Object kinetic Monte Carlo study of sink strengths

Journal of Nuclear Materials 360 (2007) 159–169

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Object kinetic Monte Carlo study of sink strengths

Lorenzo Malerba a, Charlotte S. Becquart b,*, Christophe Domain c

a SCK•CEN, Reactor Materials Research Unit, Boeretang 200, B-2400 Mol, Belgiumb LMPGM, UMR-CNRS 8517, Universite de Lille I, 59655 Villeneuve D’Ascq, France

c EDF-R&D, Department MMC, Les Renardieres, Moret-sur-Loing, France

Received 17 July 2006; accepted 2 October 2006

Abstract

The sink strength for three-dimensionally (3D) versus one-dimensionally (1D), or mixed 1D/3D, migrating defects inirradiated materials has attracted much attention in the recent past, because many experimental observations cannot beinterpreted unless 1D or mixed 1D/3D migration patterns are assumed for self-interstitial atom clusters produced incascades during irradiation. Analytical expressions for the sink strengths for defects migrating in 3D and also in 1D havebeen therefore developed and a ‘master curve’ approach has been proposed to describe the transition from purely 1D topurely 3D defect migration. Object kinetic Monte Carlo (OKMC) methods have subsequently been used to corroborate thetheoretical expressions but, although good agreement was generally found, the ability of this technique to reach the 1Dmigration limit has been questioned, the limited size of the simulation box used in OKMC studies having been mainlyblamed for the inadequacies of the model. In the present work, we explore the capability of OKMC to reproduce the sinkstrengths of spherical absorbers in a wide range of volume fractions, together with the sink strength of grain boundaries,for defects characterised by different migration dimensionality, from fully 3D to pure 1D. We show that this technique isnot only capable of reproducing the theoretical expressions for the sink strengths in the whole range of conditionsexplored, but is also sensitive enough to reveal the necessity of correcting the theoretical expressions for large sink volumefractions. We thereby demonstrate that, in spite of the limited size of the OKMC simulation box, the method is suitable todescribe the microstructure evolution of irradiated materials for any defect migration pattern, including fully 1D migratingdefects, as well as to allow for the effect of extended microstructural features, much larger than the simulation box, such asgrain boundaries.� 2006 Elsevier B.V. All rights reserved.

PACS: 82.20.Wt; 81.05.Bx

1. Introduction

The kinetic Monte Carlo (KMC) method pro-vides solutions to the master equations of a physical

0022-3115/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.jnucmat.2006.10.002

* Corresponding author. Tel./fax: +33 3 20 43 49 44.E-mail address: [email protected] (C.S. Bec-

quart).

system whose evolution is governed by a known setof transition rates between possible states, by choos-ing randomly among various possible transitionsand accepting them on the basis of appropriateprobabilities [1]. When applied to study the evolu-tion of systems of mobile species, such as atoms(atomistic KMC, AKMC) [2–4] or defects formedunder irradiation (object KMC, OKMC) [5–7] it

.

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160 L. Malerba et al. / Journal of Nuclear Materials 360 (2007) 159–169

has the advantage of going beyond the mean-fieldapproximation, by explicitly and spontaneouslytaking into account spatial correlations betweenthe elements of the physical system. As such,KMC methods are expected to implicitly reproduce,among other phenomena, the effect of sinks or trapsfor migrating species, characterised by a givengeometry and spatial distribution, i.e. these methodsare expected to provide spontaneously the sinkstrengths typically used in the rate theory todescribe the interaction of migrating defects withthe features characterising the microstructure ofthe material (e.g. voids, dislocations, grain bound-aries, . . .) [8–11].

The sink strength of each type of microstructuralfeature, k2, is proportional to the square of theinverse of the mean distance covered by the migrat-ing species before interacting with it, in general bybeing absorbed or trapped. The sink strength is apriori a function not only of the type, shape, orienta-tion, size and concentration of the sinks, as well as,in principle, of the features of the actual interaction,but also, and sometimes crucially, of the dimension-ality of the motion of the affected migrating species.In particular, the sink strength for three-dimension-ally (3D) versus one-dimensionally (1D), or mixed1D/3D migrating defects has attracted a lot of atten-tion in the past decade, as a consequence of the factthat a number of experimental observations con-cerning irradiated materials under cascade damageconditions cannot be interpreted unless 1D or mixed1D/3D migration patterns are assumed for self-interstitial atom clusters [12–14], which are by nowwell known to be produced directly in displacementcascades [15–17]. These types of migration patternshave been amply confirmed by a number of mole-cular dynamics simulation studies in a-Fe and Cu[18,19]. Thus, analytical expressions for the sinkstrengths for defects migrating not only in 3D [11],but also in 1D [20], have been developed for use inrate equation models of microstructure evolutionunder irradiation [13,14]. In addition, a ‘mastercurve’ approach has been proposed to describe thetransition of the sink strength from purely 1D topurely 3D defect migration, as a function of thefrequency of change of 1D motion direction [21].OKMC methods have subsequently been used tocorroborate the theoretical expressions [22–24].Although in general good agreement has been foundbetween analytical theory and simulation, in thecited work it has been pointed out that the OKMCmethods are only of limited applicability to verify

sink strength values. More specifically, the abilityof this technique to reach the sink strength 1Dmigration limit has been questioned [25,24] and alsoapparent discrepancies between theory and simula-tion in the case of 3D migrating defects haveremained unexplained [22]. The necessarily limitedsize of the simulation box used in OKMC studieshas been mainly blamed for these inadequacies.

In the present work, we explore the capability ofthe OKMC technique to reproduce the sinkstrengths of spherical, unsaturable absorbers in alarge range of volume fractions, as well as the sinkstrength of grain boundaries, for defects characte-rised by a varying motion dimensionality, fromthe fully 3D limit to the pure 1D limit. We show thatthis technique is not only capable of reproducing thetheoretical expressions for the sink strengths in thewhole range of conditions explored, but is alsosensitive enough to reveal the necessity of appropri-ately correcting the theoretical expressions for largesink volume fractions. We thereby demonstratethat, in spite of the limited size of the OKMCsimulation box, the method is indeed suitable tosolve the equations governing the microstructureevolution of irradiated materials for any defectmigration pattern, including fully 1D migratingdefects, as well as to allow for the effect of extendedmicrostructural features, much larger than the simu-lation box, such as grain boundaries. In Section 2 thecomputational method is described in detail. In Sec-tion 3 our results are presented, distinguishingbetween the cases of spherical absorbers for 3D,1D and mixed 1D/3D migrating defects and grainboundaries. In Section 4 the capability of the OKMCtechnique to reliably evaluate sink strengths is dis-cussed in the light of present and previous work.The main conclusions are summarised in Section 5.

2. Computational method

The general features of the OKMC code used inthe present work, LAKIMOCA, have been exten-sively described in a previous publication [7].Briefly, the model treats radiation produced defects(vacancies, self-interstitials atoms – SIA – and clus-ters thereof) as objects with specific positions in asimulation box and with associated reaction vol-umes. Each object can migrate and participate in aseries of predefined reactions. The probabilities forphysical transition mechanisms, which are basicallymigration jumps and emission from larger defects orfrom traps, are calculated in terms of Arrhenius

L. Malerba et al. / Journal of Nuclear Materials 360 (2007) 159–169 161

frequencies for thermally activated events, Ci ¼mi exp Ea;i

kBT

� �, where mi is the attempt frequency

(prefactor) for event i, Ea,i is the corresponding acti-vation energy, kB is Boltzmann’s constant and T isthe absolute temperature. The Monte Carlo algo-rithm [26] is used to select at each step the event thatis going to take place, based on the correspondingprobabilities, by extracting random numbers. Aftera certain event is chosen, time is increased according

to the residence time algorithm, Ds ¼ 1=PN th

ei¼1CiþPN ext

ej¼1 P j [27], where Pj are the probabilities of exter-

nal events, such as the appearance of a cascade or ofisolated Frenkel pairs produced by impinging parti-cles. The choice of this expression is in the long termequivalent to choosing Ds 0 = �lnR Æ Ds, where R isa random number between 0 and 1 [28]. In addition,the model includes non-thermally activated events,such as the annihilation of a defect after encounter-ing either a defect of opposite nature (i.e. a SIAencountering a vacancy) or a sink, as well aggrega-tion, either by adding a point-defect to a cluster orby forming a complex between a defect and a trapfor it. These events occur only on the basis of geo-metrical considerations (overlap of reaction vol-umes) and do not participate in defining theprogressing of time. The possibility of introducingdifferent classes of immobile traps and sinks, char-acterised by specific geometrical shapes (spheres,infinite cylinders, surfaces, . . .) and suitable to mimicvoids or other trapping nano-features, as well asdislocations and grain boundaries, is also imple-mented. The code is therefore equipped to mimicrealistic microstructures and irradiation conditions.

In the present work, however, the model is usedto explore only idealised situations, as was doneby Heinisch and co-workers [22,24], where onlyone migrating defect at a time is present in the sim-ulation box, in a microstructure defined by only oneor at the most two classes of sinks of precise geo-metry. The trajectory of the defect is followed untilit is absorbed by a sink and at that point a newdefect of the same type is introduced in the simula-tion box. The sink strength is obtained in this wayas

k2 ¼ 2n

d2j hnji

; ð1Þ

where hnji is the average number of jumps per-formed by the defects, introduced one by one inthe box, before being annihilated at the sink; n isthe dimensionality of the motion and dj is the jump

distance. Since the defects are assumed to migrate ina body centered cubic (bcc) lattice, dj =

p3/2a0,

where a0 is the lattice parameter (in practice, thevalue for a-Fe has been used, i.e. a0 = 0.287 nm).Note that, following the rate theory, the choice ofn is not necessarily related to the actual dimension-ality of the motion of the concerned defect, butrather to the choice of using, in the rate equations,the 3D or the 1D diffusion coefficient (D3 or D1)in the term reproducing the rate of annihilation atsinks, Dck2, where c is the defect concentration[12,14,20]. Since in the present work the whole rangeof motion dimensionality, from 3D to 1D, isexplored, in order to highlight the transition wemake the consistent choice of using in all cases the3D diffusion coefficient, i.e. throughout the papern = 3 (unless otherwise stated). The histories of atleast 1000 defects have been tracked for each condi-tion in order to obtain the average hnji that appearsin Eq. (1) and in most cases (all cases concerning 3Dmigrating defects and many concerning 1D migrat-ing defects) the number was as high as 10000, oreven more. In addition, it was always verified thatthe value had actually converged, i.e. that the aver-age did not change significantly by increasing thenumber of followed defect histories. If this did nothappen, the simulation was rejected and repeatedwith a larger number of sampled defects.

The dimensionality of the motion of the simu-lated defects has been defined in two ways. Oneway consisted in deciding that the defect mustchange direction of motion after a fixed number ofjumps n0

j , i.e. after having travelled a mean lengthlch ¼ dj

pn0

j . If n0j ¼ 1 the migration is fully 3D;

the larger n0j , the closer the path becomes to being

1D. This is roughly the same scheme as in [22,24].The other way consists in assigning an energy ofrotation, Er, whereby the probability of changingdirection of motion is expressed as exp(Er/kBT).With this definition, the change of direction issimply another possible stochastic event managedby the Monte Carlo algorithm. Thus, the numberof jumps executed by the defect before changingdirection is not always the same, but in average itleads to a characteristic average segment length,lch = dj

pexp(Er/kBT). Er = 0 provides a fully 3D

path, while at the chosen simulation temperatureof 573 K, Er = 1 eV is enough to have a fully 1Dpath. Intermediate values (0.3, 0.35, 0.4, 0.45, 0.6,0.7 and 0.8 eV were considered) correspond tomixed 1D/3D migration. Between each re-orienta-tion, the defect moves along a h111i direction.


Non-cubic boxes with periodic boundary conditions(PBC) have been used for all calculations. The useof non-cubic boxes was found instrumental in orderto simulate correctly the 1D migrating defects, asdiscussed in Section 4.

In the case of the spherical absorbers, boxes of300 · 350 · 400 lattice parameters, equivalent to avolume of about 106 nm3 if the lattice parameterof a-Fe is adopted, were used. The absorbers wererandomly distributed in the simulation box, but carewas taken to avoid overlap, in order to respect theassumptions made concerning their geometry, sizeand density. Their radius, R, was varied from 0.75to 10.2 nm and their number density, N, from 1016

to 1.5 · 1017 cm�3, thereby spanning volume frac-tions, fV, from 1.8 · 10�5 to 4.1 · 10�1. The defectswere introduced one by one in randomly selectedpositions within the simulation volume and thecases corresponding to defects created inside theabsorbers were automatically excluded from the cal-culation of the average in Eq. (1).

The effect of the presence of spherical grainboundaries was introduced by assigning to eachdefect two positions: the relative position insidethe simulation box, s, and the absolute positioninside the grain, S. Two close-by defects in thesimulation box will have similar relative positions,s1 and s2, but may have a priori completely differentabsolute positions in the grain, S1 and S2. Each timethe defect crosses the box boundaries and PBC areapplied to its relative position, S is corrected in sucha way that the displacement of the defect to theimage box beside is accounted for. If S gets to lieon the surface of the spherical grain, the defectdisappears. In this scheme, the box size is totallydecoupled from the grain size and the effect of thepresence of large (spherical) grains can be allowedfor, even using small simulation boxes. In the pres-ent case, boxes of 80 · 120 · 150 lattice parameters(3.4 · 103 nm3) have been used and the grain sizehas been increased from 30 nm (the radius of thesphere enclosing the simulation box) to 1 lm andbeyond, in some cases also with the addition ofspherical absorbers in it.

3. Results

3.1. Spherical absorbers

According to theory, the sink strength of unsa-turable spherical absorbers of radius R and densityN for 3D migrating defects, k2

3;s:a:, can be expressed

using the following, recursive expression (in defectof a bias factor) [8,11]:

k23;s:a: ¼ 4pNR 1þ R

ffiffiffiffiffiffiffiffiffiffik2

3;s:a:

q� �: ð2Þ

This expression is, however, customarily truncatedat the first order, k2

3;s:a: ¼ 4pNR. In the case of 1Dmigrating defects, the expression for the sinkstrength, k2

1;s:a:, is exact and is given by [20]

k21;s:a: ¼ 6ðpR2NÞ2: ð3Þ

(Always with the convention of using the 3D diffu-sion coefficient, D3, in the equations, which explainsthe factor 6 = 3 · 2 in front of the parenthesis.)Given a defect that migrates following a path notcoinciding with either the 3D or the 1D limit, thecorresponding sink strength of spherical absorbers,k2

1–3;s:a:, can be expressed as a function of the twolimiting case sink strengths and of the typical lengthfor change of direction, lch [21]. After defining twodimensionless variables:

x2 ¼l2

chk21;s:a:

12þ

k41;s:a:

k43;s:a:

ð4aÞ

and

y ¼k2

1–3;s:a:

k21;s:a:

; ð4bÞ

the master curve relating the general case to thelimiting cases is

y ¼ 1

21þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4

x2

� �s" #: ð4cÞ

These expressions will be used to benchmark theresults of the simulation presented in the followingsubsections. Throughout the section, the resultsfor the sink strengths will be represented in thefigures as functions of the absorbers’ volume frac-tion, fV.

3.1.1. 3D migrating defectsFig. 1 shows the cloud of simulation data points

in two different ways. In the upper panel the dataare grouped by absorber radii: points denoted bythe same symbol correspond to the same absorberradius for growing absorber densities (1.0, 3.0, 6.0,9.0, 12.0 and 15.0 · 1016 cm�3, the two highest den-sities were not considered in the case of the largestradius, though). In the lower panel, the simulationdata are compared to the values obtained using

10-5 10-4 10-3 10-2 10-1

10-4

10-3

10-2

10-1

Sin

k st

reng

th (

nm-2)

Absorber volume fraction

Data grouped by absorber radii 0.75 nm 1.25 nm 2.25 nm 4.25 nm 6.25 nm 8.25 nm 10.25 nm

10-5 10-4 10-3 10-2 10-1

10-4

10-3

10-2

10-1

Sin

k st

reng

th (

nm-2)


Simulation1st order approximation2nd order approximation

Fig. 1. Cloud of simulation data points for the sink strength of3D migrating defects versus spherical absorber volume fraction.Above: grouped by absorber radii (the lines are simply guides forthe eyes). Below: comparison with analytical expression (Eq. (2))to first and second order.

10-4 10-3 10-2 10-1

-30

-20

-10

0

10

20

30

40

50

Err

or (

%)


Fig. 2. Percentage error committed by using the simulation for3D migrating defects results instead of the theoretical expressiongiven in Eq. (2), in its second order approximation, versusspherical absorber volume fraction.


the first order approximation expression for the sinkstrength from Eq. (2), k2

3;s:a: ¼ 4pNR, denoted bysmall black dots, and also the second order approx-imation, k2

3;s:a: ¼ 4pNRð1þ Rp

4pNRÞ, indicated bycrosses. Despite a small scatter on the data ofFig. 1, due to statistical fluctuations, it appearsclearly that, in order for an agreement to be foundbetween simulation and theory for volume fractionsabove 10�3 it becomes necessary to use the secondorder approximation of the formula, as the firstorder approximation largely underestimates theactual sink strength. Above fV = 10�1 even the sec-ond order approximation becomes insufficient andmore terms should probably be recursively added.Fig. 2 suggests in a fairly clear way that this isindeed the case: the relative error, defined as per-centage ratio of the difference between simulationand theory versus theory, using the second order

approximation as reference value, remains low inabsolute value and negative in sign for low fV, startsgrowing in absolute value above fV = 10�2 andabove fV = 10�1 it even changes sign, becomes posi-tive and grows very rapidly. The latter is clearly theconsequence of the need for an additional correc-tion in order to obtain agreement between theoryand simulation data.

3.1.2. 1D migrating defects

Fig. 3 is the equivalent of Fig. 1 for 1D migratingdefects. In the upper panel the simulation datapoints grouped by absorber radii are shown; in thelower one the same points are compared with theresult of using Eq. (3). The equation is used toextend the data points to the whole range of volumefractions that was explored in the 3D case. As amatter of fact, it is impossible to produce statisti-cally reliable results for 1D migrating defects whenonly very few small absorbers are present, forcomputing time reasons. Even after more than1012 Monte Carlo time steps (corresponding tomonths of calculations on the used cluster ofPCs), for volume fractions less than 10�3 the num-ber of followed defect histories remained below1000 and the sink strength value could be hardlysaid to have converged.

This is a direct consequence of the quadraticdependence of k2

1;s:a: on the NR2 product (Eq. (3)),to be compared with the 1st approximation lineardependence on NR in the 3D case (Eq. (1)), whichmakes the probability for the defect to encounterabsorbers along its 1D path negligibly small for

10-3 10-2 10-110-6

10-5

10-4

10-3

10-2

10-1

Sin

k st

reng

th (

nm-2

)


Data grouped by absorber radii 2.25 nm 4.25 nm 6.25 nm 8.25 nm 10.25 nm

10-5 10-4 10-3 10-2 10-110-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Sin

k st

reng

th (

nm-2

)


Simulation Eq. (2)

Fig. 3. Cloud of simulation data points for the sink strength of1D migrating defects versus spherical absorber volume fraction.Above: grouped by absorber radii (the lines are simply guides forthe eyes). Below: comparison with analytical expression (Eq. (3)).The theoretical points are extended to the same range of volumefractions as studied in the 3D case.

10-2 10-1-1

0

1

2

3

4

Cor

rect

ion

fact

or


Fig. 4. Correction factor to be applied to Eq. (3) in order for it tocoincide with the simulation results for 1D migrating defects,versus spherical absorber volume fraction.


small values of NR, compared to a 3D migratingdefect, as is also intuitive to imagine. By comparingthe lower panel of Fig. 3 with the correspondingpanel in Fig. 1 it can be seen that, for the same smallvolume fraction (�10�5), the absorber sinkstrengths for 3D migrating defects remains closeto 10�4, while it drops to 10�9 in the 1D case! Theunlikelihood of encountering sinks along their pathis, on the other hand, exactly the distinctive featureof 1D migrating defects, which defines their specific-ity and allows, trough the postulation of their exis-tence, a number of experimental facts concerningmicrostructure evolution under irradiation to beunderstood [12,14].

This is also the reason why in general, and notonly for small volume fractions, it is more difficult

to obtain, by simulation of 1D migrating defects,sink strength results that converge to an averagewithin a narrow enough range of fluctuations. Thisexplains why the simulation data points in Fig. 3 aremore scattered and less precise than in the 3D case.Fig. 4 shows the ‘correction’ factor that should beapplied to the theoretical values from Eq. (3) inorder to provide the same value as the simulation.This factor lies between 1 and 1.5 in most cases,i.e. the simulation tends to overestimate the sinkstrength value. This is the consequence of the factthat, unless an unachievably large amount of defecthistories is followed, the statistics is always biasedtowards higher sink strengths, since in the limitedtime available to the simulation, if convergence isreached, histories that ended with an early annihila-tion in a sink will remain more likely to be sampledthan histories leading to late annihilation; so theaverage will tend to be larger rather than smallerthan the theoretical value. Nonetheless, the overallagreement is satisfactory, except for fV > 10�1,where an increase of this correction factor is regis-tered. Since, however, the increase of the volumefraction is expected to improve, rather than worsen,the statistical significance of the sampling done inthe simulation, the rapid increase of the correctionfactor for large volume fractions cannot be ascribedto lack of convergence. Two possible explanationscan be put forward. One is that, similarly to the3D case, for large volume fractions Eq. (3) ceasesto be valid and corrections to it should be added.The other is that the use of periodic boundaryconditions, in the case of many and large absorbers,


introduces a certain degree of order in the simula-tion, which determines a departure from the theo-retical expression, which is valid for randomlydistributed sinks.

3.1.3. Transition between 3D and 1D regime

In Fig. 5, the simulated sink strength of differentdensities of spherical absorbers for defects thatchange h111i direction after a fixed amount ofjumps, n0

j , is plotted versus lch, which is made tovary between 1 and 100 000 nm. The absorber radiuswas 6.25 nm in all cases. In the same figure also thecorresponding theoretical and simulation points forthe purely 1D case are shown for comparison. Thisfigure, which exemplifies the transition from 3D to1D regime, should be compared with Fig. 2 in[24]. It can be seen that, differently from the figurein the cited work, in the present case the 1D limitwas reached to very good approximation forlch = 10 lm and larger.

The same type of information, but in the mastercurve representation [21], is given in Fig. 6, wherenot only the points obtained from the simulationsat fixed number of jumps used in Fig. 5 are included,but also those coming from simulations where thechange of h111i direction was decided based onan energy of re-orientation, Er. In the latter case,groups of points characterised by the same Er (andtherefore by the same lch) correspond to differentdensities and radii, in the range specified above for1D simulations. It is noteworthy that all points fulfil

0.000001

0.00001

0.0001

0.001

0.01

0.1

0.1 1.0 10.0 1

1D length before chan

Sin

k st

ren

gth

(n

m-2

)

N=1N=3N=6N=9N=12N=151D limit simulation1D limit formula

1016 cm-3

Fig. 5. Transition from 3D to 1D sink strength for different spherical acomposing the 1D/3D migration path, lch ¼ dj

pn0

j , where dj is the judirection. The corresponding 1D limit values from the simulation and

the condition embodied by the master curve, withvery little scatter only in the region of the transition(change of slope of the curve) and then, as expected,in the 1D region (where the curve remains constant).This indirectly suggests that it is irrelevant, in prac-tice, how the change of direction is imposed.

3.2. Grain boundaries

According to theory, the sink strength of a sphe-rical grain boundary of radius Rg for 3D migratingdefects, k2

3;g:b:, is [10]

k23;g:b: ¼ 14:4=R2

g: ð5Þ

In the case of 1D migrating defects, the expressionfor the sink strength, k2

1;g:b:, is also very similar [29]:

k21;g:b: ¼ 15=R2

g: ð6Þ

(With the convention of using the 3D diffusioncoefficient, D3, in the equations.) If other sinks arepresent in the bulk, with sink strength k2

s:b:, the sinkstrength of the grain boundary will be affected andcan be expressed, through the dimensionlessvariables

a ¼ ks:b:Rg ð7aÞ

and

c ¼ k2g:b:R

2g; ð7bÞ

as

00.0 1000.0 10000.0 100000.0

ge of direction, l ch (nm)

bsorber densities, as a function of the length of the 1D segmentsmp distance and n0

j the fixed number of jumps before changingfrom Eq. (3) are also indicated.

1E-3 0.01 0.1 1 10 100 1000

1

10

100

1000

y

x

Master curve By n

j

0 ByEr (eV)

1 0.3 10 0.35 100 0.4 1000 0.45 104 0.5 105 0.6 106 0.7 107 0.8 108

109

1010

Fig. 6. Master curve representation of the transition from 3D to 1D regime in the case of the sink strength of spherical absorbers. Thecurve corresponds to Eq. (4c). The points are the result of obtaining x and y by applying Eqs. (4a) and (4b), using sink strength values forthe 3D and 1D limit, as well as for the mixed 1D/3D case, taken from the simulation. The data points are grouped according to thecriterion used to decide the change of direction, fixed number of jumps, n0

j , or energy of rotation, Er (see text). Each group of points spansa range of absorber radii and densities.


c¼ 3

23þ4b�6b b�1ð Þ

a2þ 3þ4b�6bðb�1Þ

a2

� �2(*

�4ðb2�a2Þ 2�3 b�1ð Þa2

� �2)1=2+

2�3ðb�1Þa2

� �2,

;

ð7cÞ

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

0.0 1.0 2.0 3.0

R

Sin

k st

ren

gth

(n

m-2

)

Sim-3D

Sim-1D

Fig. 7. Sink strength of a spherical grain boundary as a function of its rtheoretical expressions, dots correspond to simulation results. The 1Dotherwise the two curves and dot series would have appeared superpos

where b = acotha. This is a kind of ‘master curve’for the grain boundary sink strength, explicitlyderived in the case of 3D migrating defect (we arenot aware of any specific derivation of this curvefor 1D migrating defects). These expressions will

4.0 5.0 6.0 7.0

g (μm)

Theory 3D

Theory 1D

adius for both 3D and 1D migrating defects. Lines corresponds todata have been divided by three for better legibility of the figure,ed.

14

19

24

29

34

0 2 4 6 8α

γ

Theory

lower density

intermediate density

higher density

Fig. 8. Relationship between the sink strength of a spherical grain boundary and the sink strength of bulk sinks in terms of thedimensionless variables and master curve given in Eqs. (7). Three different spherical absorber densities (2.94, 5.88 and 8.81 · 1017 cm�3)and a range of grain boundary radii (30 nm to 1 lm) have been considered.


be used to benchmark the results of the simulationpresented in what follows.

Fig. 7 shows the sink strength of a spherical grainboundary as a function of its radius. The curvescorrespond to Eqs. (5) and (6), while the dots arethe results of the OKMC simulation. For betterlegibility, in this figure the curve and dots for the1D limit have been divided by three (thereby imply-ing the use of a 1D diffusion coefficient, D1, in thecorresponding rate equations). The agreementbetween simulation and theory is self-evident.

The effect of the simultaneous presence of bulksinks and grain boundaries has been explored byconsidering three densities of 3 nm radius absorbers,namely 2.94, 5.88 and 8.81 · 1017 cm�3. 3D migrat-ing defects have been introduced one by one and thegrain boundary radius has been made to varybetween 30 nm and 1 lm. The simulation resultsare plotted in Fig. 8 in terms of the dimensionlessvariables defined in (7a) and (7b) and comparedwith the ‘master curve’ given by Eq. (7c). Althoughthe simulation results systematically lie slightlybelow the theoretical curve, the agreement is never-theless remarkable.

4. Discussion

The results we have presented lead to a series ofimportant statements. First, the OKMC technique,in spite of its inherently stochastic and discretenature, can trustfully describe the strength of sinks

of given geometry, size and density in a large rangeof sink volume fractions, in full agreement with the-oretical expressions obtained in the framework of amean-field, continuum approach. Second, in spite ofthe finiteness and relatively small size of the simula-tion box, this technique is capable of treating in areasonably correct way also 1D migrating defects,whose path is orders of magnitude longer than thetypical size of the box (provided that a non-cubicbox is used, as will be shortly discussed). Third, alsoextended sinks much larger than the size of theOKMC simulation box can be correctly describedby this technique. None of these statements couldbe obviously deduced from the existing literatureon the subject.

In [22], the sink strength of spherical absorbers ofgrowing radius for 3D migrating defects was foundto diverge from the linear behaviour providedby the first order approximation of Eq. (2), andthe authors fitted their results to a third degree poly-nomial. However no clear reason was provided forthis discrepancy. We have shown that both our sim-ulation results and those in [22] are consistent withthe theoretical expression, provided that the latteris extended to further orders of approximation.

In [25], the conclusion of a kinetic Monte Carlostudy of damage accumulation in metals under cas-cade irradiation conditions was that the applicabilityof such a technique was limited to cases where 1Ddiffusing defects have no consequential role on themicrostructure evolution. Such a conclusion was

A B

AB

Fig. 9. Pictorial explanation (in 2D) of why the use of non-cubicboxes with PBC allows the simulation of 1D migration paths of apriori any length, while cubic boxes with PBC lead to unrealisticsituations (the grey one is the simulation box, the others are theimage boxes according to PBC). Above, path A is doomed to seethe defect migrating along it to be immediately absorbed by thesink located on it, while along path B the defect can migrateindefinitely without ever encountering a sink. Below, because themigration direction remains unchanged, by stretching the samebox to a non-cubic shape it becomes possible for the defect tomigrate through many image boxes before being absorbed by asink.


reached on the basis of the fact that the application ofPBC seemed to produce a situation where the lengthof the path of 1D diffusing defects was not limited bythe distance between sinks, but rather by events suchas recombinations, which are the more frequent, thehigher the dose; different results were obtained whenapplying a mean-field approach. On the other hand,the application of pseudo-periodic boundary condi-tions was shown to inherently destroy the one-dimensionality of the motion and was thereforerejected as a solution. In the course of the presentwork we have clearly observed that, if a cubic boxis used, only two situations can be encountered inthe case of 1D migrating defects with PBC: eitherno sink is located along their path, so that the defectcan only indefinitely cross the simulation box with-out ever being absorbed; or a sink is indeed locatedalong the path of the defect, but in this case the lengthtravelled by the migrating object can only be of theorder of the size of the box. In these conditions, thecorrect simulation of the absorption of 1D migratingdefects is indeed unfeasible. If, on the other hand,non-cubic (i.e. parallelepipedic) boxes are used, thenthe application of PBC naturally provides the possi-bility for the defect to travel long distances withoutbeing absorbed in a sink and, conversely, of findingat some point a sink that can absorb it, even withoutchanging direction of motion. This is illustrated pic-torially in Fig. 9 and essentially happens because notonly these defects move in 1D, but the possible direc-tion of motion, h111i, is dictated by the crystallogra-phy and does not change if the box shape changes.Non-cubic boxes allow the defect to explore mostof the box, by repeatedly applying PBCs, therebychanging the local landscape seen by the defect,although of course in the long run a certain degreeof periodicity will appear. More generally, the keypoint is to have boxes that change dimension indirections that are not parallel to the fixed 1D motiondirection. In [25], cubic boxes were used and webelieve that this may have been the origin of thediscrepancies from the mean-field approach foundin that work. It remains nonetheless true that theOKMC simulation of 1D migrating defects requires,in order to be reliable, very long computing time, soas to produce a statistically representative sample ofdefect histories. To this regard, a point that remain tobe clarified is the growing discrepancy betweentheory and simulation in the case of 1D migratingdefects for large sink volume fractions, which maybe due to the fast establishment of a certain periodic-ity and therefore order or simply to the inadequacy

of the theoretical expression in that range of volumefractions.

Finally, in the present work we have shown thatby assigning parallel coordinates to defects, i.e.


absolute coordinates inside a grain and relativecoordinates inside the simulation box, the effect ofextended defects such as grain boundaries can beallowed for even with a relatively small simulationbox. The same approach can in principle be appliedfor other types of extended defects, a priori alsowith non-spherical shapes.

5. Conclusions

We have demonstrated that the OKMC tech-nique naturally lends itself for the simulation ofprocesses leading to radiation produced defectabsorption at sinks, in the whole range of defectmigration patterns, from fully 3D to purely 1D,and including the case of extended sinks muchlarger than the simulation box itself. In the presentwork we have dealt with sinks characterised by awell defined geometry and for which analyticalexpressions exist, so as to be able to prove the capa-bility of the simulation model to provide correctresults by comparison with those expressions. It ishowever believed that this simulation technique issuitable to provide correct results for, a priori, anysink type, shape and orientation, as well as in a largerange of sink volume fractions, thereby goingbeyond the possibilities of mean-field theoreticalapproaches. The main exception is given by thosecases where the sink volume fraction is very small,in which the production of statistically meaningfulsamples of defect histories may be difficult toachieve. In those cases only analytical expressionsbecome applicable in practice.

Acknowledgements

This work was performed in the framework ofthe Perfect IP, partially funded by the EuropeanCommission within the 6th Framework Programme(contract no. FI6O-CT-2003-508840). The assis-tance of H. Trinkaus with the intricacies of therate theory is gratefully acknowledged. The threeauthors wish also to thank S. Golubov for veryuseful discussions.

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Object kinetic Monte Carlo study of sink strengths

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