equation that admits a clear physical interpretation.The vacancy field at large distances from any point-defectcluster situated at the origin, which can be approximated asan isotropic pointlike source, is characterized by the rate ofchange of the volume of the cluster3 V:

c(x) = c∞ − dVdt

G(|x|). (14)

This statement can be proven straightforwardly for a sphericalcavity, where

dVcav

dt= d

dt

(−4π

3R3

)= −4πR2 dR

dt

= −4πDvR[c∞ − c� (R)], (15)

which, upon substitution in Eq. (14), leads to Eq. (13).For a prismatic dislocation loop we prove in Appendix A

thatdVloop

dt= −

∮�

v(x′) · (dl′ × b) = −∮

�

b · (v(x′) × dl′)

= −∮

�

vcl(x′)be(x′)dl′, (16)

3With the caveat that V is negative for vacancy clusters or voidsbecause their growth reduces the overall vacancy concentration insolution, and positive for interstitial clusters because their growthincreases the overall vacancy concentration in solution assumingthere are no free interstitial atoms.

which, assuming constant vcl(x′) and be(x′) over �, becomes

dVloop

dt= −2πRvclbe = − 4π2RDv

ln(8R/rd )[c∞ − c�(R)], (17)

giving the leading order term of Eq. (12) upon substitution inEq. (14).

To simplify the presentation, we will consider only intersti-tial dislocation loops forming the invisible cluster population.This assumption is justified by the fact that the energy gainassociated with the formation of a vacancy cluster is smallerthan that of an interstitial cluster [44,45], and in general avacancy cluster has to be of appreciable size to remain stableat a finite temperature [46].

Also, in general c∞ should depend on time, followingadiabatically the evolution of all the clusters in the system.However, in a real finite system c∞ is governed at equilibriumby the surface energy and geometry of its external boundaries(free surfaces, grain boundaries), and can thus be consideredconstant in time for practical applications, provided that themorphology of external boundaries does not change apprecia-bly during the evolution.

B. Representing the invisible dislocation loops by a meanfield fully coupled to the visible clusters

Consider N cavities and n interstitial prismatic loops inan infinite medium. The cavities are sufficiently large andthey are visible experimentally, but the prismatic loops aretoo small to be detected. In the following the evolution of thecavities will be considered explicitly as discrete objects, butthe evolution of the loops will be treated through a mean field.The evolution of the cavities and the mean field will be fullycoupled.

Let cb(x) be the vacancy concentration field obtained byconsidering only the cavities. It is given by the equation

Dv∇2cb(x) = �Jn(x)N∑

i=1

δ[�i (t )], (18)

where �i (t ) denotes the surface of the ith cavity at time t , andJn is the vacancy current normal to a cavity surface, wherethe normal direction is considered pointing from the bulktowards the center of the cavity, i.e., Jn(x) > 0 if vacanciesare entering the cavity at x. δ(�) represents a delta functionevaluated on a surface, defined as∫

V

ϕ(x)δ(�)dV =∫

�

ϕ(x)dS, (19)

where ϕ(x) is an arbitrary trial function and V is an arbitraryvolume containing the surface �.

Consider the invisible interstitial loops. Let xi and Ri

denote respectively the center and the radius of the ith loop.To simplify the notation and make clear the parametric depen-dencies of the various functions, we define the following sets:

X = {xi : i = 1, . . . , n},R = {Ri : i = 1, . . . , n},

(20)X[i] = {xj : j = 1, . . . , n; j = i},R[i] = {Rj : j = 1, . . . , n; j = i},

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and we introduce the shorthand notations

dX =n∏

i=1

dxi , dR =n∏

i=1

dRi,

dX[i] =n∏

j =i

dxj , dR[i] =n∏

j =i

dRi. (21)

The total vacancy field obtained by also considering the effectof the n dislocation loops can then be expressed as [20]

c(x;X ,R) = cb(x) −n∑

i=1

dVi

dtG(|x − xi |)

= cb(x) +n∑

i=1

ξ�(Ri )G(|x − xi |)

× [c[i](xi ;X ,R) − c�(Ri )], (22)

where ξ�(R) = 4π2RDv/ ln(8R/rd ). The vacancy field ob-tained by omitting the contribution arising from the ith loop isc[i](x;X ,R), and it is defined by the self-consistent condition:

c[i](x;X ,R) = cb(x) +n∑

j =i

ξ�(Rj )G(|x − xj |)

× [c[j ](xj ;X ,R) − c�(Rj )]. (23)

We define the probability of finding the n dislocation loopsin the n volume elements (x1 + dx1, . . . , xn + dxn), and withloop radii in the ranges (R1 + dR1, . . . , Rn + dRn), as

p(X ,R)dXdR. (24)

The average of Eq. (22) with respect to the positions and radiiof each loop can then be expressed as

c(x) =∫

c(x;X ,R)p(X ,R)dXdR

= cb(x) +n∑

i=1

∫ [Si

nl(x;X ,R)

− Sl(x; xi , Ri )]p(X ,R)dXdR, (25)

where Sinl(x;X ,R) and Sl(x; xi , Ri ) are respectively the non-

local and local contributions of the ith loop to the totalconcentration field, defined as

Sinl(x;X ,R) = ξ�(Ri )G(|x − xi |)c[i](xi ;X ,R),

Sl(x; xi , Ri ) = ξ�(Ri )G(|x − xi |)c�(Ri ).(26)

Sl is the perturbation to the vacancy field in the neighborhoodof the ith loop arising from the condition of local thermo-dynamic equilibrium. The nonlocal term Si

nl describes thecontribution to the vacancy field near loop i from all the otherloops in the system.

By introducing the one-loop probability density function,

p1(xi , Ri ) =∫

p(X ,R)dX[i]dR[i], (27)

we can express the average of the local contribution as

Sl(x) =∫

Sl(x; xi , Ri )p1(xi , Ri )dxidRi, (28)

which can be readily calculated without any knowledge of cor-relations in the positions or radii of different loops, requiringonly the single loop spatial and size distribution p1(x, R).

We now average the nonlocal contribution. Let us definethe conditional probability density p(xi , Ri |X[i],R[i] ) by therelation

p1(xi , Ri ) p(xi , Ri |X[i],R[i] ) = p(X ,R) (29)

and the effective averaged vacancy field experienced by theith loop as

c eff[i] (x; xi , Ri )=

∫c[i](x;X ,R)p(xi , Ri |X[i],R[i] )dX[i]dR[i].

(30)

By employing these definitions, we may write the average ofthe nonlocal contribution as

Snl(x)=∫

ξ�(Ri )G(|x − xi |)c eff[i] (x; xi , Ri ) p1(xi , Ri )dxidRi.

(31)

In summary, the governing equation for the vacancy concen-tration, averaged over all possible positions and radii of theinvisible interstitial loops, is given by

c(x) = cb(x) + n

∫ξ�(Ri )G(|x − xi |) p(xi ; Ri )

× [c eff

[i] (x; xi , Ri ) − c�(Ri )]dxidRi. (32)

The integral in this equation is the mean field of the invisibleloops in which the visible clusters sit. Information aboutcorrelations between the positions and sizes of the invisibleloops is contained in the effective field ceff

[i] .

C. The simplest approximation of the field ceff[i]

of Eq. (30) and screening

A formal expansion of the effective field defined in Eq. (30)is derived in Appendix B, which provides the theoreticalfoundation to treat correlations between the positions of theinvisible loops to arbitrary degrees of accuracy. In this sectionwe consider the simplest approximation, which is to neglectall these correlations. We may then characterize the loopsentirely through the size distribution function f (R), whichis related to the single-loop probability density function asfollows:

p1(x, R) = n−1f (R). (33)

In the above equation, f (R)dR is the number of dislocationloops per unit volume with radii between R and R + dR.

Provided the concentration of invisible loops is sufficientlylarge, it is reasonable to assume that the average vacancyfield seen by dislocation loops is equal to the configura-tion averaged field, i.e., c(x) ≈ c eff

[i] . In these approximationsEq. (32) for the configuration-averaged concentration takes onthe form

c(x) = cb(x) +∫

dV ′G(|x − x′|)∫ ∞

b

ξ�(R)f (R)

× [c(x′) − c�(R)]dR, (34)

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I. ROVELLI, S. L. DUDAREV, AND A. P. SUTTON PHYSICAL REVIEW E 98, 043002 (2018)

where the Burgers vector b acts as a lower bound on thepossible dislocation loop size. We now define the averages:

ξ� =∫ ∞

b

dRf (R)ξ�(R),

(35)

ξ�c� =∫ ∞

b

dRf (R)ξ�(R)c�(R),

where both have the dimensions of inverse time. Equation (34)then becomes

c(x) = cb(x) +∫

dV ′G(|x − x′|)[ξ�c(x′) − ξ�c�]. (36)

We apply the operator Dv∇2 to both sides of Eq. (36),obtaining

Dv∇2c(x)=Dv∇2cb(x) +∫

dV ′δ(x − x′)[ξ�c(x′) − ξ�c�],

(37)

which, along with Eq. (18), can be reorganized as

(Dv∇2 − ξ�)c(x) = �Jn(x)N∑

i=1

δ[�i (t )] − ξ�c�. (38)

Equation (38) has the mathematical structure of an inhomoge-neous Debye-Hückel equation.

The Yukawa Green’s function, GY (|x − x′|; ξ�), is definedby the equation

(Dv∇2 − ξ�)GY (|x − x′|; ξ�) = δ(x − x′), (39)

for which the solution is

GY (|x − x′|; ξ�) = −exp{−

√ξ�/Dv |x − x′|}

4πDv|x − x′| . (40)

Thus, the diffusive interaction between cavities in Eq. (37)is screened by the mean field of the dislocation loops. Thescreening coefficient,

√Dv/ξ�, limits the range of direct

diffusional interaction between cavities.In this continuum treatment, once the effect of the loop

population is replaced by a mean field, the medium betweenthe cavities becomes a net adsorber for vacancies. This re-flects the reality at the atomic scale where the mean freepath of propagating vacancies is reduced by the presenceof distributed sinks and sources. An analogous picture wasobtained in the case of diffusion screening theory of Ostwaldripening [21–26], where similar screened diffusion-mediatedinteractions were derived from a coarse-graining of second-phase particles. In this context, screened diffusive interactionswere originally employed as a tool to avoid divergences forfinite volume fractions of particles due to the infinite-rangeLaplacian field.

In our case, the coarse grained diffusion field that generatesscreened interactions allows us to effectively account for thelimited information about the distributions of small defectclusters.

In a dilute configuration of cavities, each of them can beapproximated as a point source with an effective sink strengthgiven by Qi = 4πR2

i Ri . A formal solution of Eq. (38) is then

c(x) =N∑

i=1

QiGY (|x − xi |; ξ�) − ξ�c�

×∫

dV ′GY (|x − xi |; ξ�) + c∞, (41)

where xi denotes the center of the ith cavity. We note that

−ξ�c�

∫dV ′GY (|x − x′|; ξ�)

= ξ�c�

ξ�

=∫ ∞b

dR ξ�(R)f (R)c�(R)∫ ∞b

dR ξ�(R)f (R)= 〈c�〉, (42)

where the average 〈. . .〉 is defined with respect to the weighteddistribution f (R) = ξ�(R)f (R).

Thus, we arrive at the following self-consistent relationbetween the rates of change of the cavity radii Ri , the vacancyfield, and the distribution of the interstitial dislocation loopsf (R):

c(x) = −N∑

i=1

R2i Ri

exp[−√

ξ�/Dv|x − xi |]Dv|x − xi | + c∞ + 〈c�〉.

(43)

To solve this equation the vacancy concentration has to beevaluated at the position of each cavity, satisfying the bound-ary condition of local thermodynamic equilibrium, i.e.,

c� (Ri ) = −N∑

j = 1j = i

R2j Rj

exp[−√

ξ�/Dv|xi − xj |]Dv|xi − xj |

− RiRi

Dv

+ c∞ + 〈c�〉, i = 1, . . . , N, (44)

where −RiRi

Dvis a finite size correction accounting for the

self-diffusional interaction of a cavity with itself, representingvacancies propagating between different points on the surfaceof the same cavity. As a comparison, we recall the analogousof the above system of equations for a system where the n

dislocation loops are explicitly considered as discrete objects,given by the set of N + n equations [15]:

c� (Ri ) = −N∑

j = 1j = i

R2j Rj

Dv|xi − xj | − RiRi

Dv

+N+n∑

k=N+1

beRkRk

2Dv|xi − xk| + c∞, i = 1, . . . , N

c�(Ri ) = −N∑

j=1

R2j Rj

Dv|xi − xj | +N+n∑

k = N + 1k = i

beRkRk

2Dv|xi − xk| −be ln

( 8Ri

rd

)Ri

2πDv

+ c∞, i = N + 1, . . . , N + n, (45)

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STATISTICAL MODEL FOR DIFFUSION-MEDIATED … PHYSICAL REVIEW E 98, 043002 (2018)

where the indices from 1 to N denote cavities and from N + 1 to N + n denote dislocation loops. By inverting the linear systemdefined by Eq. (44), the set of Ri can be then calculated provided f (R) is known.

The screening length changes with time because it dependson the evolution of the distribution of the interstitial loopsizes. Let the time-dependent size distribution function beF (R, t ). A continuity equation in particle-size space for thedistribution F (R, t ) can be written as

∂F (R, t )

∂t+ ∂[F (R, t )vil(R, t )]

∂R= n(R, t ), (46)

where n(R, t ) is the net rate at which loops of radius R

are created, as a result of coalescence of existing loops andnucleation of new loops. The growth rate vil(R, t ) of a loop of

radius R at time t is given by

vil(R, t ) = −[cavg(t ) − c�(R)]2πDv

be ln(8R/rd ), (47)

where cavg(t ) is the spatially averaged vacancy concentration.Differentiating with respect to R gives

∂vil(R, t )

∂R= 2πDv

be ln(8R/rd )

{[cavg(t ) − c�(R)]

R ln(8R/rd )

+ μbe�c�(R)

4π (1 − ν)R2kBT

[ln

(8R

rd

)− 2

]}. (48)

In the homogeneous mean field treatment, the term cavg(t ) canbe obtained by averaging Eq. (43) with respect to x:

cavg(t ) = − limV →∞

⎧⎨⎩

N∑i=1

R2i (t )Ri (t )

V Dv

∫V

d3xexp[−

√ξ�/Dv|x − xi |]|x − xi |

⎫⎬⎭ + c∞ + 〈c�〉

= 〈c�〉 + c∞ − 4πρc

ξ�

(1

N

N∑i=1

R2i (t )Ri (t )

)= 〈c�〉 + c∞ − V c(t )ρc

ξ�

, (49)

where ρc denotes the number density of cavities and V c(t )denotes the average rate of change of the volume of cavities, attime t . It is clear that in the limiting case of ξ� = 0 the aboveexpression diverges, and the integral of the Green’s functionhas to be performed up to a suitable cutoff distance.

In summary, while the mean field of interstitial loopsscreens the diffusive interaction between cavities, the rate ofgrowth of cavities also determines the evolution of the meanfield: the cavities and the mean field are coupled. In Fig. 5we show a flowchart that summarizes a scheme to computethe system evolution, highlighting the couplings between thecavities and the mean field.

Discrete cavities

Mean Field loops

Size

distribution

Compute new

size distribution

via eq. (46)

Compute effective

screening constant

via eq. (35)

N positions

N radii

Compute new

cavity radii

Compute cavity

growth rates

via eq. (44)

FIG. 5. Flowchart summarizing a scheme to compute the evo-lution of the coupled mean field of interstitial loops and discretecavities.

D. Numerical estimate of the screening length

In this section we calculate the initial value, at t = 0, of thescreening length of the effective interaction between cavitiesin the presence of a mean field of interstitial dislocation loops.It is helpful to introduce the scaled distribution function:

Aφ

(R

b, t

)= F (R, t ), (50)

where A is a normalization factor with the dimensions oflength−4, defined with respect to the size distribution at t = 0as

A = ρ(0)

(b

∫ ∞

1du φ(u, 0)

)−1

, (51)

where ρ(t ) is the number density of the mean field clusters attime t . The screening length �l =

√Dv/ξ� , at time t = 0, is

given by

�l(0) =(∫ ∞

b

dR ξ�(R)F (R, t )/Dv

)−1/2

= 1

2πb√

A

(∫ ∞

1du u φ(u, 0)/ ln(8bu/rd )

)−1/2

.(52)

For the initial loop size distribution we assume a power lawwith an exponentially decaying cutoff at small loop radii, i.e.,

φ(u, 0) = e−c1/uu−c2 , (53)

where c1 and c2 are constants. To ensure the existence of theintegrals defining ξ� and ξ�c we require c2 > 2.

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I. ROVELLI, S. L. DUDAREV, AND A. P. SUTTON PHYSICAL REVIEW E 98, 043002 (2018)

200

400

600

800

1000

10-07

10-06

10-05

10-04

Scre

enin

g length

[nm

]

Defect number density [nm-3

]

FIG. 6. Plot of the prefactor (2π )−1(ρ(0)b)−1/2 of Eq. (54) as afunction of loop number density ρ. The value of the Burgers vectorb is 0.27 nm, which is representative of tungsten.

The screening length can then be recast as

�l(0)

≈ 1

2π√

ρ(0)b

√ ∫ ∞1 du exp(−c1/u)u−c2∫ ∞

1 du exp(−c1/u)u−c2+1/ ln(8ub/rd )

≡ �(c1, c2)

2π√

ρ(0)b, (54)

where we have separated the dependence on the loop numberdensity from the shape of the size distribution contained in�(c1, c2). For realistic loop densities of 10−6–10−4 nm−3

the factor (2π√

ρ(0)b)−1

is of the order of 102–103 nm (seeFig. 6), while �(c1, c2) is of the order of unity for reasonablevalues of c1 and c2. In particular, with 0.1 < c1 < 1 and2 < c2 < 4, we have 0.68 � � � 1.3. At these loop densitiesthe screening length is comparable to the separation of theloops, which may be substantially smaller than the separationof cavities. Therefore the screening of the diffusive interactionbetween cavities by small dislocation loops plays a significantrole in the evolution of the distribution of cavity sizes duringan anneal.

IV. EXTENSION TO THIN FILM GEOMETRY

Transmission electron microscopy is providing accuratedata on the distributions of radiation defects [2,47]. SinceTEM samples are always thin films we provide in this sectionan extension of the theory to infinitely extended thin films.

Consider a region V ∈ R3 infinitely extended in the x

and y directions and of thickness H in the z direction, i.e.,V = {(x, y, z) ∈ R3 : x ∈ R, y ∈ R,−H/2 < z < H/2}.We define a L × L × H cell �(0) ={x = x1a1 + x2a2 + x3a3 : −1/2 < xα < 1/2, α = 1, 2, 3},where a1 = (L, 0, 0), a2 = (0, L, 0), and a3 = (0, 0,H ) arethe three basis vectors defining the cell. The cell �(0) containsN spherical cavities and a much larger number of smallercircular prismatic dislocation loops, with size distributionfunction f (R), normalized to the loop number density. Theprimitive cell is infinitely replicated in the x and y directions.

2H HQ3Q1

Q2

-Q3-Q1

-Q2

FIG. 7. 2D sketch of the primitive cell structure for the thin filmconfiguration. The red shaded region is a repeat cell of the thinfilm. Image clusters (portrayed with dashed boundaries) of oppositeeffective charge are constructed in the yellow shaded regions and theoriginal primitive cell is extended accordingly. Periodic boundaryconditions are imposed on the dashed boundaries and in the direc-tions perpendicular to the drawing. The size of cavities with respectto the simulation cell is exaggerated for clarity.

Let �i denote the surface of the ith cavity. We impose thefollowing boundary conditions on the vacancy concentrationfield c(x):

c(x, y,−H/2) = c(x, y,H/2) = cS,

c(x ∈ �i ) = c�ii = 1, . . . , N.

(55)

Using the terminology of electrostatics, each cavity can beassociated with an effective charge Q, related to rate of growthof the cavity volume V:

Qi = −dVi

dt= 4πR2

i

dRi

dt, (56)

where Ri is the radius of the ith cavity. As noted in theprevious section, to first order the effect of the mean field ofdislocation loops is to introduce a screened interaction withscreening length

√Dv/ξ�(t ), and to displace the resultant

concentration field by an amount

〈c�〉(t ) =∫

dRξ�(R)F (R, t )c�(R)∫dRξ�(R)F (R, t )

.

The boundary conditions at c(z = ±H/2) = cS can be sat-isfied using the method of images, by introducing an infinitenumber of periodic images of the thin film in the z directions.The images are constructed as follows: we extend the prim-itive cell by H/2 in the positive and negative z directions;for each cavity with charge Q in the original primitive celllocated at (x, y, z) we add a virtual cavity of charge −Q at(x, y, sgn(z)H − z). A new primitive cell may then be de-fined containing 2N interacting cavities: �(0) = {x = x1a1 +x2a2 + x3a3 : −1/2 < xα < 1/2, α = 1, 2, 3}, with primitivevectors a1 = (L, 0, 0), a2 = (0, L, 0), and a3 = (0, 0, 2H ).A two-dimensional (2D) sketch of such a construction ispresented in Fig. 7. This new primitive cell is charge neutralby construction, and it is repeated infinitely many timesalong z.

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The self-consistent system of equations that determines theset of {Qi}i=1,...,N at time t is therefore given by

c�i=

N∑j = 1j = i

Qj

[ ′∑mGY (|xi − m + xj |; ξ�)

−∑

m

GY (|xi − m + xj |; ξ�)

]

− Qi

4πDvRi

+ cS + 〈c�〉,i = 1, . . . , N, (57)

where m = m1a1 + m2a2 + m3a3, mα ∈ Z, α = 1, 2, 3, xj =(xj , yj , sgn(zj )H − zj ) and primed sums denote that the termm = 0 is not included when xi = xj .

The sums over m are convergent for finite values ofthe screening length. However, it is possible that the meanfield loops disappear during an anneal, leading to an infinitescreening length and therefore to a 1/r diffusive interactionbetween the cavities. In that case, the sums over m are onlyconditionally convergent.

Since the new primitive cell is charge neutral, wemay use a variant of the Ewald summation technique toderive an absolutely convergent series for all screeninglengths. This can be achieved by introducing the followingkernel:

K (xi , xj ; ε) = −∑

k

exp[−(k2 + ε2)/4β2]

V Dv (k2 + ε2)eik·xi [e−ik·xj − e−ik·xj ]

− 1

8πDv

{ ′∑m

1

|xi − xj + m|[

erfc

(β|xi − xj + m| + ε

2β

)eε|xi−xj +m|

+ erfc

(β|xi − xj + m| − ε

2β

)e−ε|xi−xj +m|

]

−∑

m

1

|xi − xj + m|[

erfc

(β|xi − xj + m| + ε

2β

)eε|xi−xj +m|

+ erfc

(β|xi − xj + m| − ε

2β

)e−ε|xi−xj +m|

]}+ δij

4πDv

[2βe−ε2/4β2

√π

− ε erfc

(ε

2β

)]. (58)

The set of self-consistent equations to be solved become

c�i=

N∑j = 1j = i

QjK (xi , xj ;√

ξ�/Dv ) − Qi

4πDvRi

+ cS + 〈c�〉, i = 1, . . . , N, (59)

which is the analog of Eq. (44) for the thin film configuration. A detailed derivation of K (xi , xj ; ε) is given in Appendix C.We note that the numerical scheme to compute the temporal evolution of cavities remains the same as that presented at the

end of Sec. III.By taking the limit in the kernel of an infinite screening length we obtain

limε→0

K (xi , xj , ε) = −∑

k

exp[−k2/4β2]

V Dvk2eik·xi [e−ik·xj − e−ik·xj ]

− 1

4πDv

[ ′∑m

erfc(β|xi − xj + m|)|xi − xj + m| −

∑m

erfc(β|xi − xj + m|)|xi − xj + m|

]+ δij

2β

4π3/2Dv

, (60)

which is the usual expression for the Ewald sumfor the unscreened Green’s function G(|x − x′|) =−(4πDv|x − x′|)−1. Thus the kernel K (xi , xj ; ε) enablesus to treat within the same set of equations the evolution ofcavities, whatever the screening length.

V. FULL MEAN FIELD DESCRIPTION

In this section we consider both dislocation loops andcavities as contributors to a mean field. This may be usefulwhenever information on cluster distributions at all length

scales is incomplete. Consider the vacancy field generatedby a spherical cavity in an infinite homogeneous medium inwhich the vacancy concentration far from the cavity is c∞:

c(r ) = c∞ − [c∞ − c� (R)]R

r

= c∞ + ξ� (R)G(r )[c∞ − c� (R)], (61)

where ξ� (R) = 4πDvR and

c� (R) = c0 exp

[2γ�

RkBT

]. (62)

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I. ROVELLI, S. L. DUDAREV, AND A. P. SUTTON PHYSICAL REVIEW E 98, 043002 (2018)

Consider the case of a spatially homogeneous, uncorrelateddistribution of loops and cavities. We define the numberdensity respectively of vacancy loops, interstitial loops, andcavities, with radii in the range (R + dR), as fv(R)dR,fi(R)dR, and fc(R)dR.

Following a similar treatment to that used in Sec. III, wecan write a mean field equation analogous to that of Eq. (34):

c(x) = c∞ +∫

dV ′G(|x − x′|)∫ ∞

b

dR {ξ�(R)fv(R)

× [c(x′) − cv�(R)] + ξ�(R)fi(R)[c(x′) − ci

�(R)]

+ ξ� (R)fc(R)[c(x′) − c� (R)]}, (63)

where the boundary condition cv�(R) for vacancy loops differs

from that for interstitial loops, ci�(R), by the sign in the

exponent, i.e., cv�(R)ci

�(R) = c20.

We define the quantities c1 = cv�, c2 = ci

�, c3 = c� , f1 =fv , f2 = fi , f3 = fc, ξ1 = ξ2 = ξ�, and ξ3 = ξ� . Equa-tion (63) then assumes the compact form:

c(x) = c∞ +3∑

α=1

∫dV ′G(|x − x′|)

×∫ ∞

b

dR ξα (R)fα (R)[c(x′) − cα (R)]. (64)

We define

ξ =3∑

α=1

ξα =3∑

α=1

∫ ∞

b

dR ξα (R)fα (R),

ξc =3∑

α=1

ξαcα =3∑

α=1

∫ ∞

b

dRξα (R)fα (R)cα (R),

(65)

so that

c(x) = c∞ + ξ

∫dV ′G(|x − x′|)c(x′)

− ξc

∫dV ′G(|x − x′|). (66)

In momentum space, using the Fourier transform definitionf (x) = (2π )−3/2

∫dqeiq·xf (q), this becomes

˜c(q) = c∞ + ξG(q) ˜c(q) − (2π )3/2ξcG(q)δ(q), (67)

leading to the solution

˜c(q) = c∞ − (2π )3/2ξcG(q)δ(q)

1 − ξG(q). (68)

In a homogeneous infinite medium, we have G(|x − x′|) =−1/(4πDv|x − x′|) leading to

˜c(q) = δ(q)Dvc∞ + (2π )3/2ξc/q2

Dv + ξ/q2

= δ(q )

2π

Dvc∞ + (2π )3/2ξc/q2

Dvq2 + ξ, (69)

and, transforming back in real space,∫dq

(2π )3/2eiq·x ˜c(q)

= 1

(2π )5/2

∫ ∞

0dq

∫ π

0sin φ dφ

∫ π

−π

dθδ(q )

× Dvq2c∞ + (2π )3/2ξc

Dvq2 + ξeiqr cos φ = ξc

ξ, (70)

which implicitly depends on time through the size distributionfunctions fα (R). We point out that for fα (R) = δα2f (R) werecover the result of the previous sections.

The evolution laws for the size distributions are analogousto Eq. (46), where the growth velocities are given by

vvl(R, t ) = [c(t ) − cv

�(R)] 2πDv

b ln(

8Rrd

) ,vil(R, t ) = −[

c(t ) − ci�(R)

] 2πDv

b ln(

8Rrd

) ,vc(R, t ) = −[c(t ) − c� (R)]

Dv

R,

(71)

respectively for vacancy loops, interstitial loops, and cavities.As a consequence, the average vacancy concentration de-

pends on time through the time-dependent size distributionfunctions Fα (R, t ) as

c(t ) = ξc(t )

ξ (t )=

∑3α=1

∫ ∞b

dR ξα (R)Fα (R, t )∑3α=1

∫ ∞b

dR ξα (R)Fα (R, t )cα (R). (72)

A numerical scheme for the computation of the cluster sizedistribution as a function of time can then be summarized asfollows:

(1) Starting from the size distributions fv(R, t0), fi(R, t0),and g(R, t0), compute c(t0) via Eq. (70).

(2) Compute the new size distributions at time t0 + dt

using Eq. (46).(3) Reiterate from step (1).

VI. NUMERICAL SIMULATIONS

We present simulations of an anneal of an infinite thinfilm of thickness H = 200 nm with 30 cavities per periodiccell, using the technique presented in Sec. IV, for tungsten,iron, and beryllium. The sizes L of the primitive cell inboth the x and y directions are respectively 133.64, 422.58,and 2112.89 for the simulations with cavity densities of10−5 nm−3, 10−6 nm−3 and 4 × 10−8 nm−3.

We compared the effect of three mean field conditions:containing only interstitial loops, or only vacancy loops, or noloops. The initial size distribution of the loops was φ(R) =(b/R)c2 exp(−c1b/R), c1 = 0.2, c2 = 3 and normalizationsuch that the loop number density at t = 0 was 10−4 nm−3,therefore the average separation between loops was ∼ 22 nm.

The cavities were assigned to random positions in theperiodic cell, satisfying the number density and a minimumseparation between cavities of 10 nm, with sizes taken froma Gaussian distribution of mean 1 nm and standard deviation0.1 nm. No vacancy supersaturation was assumed in the film,so that its free surfaces were assigned a constant vacancyconcentration equal to the equilibrium value c0.

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10-4

10-2

100

102

104

106

0.25 0.3 0.35 0.4 0.45 0.5 0.55

Annih

ilation t

ime [

s]

T/Tm

W, vac.W, int.W, no MFFe, vac.Fe, int.Fe, no MFBe, vac.Be, int.Be, no MF

0.3 0.35 0.4 0.45 0.5 0.55

T/Tm

W, vac.W, int.W, no MFFe, vac.Fe, int.Fe, no MFBe, vac.Be, int.Be, no MF

0.3 0.35 0.4 0.45 0.5 0.55

T/Tm

W, vac.W, int.W, no MFFe, vac.Fe, int.Fe, no MFBe, vac.Be, int.Be, no MF

(a) (b) (c)

FIG. 8. Time taken to remove all cavities in thin films of thickness 200 nm in tungsten (red, circles), bcc iron (orange, triangles), andberyllium (blue, diamonds), as a function of annealing homologous temperature, for three number densities of cavities: (a) 10−5 nm−3,(b) 10−6 nm−3, and (c) 4 × 10−8 nm−3. In each case there are either no dislocation loops present (solid lines), or only vacancy loops (dashedlines) or only interstitial loops (dotted solid-dashed lines). Diffusion is assumed to occur by a vacancy mechanism only.

In Fig. 8 we plot the time taken to remove all the cavities,as a function of homologous annealing temperature T/Tm.

We see that the time-scale estimates given in Sec. II arein broad agreement with numerical simulations, for all threeinvestigated materials. As already noted in our previous work[15], we see that as the number density of cavities increasesthe time to annihilate all cavities also increases. This is dueto the diffusive interaction between cavities: as the distancesbetween cavities are reduced, the local vacancy concentrationbetween cavities increases. The driving force for vacancyemission from each cavity is then reduced compared to thecase of a dilute configuration of cavities.

The presence of a mean field of dislocation loops, aspreviously discussed, introduces a screening of the diffusiveinteraction between cavities and increases of the overall back-ground vacancy concentration. Screening accelerates the anni-hilation of cavities, while the increased vacancy concentrationhas the opposite effect, and can even induce a transient phaseof cavity growth. Whether one effect or the other dominatesdepends on the type of dislocation loops that make up themean field.

We have assumed that the only mobile point defects arevacancies. The concentrations of these point defects just out-side interstitial and vacancy loops are highly asymmetricaldue to the exponential dependence on the climb force and thechange of sign of the Burgers vector between the loops. As aconsequence, the vacancy concentration near interstitial loopsis much less than for vacancy loops. Therefore the rate of evo-lution of vacancy loops is faster than that of interstitial loops.

The screening factor and the larger background vacancyconcentration provided by vacancy loops suggest that theymight affect the evolution of cavities to a higher degree thaninterstitial loops. However, it should also be kept in mind thatthe evolution by vacancy diffusion of vacancy loops is fasterthan that of interstitial loops with the same initial distributionsof loop sizes, and by the end of the simulation a large fractionof the initial vacancy loop population has evaporated. Toillustrate this behavior, we plot in Fig. 9 the time requiredfor the vacancy loops screening coefficient

√ξ/Dv to halve

with respect to its initial value, as a function of simulated tem-perature. We point out that the “half-lives” of the screeningcoefficient are always much shorter than the time required forall cavities to evaporate. The evolution of the vacancy loopsmean field does not appear to be noticeably affected by thenumber density of cavities, at least in the investigated densityrange from 4 × 10−8 to 10−5 nm−3.

In simulations with only interstitial loops as part of themean field, on the other hand, the initial size distributionfunction, the screening coefficient, and the shift to the back-ground vacancy concentration are practically constant withrespect to the evolution of cavities.

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

0.3 0.35 0.4 0.45 0.5 0.55

Scre

enin

g facto

r half-l

ife [

s]

T/Tm

WFeBe

FIG. 9. Time taken to halve the screening factor√

ξ/Dv withrespect to its initial value, in thin films of thickness 200 nm intungsten (red, circles), bcc iron (orange, triangles), and beryllium(blue, diamonds), with a mean field of vacancy loops, as a func-tion of annealing homologous temperature. The initial cavity anddislocation loop number densities are respectively 10−5 nm−3 and10−4 nm−3.

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12

(a) (b)

14 16

R [

nm

]

t [s]

Int. loops MFVac. loops MF

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.5 1 1.5 2 2.5

t [s]

Int. loops MFVac. loops MF

FIG. 10. Evolution of the radius of a single cavity at z = 0.39H as a function of time for tungsten at 1800 K and for a cavity numberdensity of 10−6 nm−3. Results are compared for the case of an interstitial loop mean field (solid red lines), and a vacancy loop mean field(dashed blue lines). Plot (b) is a magnification of the boxed area of plot (a).

According to our simulations, when the mean field ismade of interstitial loops there is a systematic decrease ofcavity evaporation time scales with respect to the referencevalues, for every investigated cavity number density. We mustconclude that the screening effect, partially suppressing thediffusive interaction between cavities, is the dominant onefor interstitial loops. The opposite holds true for the vacancyloops mean field simulations, where a systematic and verypronounced increase in cavity evaporation time scales sug-gests that the extra background vacancy concentration playsthe most important role.

In order to further elucidate this point, we show in Fig. 10the evolution in time of the size of a single cavity in tungstenat 1800 K and with a cavity number density of 10−6 nm−3.The cavity is located at z = 0.39H . We note that in theinitial phases of the simulation the additional backgroundvacancy concentration induced by vacancy loops results inthe cavity rapidly adsorbing vacancies. As the mean fieldevolves, vacancy loops decrease in number and the extravacancy concentration is reduced until the surface energyof cavities becomes the dominant driving force, leading totheir evaporation. Due to the initial phase of growth of thecavity, the total annihilation time with the vacancy loopsmean field is therefore larger than in the case of the inter-stitial loops mean field. While we found this to hold truefor all investigated cavity densities, the picture is slightlymore complex when only considering the evolution of asingle cavity of the ensemble. In particular, the same cavitywill not always evaporate faster with an interstitial loopmean field, especially in simulations with a high cavitydensity.

To illustrate such a case, we present in Fig. 11 the evolutionof a single cavity at z = 0.65H , with a cavity number density10−5 nm−3 and at a temperature of 1800 K. As we can see,the vacancy loops mean field still induces an initial phaseof cavity growth (Fig. 11), but the cavity evolves erraticallyin the interstitial loops mean field simulation. By comparingFig. 11(b) with a real-space view of the simulation ( Fig. 12)we find this behavior is most likely due to local interactionbetween cavities.

In particular, the erratic size increases and decreases ofthe cavity seem to be correlated with the evaporation ofcavities in its neighborhood. Such local effects are muchless important with the vacancy loops mean field because,even though the shift in background vacancy concentrationdominates, the screening coefficient is still substantially largerthan for interstitial loops, as already mentioned. This suggeststhat at higher densities local interactions between cavities playa large role and might lead the dynamics of single cavities todeviate from the general behavior with respect to mean fieldconditions.

VII. CONCLUSIONS

In this paper we derived a hybrid mean field and real-space model that couples our earlier [15] nonlocal model ofevolution of cavities produced by irradiation with a meanfield representing dislocation loops smaller than the experi-mental detection limit. The main result is that the mean fieldscreens the diffusive interactions between cavities and addsto, or subtracts from, the vacancy concentration dependingon whether the loops are vacancy or interstitial characterrespectively. We presented a general scheme to implementhigher order corrections to the model. The model was initiallyderived for an infinite medium, but it was then modifiedto treat the case of an infinitely extended thin film, whichis more useful for comparing with available experimentaldata obtained by transmission electron microscopy. Throughpreliminary numerical simulations we discussed some of thefeatures of the model, highlighting in particular the interplaybetween the evolution of the mean field and the cavities,and the roles of the cavity number density and mean fieldscomprising vacancy vs interstitial loops.

Finally, we mention some avenues for development ofthe model. In its current state, we have assumed cavitiesand dislocation loops are immobile, with only their sizes asdynamical variables. However, nanometric dislocation loopsin bcc metals have been experimentally observed [48] toperform fast, one-dimensional diffusion. Theoretical inves-tigations [49] have also shown that interstitial loops in bcc

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0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

R [

nm

]

t [s]

Int. loops MFVac. loops MF

1.06

1.07

1.08

1.09

1.1

1.11

1.12

1.13

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t [s]

Int. loops MFVac. loops MF

(a) (b)

FIG. 11. Same as Fig. 10 but for a cavity number density of 10−5 nm−3. The cavity is located at z = 0.65H .

metals might be extremely mobile, with collective diffusioncoefficients comparable to single interstitials. Therefore, itwould be useful to incorporate the mobility of dislocation

loops in the model. The introduction of elastic interactionsbetween dislocation loops, cavities, and surfaces would alsobe a valuable addition, because they have been shown to

FIG. 12. Snapshots of the evolution of the cavity of Fig. 11, shaded in orange and white in the cell center, and neighboring cavities in blue;(a) initial configuration, (b) t = 0.208 s, (c) t = 0.435 s, (d) t = 0.586 s, (e) t = 0.632 s, (f) t = 2.05 s. The plots show only a small portionof the periodic cell. The sizes of the cavities have been increased by a factor of 2 for greater clarity.

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I. ROVELLI, S. L. DUDAREV, AND A. P. SUTTON PHYSICAL REVIEW E 98, 043002 (2018)

be a critical factor in determining the large scale spatialdistribution of dislocation loops [50]. Additionally, elastic in-teractions between large numbers of dislocation loops are alsoresponsible for partially suppressing their one-dimensionaldiffusion, as loops may to be confined in local minima ofinteraction energy.

In this work we assumed the mobile point defects areexclusively monovacancies. It is possible to adapt the modelto the case of interstitial-only diffusion with relative ease.Doing so would generate similar dynamics, with a perfectlysymmetrical inversion between the behavior of interstitial andvacancy loops, while cavities would evolve at a considerablyreduced rate. It would be more involved to treat both mono-vacancies and single interstitials as diffusing species, sinceit would require the introduction of reaction rates for theannihilation of the two species in the bulk. It would also benecessary to introduce bias factors between cluster-vacancyand cluster-interstitial interactions because the evolution ofthe cluster would no longer be determined by a unique point-defect absorption or emission mechanism.

ACKNOWLEDGMENTS

This work has been carried out within the framework ofthe EUROfusion Consortium and has received funding fromthe Euratom Research and Training Programme 2014–2018under Grant Agreement No. 633053. Also, it has been par-tially funded by the RCUK Energy Programme (Grant No.EP/P012450/1). The views and opinions expressed herein donot necessarily reflect those of the European Commission.I.R. was also supported by the EPSRC Centre for DoctoralTraining on Theory and Simulation of Materials at ImperialCollege London under Grant No. EP/L015579/1. A.P.S. isgrateful to the Blackett Laboratory for the provision of lab-oratory facilities.

APPENDIX A: RELAXATION VOLUME OF ADISLOCATION LOOP AND ITS EVOLUTION IN TIME

Let us consider an arbitrarily shaped dislocation loop, notnecessarily planar, with Burgers vector b and bounded by apiecewise linear dislocation line � divided in N segments.Given an arbitrary orthogonal coordinate system, let Ri andRi+1 = Ri + �Li be the vectors denoting the extremes of theith segment on �, with �Li parallel to the dislocation linedirection, and RN+1 = R1. The vector area of the triangleidentified by the origin O, Ri , and Ri+1 is given by

Ai = 12 (Ri × Ri+1). (A1)

Summing the individual contributions given by the N trian-gles, we obtain the total vector area of the loop:

A =N∑

i=1

Ai = 1

2

N∑i=1

(Ri × Ri+1), (A2)

which, in order to be a well-defined quantity, has to beindependent from the choice of the origin of coordinates. Inorder to prove this statement, let us shift the coordinate system

O

Γ

Ri

Ri+1

ΔLi

FIG. 13. Sketch of the vectors Ri , Ri+1, and �Li with respect tothe boundary � of the dislocation loop. The arrows on � denote thedirection of the dislocation line.

by an arbitrary vector R0 and compute the new A′:

A′ = 1

2

N∑i=1

[(Ri + R0) × (Ri+1 + R0)]

= 1

2

N∑i=1

(Ri × Ri+1) + 1

2R0 ×

N∑i=1

(Ri+1 − Ri )︸ ︷︷ ︸0

= A.

(A3)

Therefore A is indeed a well-defined measure for the vectorarea of a dislocation loop. A formula for the vector area of adislocation loop can also be written in the form of a contourintegral, see Eq. (27.11) of Ref. [41], as

A = 1

2

∮(x × dl), (A4)

where x ∈ � and dl is everywhere tangential to the dislocationline. In the notations of Fig. 13 the above equation corre-sponds to the limit N → ∞.

Using the convention for the Burgers vector and the dislo-cation line tangential vector adopted in [33], we now definethe relaxation volume of the loop as follows:

V = −(b · A) = −1

2b ·

N∑i=1

(Ri × Ri+1), (A5)

which is correctly negative for vacancy prismatic loops, whereb is parallel to A, and positive for interstitial prismatic loops,where b is antiparallel to A. We now consider the continuouslimit for |�Li | → 0 and N → ∞, leading to the line integral:

V = −1

2

∮�

b · (x × dl). (A6)

Let us assume that the line �, defining the perimeter of thedislocation loop, is parametrized by variable ϕ ∈ [0, 1) andthat it evolves with time, so that for x ∈ � we have x = x(ϕ, t )and dl = ∂x

∂ϕdϕ. We can then write V(t) as

V (t ) = −1

2b ·

∫ 1

0

[x(ϕ, t ) × ∂x

∂ϕ(ϕ, t )

]dϕ (A7)

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and its temporal rate of change as

dVdt

= −1

2b ·

{∫ 1

0

[dxdt

× ∂x∂ϕ

]dϕ +

∫ 1

0

[x × ∂

∂ϕ

(dxdt

)]dϕ

}. (A8)

Using integration by parts, the second integral in the right-hand side can be rearranged as∫ 1

0

[x × ∂

∂ϕ

(dxdt

)]dϕ =

[x × dx

dt

]∣∣∣∣ϕ=1

ϕ=0︸ ︷︷ ︸0

−∫ 1

0

[∂x∂ϕ

× dxdt

]dϕ =

∫ 1

0

[dxdt

× ∂x∂ϕ

]dϕ (A9)

so that

dVdt

= −b ·∫ 1

0

[dxdt

× ∂x∂ϕ

]dϕ = −

∮�

b ·[dxdt

× dl], (A10)

which is a slightly modified form of Eqs. (4)–(2) (where it was given without proof) found in [33]. We can recast Eq. (A10) in aform more convenient for our applications:

dVdt

= −∮

�

b ·(

dxdt

× dl)

= −∮

�

b · (v × dl) = −∮

�

v(x) · (dl × b) = −∮

�

vcl(x)be(x)dl, (A11)

where v(x) = dx/dt is the vector velocity of a point x ∈ �, vcl is the (scalar) dislocation climb velocity, be is the edge componentof the Burgers vector, and dl is the differential arc length on �.

APPENDIX B: PERTURBATIVE EXPANSION OF THE FORMAL SCATTERING SERIES

Let us combine Eqs. (22) and (23) in order to obtain a formal scattering series for the concentration field:

c(x;X ,R) = cb(x) +n∑

i=1

ξ�(Ri )G(x, xi )[cb(xi ) − c�(Ri )]

+n∑

i=1

∑j =i

G(x, xi )ξ�(Ri )G(xi , xj )ξ�(Rj )[cb(xj ) − c�(Rj )]

+n∑

i=1

∑j =i

∑k =j

G(x, xi )ξ�(Ri )G(xi , xj )ξ�(Rj )G(xj , xk )ξ�(Rk )[cb(xk ) − c�(Rk )] + · · · (B1)

We can now recast Eq. (B1) in a manner similar to classical scattering theory:

c(x;X ,R) = cb(x) +n∑

i=1

∫dx′dx′′dR G(x, x′)Ti (x′, x′′, R)[cb(x′′) − c�(R)]

+n∑

i=1

∑j =i

∫dx′dx′′dx′′′dxivdRdR′G(x, x′)Ti (x′, x′′, R)G(x′′, x′′′)Tj (x′′′, xiv, R′)[cb(xiv ) − c�(R′)]+ · · · , (B2)

where T is a scattering operator defined by

Ti (x, x′, R) = ξ�(R)δ(x − xi )δ(x′ − xi )δ(R − Ri ). (B3)

Equivalently, in momentum space, using the Fourier transform definition f (x) = (2π )−3/2∫

dqeiq·xf (q), we have

c(q;X ,R) = cb(q) +n∑

i=1

∫-dq′dR G(q)Ti (q − q′, R)[cb(q′) − (2π )3/2δ(q′)c�(R)]

+n∑

i=1

∑j =i

∫-dq′q′′dRdR′G(q)Ti (q − q ′, R)G(q′)Tj (q′ − q′′, R′)[[cb(q′′) − (2π )3/2δ(q′′)c�(R′)] + · · · , (B4)

where

Ti (q, R) = ξ�(R)δ(R − Ri )e−iq·xi (B5)

and -dq = (2π )−3/2dq.

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We can similarly write a corresponding equation for the configuration-averaged concentration:

c(x) = cb(x) +∫

dx′dx′′dR G(x, x′)S (x′, x′′, R)[c(x′′) − c�(R)]

= cb(x) +∫

dx′dx′′dR G(x, x′)S (x′, x′′, R)[cb(x′′) − c�(R)]

+∫

dx′dx′′dx′′′dxivdRdR′G(x, x′)S (x′, x′′, R)G(x′′, x′′′)S (x′′′, xiv, R′)[cb(xiv ) − c�(R′)] + · · · , (B6)

which also serves as a definition for the “self-energy” S , physically representing the interaction of the mean field with itself. Ananalogous equation in momentum space takes the form

˜c(q) = cb(q) +∫

dR G(q)S (q, R)[ ˜c(q′) − (2π )3/2δ(q′)c�(R)]

= cb(q) +∫

dR G(q)S (q, R)[cb(q′) − (2π )3/2δ(q′)c�(R)]

+∫

dRdR′G(q)S (q, R)G(q)S (q, R′)[cb(q′) − (2π )3/2δ(q′)c�(R)] + · · · . (B7)

In order to determine the function S (q, R) we now have to perform the configuration average in terms of the scattering operatorsTi (q, R), and compare the resulting expression with Eq. (B7), using an approach similar to the one implemented by Marquseeand Ross [21].

By introducing the notation f = ∫f (X ,R)p(X ,R)dXdR, the configuration average of Eq. (B4) takes the form

˜c(q) = cb(q) +n∑

i=1

∫-dq′dR G(q)Ti (q − q′, R)[cb(q′) − (2π )3/2δ(q′)c�(R)]

+n∑

i=1

∑j =i

∫-dq′-dq′′dRdR′G(q)Ti (q − q′, R)G(q′)Tj (q′ − q′′, R′)[[cb(q′′) − (2π )3/2δ(q′′)c�(R′)] + · · · . (B8)

Let us define Sj (q, R) as the contribution to S containing the product of exactly a number j of T operators, i.e., S = ∑∞j Sj .

By comparing Eqs. (B7) and (B8) we can determine each order of the self-energy recursively:

S1(q, R) =n∑

i=1

∫-dq′Ti (q − q′, R),

S2(q, R) =∫

-dq′′-dq′′dR′n∑

i=1

∑j =i

Ti (q − q′, R′)G(q′)Tj (q′ − q′′, R) −∫

dR′S1(q, R′)G(q)S1(q, R),

S3(q, R) =∫

-dq′-dq′′-dq′′′dR′dR′′n∑

i=1

∑j =i

∑k =j

Ti (q − q′, R′′)G(q′)Tj (q′ − q′′, R′′)G(q′′)Tk (q′′ − q′′′, R)

−∫

dR′dR′′S1(q, R′′)G(q)S1(q, R′)G(q)S1(q, R)

−∫

dR′[S1(q, R′)G(q)S2(q, R) + S2(q, R′)G(q)S1(q, R)] (B9)

and so on.The general pattern is evident: Sn contains a term of the form T GT GT . . ., with respectively n and n − 1 T and G operators,

minus all the possible unique combinations of lower order S and G terms containing exactly a number n of T operators. Let usassume that the loops are homogeneously distributed in space and that there are no correlations either in the positions or radii ofdifferent loops, allowing us to write

p1(x, R) = n−1f (R), (B10)

where f (R)dR is the number density of dislocation loops with radii in the range (R,R + dR). We also consider thethermodynamic limit of n → ∞, with constant number density. The first contributions to S can then be explicitly expressedas

S1(q, R) = ξ�(R)f (R), S2(q, R) = −ξ�G(q)ξ�(R)f (R), S3(q, R) = ξ�ξ 2�(R)f (R)

∫-dq′′G2(q). (B11)

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APPENDIX C: EWALD SUMMATION OF THE DIFFUSIVE INTERACTIONS IN A THIN FILM GEOMETRY

Consider a periodic system in three dimensions with N clusters with effective charges {Qi}Ni=1 and N “image” clusters witheffective charges {−Qi}Ni=1 per primitive cell. The primitive cell is defined as �(0) = {x = x1a1 + x2a2 + x3a3 : −1/2 < xα <

1/2, α = 1, 2, 3}, with primitive vectors a1 = (L, 0, 0), a2 = (0, L, 0), and a3 = (0, 0, 2H ).Let m = m1a1 + m2a2 + m3a3, mα ∈ Z, and xj = (xj , yj , sgn(zj )H − zj ) and GY (x, x′; ξ�) =

−e−√

ξ�/Dv (4πDv|x − x′|)−1. We want to reformulate the expression

K (xi , xj , ξ�) =′∑

mGY (xi , m + xj ; ξ�) −

∑m

GY (xi , m + xj ; ξ�) i = 1, . . . , N (C1)

as an absolutely convergent series in real and reciprocal space.We first define a “primitive cell effective Green’s function” as follows:

G�Y (x; ξ�) =

∑m

GY (x + m; ξ�), (C2)

which satisfies the equation

(Dv∇2 − ξ�)G�Y (x; ξ�) = δ� (x) =

∑m

δ(x + m). (C3)

We split the primitive cell Green’s function: G�Y = G�

Y,F + G�Y,D , where

(Dv∇2 − ξ�)G�Y,F (x; ξ�) =

∑m

λρβ (x + m),

(Dv∇2 − ξ�)G�Y,D (x; ξ�) =

∑m

[δ(x + m) − λρβ (x + m)],(C4)

and ρβ (x) is a normalized Gaussian distribution:

ρβ (x) =(

β2

π

)3/2

e−β2|x|2 . (C5)

Let us introduce a reciprocal lattice with basis vectors: b1 = (1/L, 0, 0), b2 = (0, 1/L, 0), and b3 = (0, 0, 1/2H ), and thegeneral reciprocal space vector as k = k1b1 + k2b2 + k3b3, kα ∈ Z, α = 1, 2, 3. In particular, G�

Y,F is more convenientlyexpressed in terms of reciprocal space functions. By Fourier transforming ρβ and using the relation

∑m δ(x + m) = ∑

k exp(ik ·x), we have

G�Y,F (x, ξ�) = − λ

V

∑k

exp(ik · x − k2/4β2)

Dvk2 + ξ�

, (C6)

where V is the volume of the primitive cell. On the other hand, we have

G�Y,D (x; ξ�) = G�

Y (x; ξ�) − G�Y,F (x; ξ�) =

∑m

[GY (x + m; ξ�) − λψβ (x + m)], (C7)

where the potential ψβ (x) is given by

ψβ (x) =∫

dx′GY (x, x′, ξ�)ρβ (x′) = −eξ�/4Dvβ2

4πDv

⎧⎨⎩e−

√ξ�/Dv |x|

2|x|

⎡⎣erf

⎛⎝β|x| +

√ξ�/Dv

2β

⎞⎠

+ erf

⎛⎝β|x| −

√ξ�/Dv

2β

⎞⎠⎤⎦ − sinh(

√ξ�/Dv|x|)erfc

⎛⎝β|x| +

√ξ�/Dv

2β

⎞⎠⎫⎬⎭. (C8)

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I. ROVELLI, S. L. DUDAREV, AND A. P. SUTTON PHYSICAL REVIEW E 98, 043002 (2018)

The parameter λ should be chosen in a way that minimizes the long tail of G�Y,D . In particular, as |x| → ∞, ψβ (x) � eξ�

2/4β2−ξ� |x||x| ,

therefore the optimal choice is λ = e−ξ�2/4β2

and G�Y assumes the form

G�Y (x; ξ�) = −

∑k

exp[−(k2 + ξ�/Dv )/4β2]

V (Dvk2 + ξ�)eik·x − 1

8πDv

∑m

1

|x + m|

⎡⎣erfc

⎛⎝β|x + m| +

√ξ�/Dv

2β

⎞⎠e

√ξ�/Dv |x+m|

+ erfc

⎛⎝β|x + m| −

√ξ�/Dv

2β

⎞⎠e−

√ξ�/Dv |x+m|

⎤⎦. (C9)

We now have to remove the spurious self-interactions:′∑

mGY (xi , m + xi ; ξ�) = lim

x→xi

[G�

Y (x, xi ; ξ�) − GY (x, xi ; ξ�)]

= −∑

k

exp[−(k2 + ξ�/Dv )/4β2]

V (Dvk2 + ξ�)

− 1

8πDv

∑m =0

1

|m|

⎡⎣erfc

⎛⎝β|m| +

√ξ�/Dv

2β

⎞⎠e

√ξ�/Dv |m| + erfc

⎛⎝β|m| −

√ξ�/Dv

2β

⎞⎠e−

√ξ�/Dv |m|

⎤⎦

− 1

4πDv

limr→0

1

r

⎡⎣erfc

⎛⎝βr +

√ξ�/Dv

2β

⎞⎠e

√ξ�/Dvr

2+ erfc

⎛⎝βr −

√ξ�/Dv

2β

⎞⎠e−

√ξ�/Dvr

2− e−

√ξ�/Dvr

⎤⎦. (C10)

Expanding to first order with respect to r ,

erfc

⎛⎝βr +

√ξ�/Dv

2β

⎞⎠e

√ξ�/Dvr

2+ erfc

⎛⎝βr −

√ξ�/Dv

2β

⎞⎠e−

√ξ�/Dvr

2− e−

√ξ�/Dvr

≈⎡⎣√

ξ�

Dv

erfc

( √ξ�

2√

Dvβ

)− 2e

− ξ�

4Dvβ2

√π

⎤⎦r + O(r2). (C11)

We can now combine all the terms, yielding to an absolutely convergent series in real and reciprocal space:

K (xi , xj ; ε) = −∑

k

exp[−(k2 + ε2)/4β2]

V Dv (k2 + ε2)eik·xi [e−ik·xj − e−ik·xj ]

− 1

8πDv

{ ′∑m

1

|xi − xj + m|[

erfc

(β|xi − xj + m| + ε

2β

)eε|xi−xj +m|

+ erfc

(β|xi − xj + m| − ε

2β

)e−ε|xi−xj +m|

]

−∑

m

1

|xi − xj + m|[

erfc

(β|xi − xj + m| + ε

2β

)eε|xi−xj +m|

+ erfc

(β|xi − xj + m| − ε

2β

)e−ε|xi−xj +m|

]}+ δij

4πDv

⎡⎣2βe

− ε2

4β2

√π

− ε erfc

(ε

2β

)⎤⎦. (C12)

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