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Acta Materialia 118 (2016) 342e349

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Conservative climb motion of a cluster of self-interstitial atoms towardan edge dislocation in BCC-Fe

T. Okita a, *, S. Hayakawa b, M. Itakura c, M. Aichi a, S. Fujita b, K. Suzuki a

a Research into Artifacts, Center for Engineering, The University of Tokyo, Kashiwa, Tokyo, 2778568, Japanb School of Engineering, The University of Tokyo, Tokyo, 1138568, Japanc Center for Computational Science & e-Systems, Japan Atomic Energy Agency, Kashiwa, 2770871, Japan

a r t i c l e i n f o

Article history:Received 29 April 2016Received in revised form27 July 2016Accepted 1 August 2016

Keywords:Monte Carlo simulationMolecular dynamicsDislocation loopIrradiated metals

* Corresponding author. Research into Artifacts,University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Ch

E-mail addresses: okita@race.u-tokyo.ac.jp (T. Okac.jp (S. Hayakawa), itakura.mitsuhiro@jaea.go.jp (M.ac.jp (M. Aichi), fujita@n.t.u-tokyo.ac.jp (S. Fujit(K. Suzuki).

http://dx.doi.org/10.1016/j.actamat.2016.08.0031359-6454/© 2016 Acta Materialia Inc. Published by

a b s t r a c t

We have developed a model for the conservative climb motion of a hexagonal self-interstitial atom (SIA)cluster that is under the influence of the stress field of a nearby dislocation. The parameters required forthe calculations were derived by molecular simulations of body-centered-cubic Fe. Using these param-eters, kinetic Monte Carlo simulations were conducted to simulate the absorption process of an SIAcluster through the conservative climb to an edge dislocation that has the parallel Burger vector. Thevelocity of the conservative climb was found to be proportional to the inverse square of the distancebetween the cluster and dislocation, which is the same dependency observed in a previous analysis.However, the absolute value of the velocity strongly depends on the number of extra atoms or vacantsites that inherently exist on the edges of the cluster, because they enhance the formation of anotherdefect pair that will induce the conservative climb. The activation energy for the conservative climb ismuch higher than the activation energy for pipe-diffusion of an atom along the edge of the cluster,although the latter is an elementary process of the former. The velocity of the conservative climb isnearly proportional to the inverse eighth power of the cluster size. Both of these findings are significantlydifferent from those of the elasticity analysis.

© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction

One of the characteristic features of structural materials of nu-clear power plants is that highly energetic neutrons produced bynuclear reactions cause damages through collision cascades. Mo-lecular dynamics (MD) simulations have been used to clarify thisprocess; so far, they have shown that clusters of self-interstitialatoms (SIAs), vacancy clusters, mono-SIAs, and mono-vacanciesare formed within a few picoseconds during collision cascades[1]. The number of SIAs in these clusters varies from 2 to 10 ormore,and the maximum size tends to increase with the energy of theprimary knock-on atom (PKA) [2]. These clusters have a form that isbest described as a small SIA dislocation loop [3], and some of themare the crowdion-type, i.e., the Burgers vector is represented by

Center for Engineering, Theiba, 2778568, Japan.ita), hayakawa@race.u-tokyo.Itakura), aichi@race.u-tokyo.a), katsu@race.u-tokyo.ac.jp

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b!¼ a0=2⟨111⟩, where a0 is lattice constant for body-centered cu-bic (BCC) metals and b

!¼ a0=2⟨110⟩ for face-centered cubic metals[3]. Since the migration energy is very low and almost independentof the size, these clusters are highly one-dimensionally mobilealong the direction of their Burgers vector [4]. They are energeti-cally attracted to their stable positions through elastic interactionwith internal stress originating mainly from a dislocation [5].Indeed, it has been experimentally confirmed that small SIA clus-ters are accumulated in high density and trapped near dislocationswith a very small separation as a result of collision cascades [6]. Inthe temperature range where swelling occurs significantly, thesetrapped SIA clusters are most likely absorbed in nearby disloca-tions. For the case of parallel Burgers vectors, in which the inter-action is strongest and the probability of trapping is thus thehighest, the absorption of these clusters requires a motion fromtheir trapped position toward the core of the dislocation, the di-rection of which is different from their original glide direction. Thecritical feature is that the nature of this motion is size-dependent,i.e., the motion is completely different depending on cluster size.Previous studies showed that small clusters containing 2 to 3 SIAs[7], or up to 5 SIAs or less [8] for BCC-Fe, frequently change their

Fig. 1. (a) The periphery and larger hexagonal border of a hexagonal SIA cluster. (b)Formation of a defect pair.

1 Through this process, an SIA-like defect is not formed at the first-nearest seg-ments, but at one of the second-nearest segments. This happens because an SIA-likedefect at the first-nearest segment would have only one adjacent inner atom, whichis not permitted according to the second assumption for the jump process.

T. Okita et al. / Acta Materialia 118 (2016) 342e349 343

glide direction. Hence, the motion of these small clusters towardthe dislocation mainly involves the change in the Burgers vector tothe non-parallel direction. However, Terentyev et al. showed that acluster containing more than 5 SIAs does not change the directionat least within the MD time scale for BCC-Fe [8]. Unless the clustermakes a contact interaction with a dislocation, even the stress fieldoriginating from a dislocation does not induce a change in theBurgers vector in its vicinity [9,10]. More detailed calculations havebeen conducted by Gao and co-workers [11]. They showed thatclusters containing more than 7 SIAs may not change the Burgersvector before they dissociate into small clusters, because thebinding energy per SIA is smaller than the energy per SIA requiredfor the directional change. The change in the Burgers vector wouldbe a rare event, if any, for larger clusters containingmore than 5 to 7SIAs. The probability of the formation of these larger clusters duringa collision cascade becomes higher with increasing PKA energy[12]. Obviously, the impact on swelling is greater with increasingsize of the SIA cluster absorbed by a dislocation. To construct apredictive model for swelling, especially for structural materials ofnuclear reactors that experience high PKA energy, it is thereforenecessary to clarify and quantify the motion of larger SIA clusterstoward the core of dislocations.

A conservative climb is a possible process for this motion. Oftenreferred to as self-climb, it is the climb motion of a cluster duringwhich its number of defects is preserved. There have been severalexperimental studies that confirm this motion of either SIA clustersor vacancy clusters, namely ion-irradiated UO2 by annealing [13],neutron-irradiated Mo by annealing [14], quenched Al by annealing[15], and pure Fe during ion irradiation [16]. Theoretical studieshave also been conducted. Kroupa and Price showed, for the firsttime, the possible mechanism for the conservative climb of a va-cancy cluster toward an edge dislocation [17]. They showed that thegradient in the stress field inside the cluster originating from adislocation causes a transfer of vacancies from one end of thecluster to the other by pipe-diffusion along the core of the cluster.Later, Kroupa and co-workers conducted another calculation for theconservative climb of a smaller cluster to a larger cluster byincluding clusterecluster interactions and clusterevacancy in-teractions [18]. Here, they simplified the calculations by two ap-proximations: (i) the cluster was assumed to be a square shape,which simplified the relationship between the vacancy flux and thevelocity of the conservative climb; (ii) an infinitesimally smallcluster approximation was used, and it was assumed that the forceon the cluster changed linearly from the nearest segment to thefarthest segment. A few years later, Turnbull conducted moredetailed calculations for the conservative climb of two circularclusters formed by ion-irradiated UO2 [13]; however, the effect ofdilatation caused by vacancies was not included because it is muchsmaller in ionically bound UO2 than in metallic materials.

Here, we report the construction of a model that describes theconservative climb motion of an SIA cluster to an edge dislocationby incorporating atomistic behavior. The model is applied to theparallel Burgers vector case in BCC-Fe, where the parametersnecessary for the calculation are obtained either by molecularstatics (MS) or MD simulations. The rest of this paper is organizedas follows. The calculation method for the conservative climb isdescribed in the second section, and the procedures and results forMS/MD simulations are shown in the third section. Finally, thecalculation results for the conservative climb and discussion aredescribed in the fourth section, followed by the conclusions in thelast section.

2. Calculation method

We begin with a hexagonal SIA cluster whose edge contains at

least three atoms (l � 3), which consequently comprises 19 SIAs ormore in total. The periphery of the cluster (denoted by the dottedline in Fig. 1(a)) is divided into segments with the length of oneatomic diameter. Each segment is a possible location for a vacancy-like defect. We assume another hexagonal border lying one layerabove the periphery (hereinafter, we refer to it as the “larger hex-agonal border,” denoted by the dashed line in Fig. 1(a)), which isdivided in the same manner as that of the periphery. Each segmentis a possible location for an SIA-like defect. We assume that avacancy-like defect and an SIA-like defect can exist only on theperiphery and larger hexagonal border, respectively.

An atom on the periphery occasionally jumps to one of thesecond-nearest segments on the larger hexagonal border, throughwhich a pair of vacancy-like and SIA-like defects is generated on theperiphery and larger hexagonal border, respectively (Fig. 1(b)) (seeFootnote1). In some cases, we consider an irregular hexagonalcluster, where some extra atoms are initially placed on the largerhexagonal border, or some atoms on the periphery are initiallyremoved; they are also treated as SIA-like defects or vacancy-likedefects, respectively.

The pipe-diffusion of SIA-like defects begin along the largerhexagonal border with activation energy of Em. In the meantime,vacancy-like defects also diffuse along the periphery. As we shallsee in the next section, the activation energy for pipe-diffusion of avacancy-like defect is essentially equal to that of an SIA-like defect.Hence, we treat the pipe-diffusion of a vacancy-like defect as thejump of an atom that exchanges its position with the vacancy-likedefect in a direction opposite to that for the defect. Thereby, it ispossible to treat the pipe-diffusion of both types of defects as thejump of atoms on the outermost layer.

There are two assumptions for jumps of an atom. The first one isthat the outermost layer has to be either the periphery or the largerhexagonal border (Fig. 2(a)). An atom on the larger hexagonalborder cannot jump to a position on top of another atom on thelarger hexagonal border, because it would create an SIA-like defectabove the larger hexagonal border (Fig. 2(a-2)). A jump of atomslocated inside the periphery is also not permitted, because it wouldcreate a vacancy-like defect on the layer below the periphery(Fig. 2(a-3)). This assumption is set for the sake of simplicity.However, the results will show that owing to this assumption, thereis a limit to the initial number of SIA-like defects (Ndefect > 0) orvacancy-like defects (Ndefect < 0) in our model. The secondassumption is that all the atoms except those at the corners must

Fig. 2. (a-1)e(a-3) Examples of the first assumption and (b-1)e(b-4) examples of the second assumption for the jump process.

T. Okita et al. / Acta Materialia 118 (2016) 342e349344

have contact with two adjacent inner atoms (Fig. 2(b-1)e2(b-3)).The atoms at the corners, which have only one adjacent inner atomand two neighboring atoms in nature, must have contact with theadjacent atom and at least one of the two neighboring atoms(Fig. 2(b-4)). The assumption is necessary because an atom on theoutermost layer that does not match the assumption (blue atoms inFig. 2(b)) would be unstable. Jumps of an atom that would notfollow these two assumptions are not permitted.

The jump of an atom is affected by two factors: the change ininteraction energy Eint of the atom with the stress field of thedislocation through its position dependence (DEint); the change intotal energy Etot caused by the change in the cluster shape (DEtot).When we define DE ¼ DEint þ DEtot, the activation energy formigration is described as Em þ DE/2 [19]. Even the defect formationand its reverse process, namely defect annihilation, can bedescribed by the migration of an atom on the outermost layer thatinvolves a change in energy.

The occurrence of a conservative climb is evaluated after eachmigration event. After tentatively setting the new center at one ofthe neighboring atoms of the original center that are nearer thedislocation, the periphery and larger hexagonal border are drawnwith the new center. If this process reduces the sum of the numberof SIA-like defects and vacancy-like defects, a conservative climb isconsidered to have occurred. Then, we restart the calculations bysetting the new center and modifying the number and positions ofthe defects.

One example is shown in Fig. 3(a) In this case, we setNdefect to be6 (Fig. 3(a-I)). After several migrations in Fig. 3(a-II-1), there are 8SIA-like defects and 2 vacancy-like defects with the original centerof C. When a periphery and a larger hexagonal border with thesame size are drawn with the new center of C0 next to the originalcenter C (Fig. 3(a-II-2)), there will be 7 SIA-like defects and 1vacancy-like defect. Since the sum of the number of the defects isdecreased by setting the new center, we modify the position andthe number as shown in Fig. 3(a-II-2) for the next step in the cal-culations. The climb distance hclimb is then decreased by the pro-jection of the distance between the original and new centers to thehclimb direction (approximately 0.20 nm). A similar situation isshown in Fig. 3(a-III). If setting the new center would not follow theassumption that the outermost layer is either the periphery or thelarger hexagonal border, we do not employ that particular newcenter.

Fig. 3(b) shows the trajectory of the cluster near the dislocation.The initial hclimb is set at ~10 nm, where the interaction energy peratom is equivalent to the thermal energy at 673 K. The cluster startsto move toward the dislocation in a zigzag motion along its stableposition with a combination of glide and conservative climb. At a

certain hclimb, a stable position of the cluster along the glide cyl-inder (hglide) can be derived by calculating the interaction energybetween the cluster and the dislocation. This calculation is con-ducted using the classical elasticity as follows:

hglide ¼ 0:58$hclimb: (1)

When the segment of the larger hexagonal border that is nearestthe dislocation reaches the glide plane of the dislocation, we definethat the cluster is absorbed by the dislocation. Since the glidemotion of the cluster to the stable position occurs immediately, weassume that the time required for the cluster to be absorbed isequal to the total time required for the conservative climb motiononly.

Kinetic Monte Carlo (kMC) simulations are then conducted tosimulate the time evolution of the conservative climb process [20].In this study, defects that flow into the cluster or out to the bulk arenot taken into account, and the number of the atoms in the clusteris preserved. The rate for migration of atoms is given by

v ¼ v0 exp�� Em þ DE=2

kT

�(2)

where n0 is set to 1013 Hz. The parameters that are necessary for thekMC simulations are the activation energy for pipe-diffusion (Em),the position-dependent interaction energy of an atom with adislocation (Eint (r)), and the change in total energy caused by thechange in the cluster shape (DEtot), which will be evaluated in thenext section. In our study, 40 repeated kMC simulations wereperformed at each condition, and the average time required for oneconservative climb motion was obtained.

3. Deriving kMC parameters with MS and MD simulations

All the MS and MD simulations were carried out by using theLarge-scale Atomic/Molecular Massively Parallel Simulator(LAMMPS) code [21] and the interatomic potential for BCC-Fedeveloped by Mendelev and co-workers [22]. A schematic dia-gram of the simulation cell is shown in Fig. 4. The axes of the cellwere set to [e2 1 1], [0 1e1], and [1 1 1], and the length of the cellwas approximately 4.1 nm, 4.4 nm, and 3.6 nm for the x, y, and zdirections, respectively. Periodic boundary conditions wereemployed in all directions. An extra xey plane with a long rect-angular strip measuring approximately 4.1 nm � 2.0 nm wasinserted in the center of the cell to simulate the edge of the cluster.The cell length in the z direction was expanded to eliminate thestress induced by the extra plane. The temperature was set at 0 K,

Fig. 3. (a) Example of changes in the cluster shape during a conservative climb. (b) Trajectory of the cluster.

y [0 1 -1]

z [1 1 1]

x [-2 1 1]

4.1 nm

2.0 nm

4.1 nm

3.6 nm

4.4 nm

Fig. 4. Schematic of the MD and MS simulation cell.

T. Okita et al. / Acta Materialia 118 (2016) 342e349 345

except for the calculation of the activation energy for pipe-diffusion.

Fig. 5. Arrhenius plot of MD results for jump frequencies of an SIA-like defect and avacancy-like defect along the line of the extra plane.

3.1. Activation energy for pipe-diffusion, Em

After either an SIA-like defect or a vacancy-like defect wasplaced along the line of the extra plane, the temperature depen-dence of the jump frequencies was calculated for each type ofdefect. Since their jump was quite a rare event compared to that inthe bulk, we set the temperature to be extremely high at 1100,1200, 1350, and 1500 K. The simulation was conducted with aconstant time step of 1.0 � 10�15 s until more than 400 jumps weredetected.

The values of Em evaluated by the Arrhenius plots are 1.02 eV forboth types of defects, as shown in Fig. 5. It shows that Em of an SIA-like defect along the line is much higher than the activation energyfor bulk diffusion (Emi ¼ 0.36 eV for a dumbbell SIA [23]), possiblybecause the jump along the line is not along the ⟨111⟩ crowdiondirections, which requires several rotations. Based on the fact thatthe activation energy for pipe-diffusion of a vacancy-like defect isequal to that of an SIA-like defect, we treat the pipe-diffusion of

both types of defect as the jump of an atom on the outermost layer.In other words, an SIA-like defect is treated as an atom on the largerhexagonal boarder, which jumps onto one of the nearest segmentson the boarder. It occasionally jumps onto a segment on the pe-riphery if there is no atom on the segment. A vacancy-like defect, onthe contrary, is treated as a location to which the nearest atom onthe periphery or larger hexagonal border can jump. Its jump is nottreated explicitly in the calculation, although vacancy-like defectsare treated in the analysis. Thereby, what we need to quantify isonly the interaction energy of an atom on the outermost layer withthe stress field of a dislocation.

T. Okita et al. / Acta Materialia 118 (2016) 342e349346

3.2. Interaction energy, Eint

The interaction energy (Eint) is estimated by calculating thechange in the formation energy with respect to the local stress.Since the formation energy of an atom on the outermost layer of a[111] cluster is thought to be affected by hydrostatic and/or [111]stress, we apply these two types of strain. The interaction energycan be expressed by Eq. (3):

Eint ¼ VH$sH þ V½111�$s½111�; (3)

where sH and s[111] denote the hydrostatic and [111] uniaxialcomponents of the stress, respectively. Using linear regression, thecoefficients were estimated as VH ¼ 2.15 � 10�30 m3 andV[111] ¼ 1.51 � 10�29 m3, which indicates that the [111] uniaxialstress rather than the hydrostatic stress is the dominant factor ofthe interaction with an atom on the outermost layer of a [111]cluster.

Eint (r) is evaluated by Eq. (3), with the stress value at eachsegment derived according to the isotropic elasticity. The effectiveisotropic shearmodulus and the Poisson's ratio were set at 65.5 GPaand 0.323, respectively. The difference in Eint between segmentsbefore and after the jump of an atom (DEint) is incorporated into theactivation energy for migration.

3.3. Change in total energy caused by change in cluster shape, DEtot

Fig. 6 shows all the possible changes in atomic configurationsthat involve DEtotwith one jump of an atom on the outermost layer.The change in the energy in the absence of a jog is as large as 1.74 eV(Fig. 6(a)), while that in the presence of a jog is comparably lower,varying from 0.30 to 1.03 eV (Fig. 6(bee)) depending on theconfiguration.

The value of DEtot at a corner can be evaluated by utilizing theresults at a jog. It is based on the fact that DEtot is mainly

Fig. 6. Changes in atomic configurations that involve DEtot (a) on the line, (b)e(e) at ajog, and (f, g) at a corner. (a) DEtot ¼ 1.74 eV; (b) DEtot ¼ 1.03 eV; (c) DEtot ¼ 0.91 eV; (d)DEtot ¼ 0.30 eV; (e)DEtot ¼ 0.72 eV; (f) DEtot ¼ 0.30 eV; (g) DEtot ¼ 0.91 eV.

determined by configurations of the first-nearest-neighbor atoms,whereas those of the second-nearest-neighbor atoms affect DEtoton the order of 0.01 eV, which is negligibly small. Hence, DEtot at acorner in Fig. 6(f) or (g) is quantified by the results of Fig. 6(d) or (c),respectively. For a cluster that has no jogs, DEtot of 0.91 eV at acorner is much lower than DEtot of 1.74 eV on the line. It should benoted that an extra atom existing near a corner strongly reducesDEtot from 0.91 to 0.30 eV, promoting formation of another defectpair as a result.

The reverse changes in the configuration shown in Fig. 6 are alsoincorporated. All the values of DEtot associated with the forwardand reverse changes in the configuration were used as input to Eq.(2) to treat migrations of atoms on the outermost layer.

4. Results and discussion

Fig. 7 shows an example of the velocity of the conservative climbat 673 K. The velocity is so low at high values of hclimb that oneconservative climb motion takes as long as 100 s. As the distancedecreases, it becomes faster, and the maximum velocity reaches~0.07 nm/s at hclimb < 2 nm, where one conservative climb motiontakes only several seconds. When the distance further decreases,the velocity suddenly drops because the stress of the dislocation inthis nearby region is not precisely described without cluster rota-tion [5]. Hence, the current model can be applicable when hclimb isequal to or larger than the cluster diameter. The velocity was foundto be inversely proportional to the square of the distance. This is thesame dependency predicted by the Einstein equation, in which thevelocity is determined by the partial derivative of the interactionenergy in terms of the distance.

However, the absolute value of the velocity of the conservativeclimb differs with Ndefect. Fig. 8 shows the dependence of the ve-locity at 973 K on Ndefect for a cluster with l ¼ 4. For a perfecthexagonal cluster (Ndefect ¼ 0), the velocity is very low because oneconservative climb motion does occur after the l SIA-defects andvacancy-like defects are generated and migrate to the edge nearerthe dislocation and the edge on the opposite side, respectively. Itshould be noted that these defect pairs are preferentially formed atthe six corners because of the lower DEtot, and they need to escapefrom the recombination and annihilation in order to contribute tothe occurrence of the conservative climb. The velocity of the con-servative climb is even lower at lower temperature, which isbeyond our computing time. Low velocity was also observed for a

Fig. 7. Velocity of the conservative climb at 673 K as a function of hclimb and (hclimb)�2.

Fig. 8. (a) Velocity of the conservative climb of a cluster with l ¼ 4 at 973 K as a function of Ndefect. (b) Enlarged figure for better visualization of the data at �5 � Ndefect � 3.

Fig. 9. Arrhenius plot showing Dcc.

T. Okita et al. / Acta Materialia 118 (2016) 342e349 347

cluster containing very few Ndefect (Fig. 8(b)). These clusters prob-ably absorb defects from the bulk before their conservative climbmotion. When Ndefect reaches a certain number, the conservativeclimb occurs at sufficiently high velocity.

The dependence of the velocity on Ndefect is not symmetricacross Ndefect ¼ 0; SIA-like defects (Ndefect > 0) enhance conserva-tive climb more significantly than vacancy-like defects (Ndefect < 0)because there are several cases of an SIA-like defect reducing theincrease in total energy, causing formation of another defect pair(Fig. 6(b), (d), and (e)), while a vacancy-like defect prefers to betrapped at the corners with DEtot ¼ �0.72 eV as the reversemigration of Fig. 6(e). However, the increase in the number of SIA-like defects does not monotonically increase the velocity; while thevelocity increases up to Ndefect � 4, it decreases when Ndefectchanges from 5 to 6. Based on these results, we can categorizeNdefect into five groups at any cluster size l:

Group #1, e2(l e 1) � Ndefect � �1: Although the velocity islower than that for Ndefect > 0, it slightly increases with increasingnumber of vacancy-like defects.

Group #2, 0 � Ndefect � l e 2: Owing to the few number of SIA-like defects, the velocity is very low.

Group #3, Ndefect ¼ l e 1, or 2(l e 1): Extra atoms tend to occupyall the segments on certain edges of the larger hexagonal border;likely the edge nearest the dislocation for Ndefect ¼ l e 1, and theones nearest two edges for Ndefect ¼ 2(l e 1). The shape of thecluster will be similar to a hexagon, which suppresses the conser-vative climb. In Fig. 8,Ndefect¼ 6, i.e., 2(le 1), is categorized into thisgroup, where the velocity is lower despite the higher Ndefect.

Group #4, l� Ndefect � 2(l e 1) e 1: Conservative climb occurs atquite a high velocity. Even though all the segments on one edge ofthe larger hexagonal border are occupied with extra atoms, thereare still several extra atoms that enhance formation of anotherdefect pair.

Group #5, Ndefect � 2(l e 1) þ 1 or Ndefect � �2(l e 1) e 1:Although the velocity is quite high, it may be underestimated bythis model. For the case of Ndefect � 2 (l e 1) þ 1, a situation isfrequently observed in which all the segments on two adjacentedges of the larger hexagonal border (neither of which is nearestthe dislocation) are occupied with extra atoms. According to ourmodel, a conservative climb does not occur in this situation.Furthermore, the corner between the two edges of the largerhexagonal border cannot serve as a preferential site for formationof defect pairs because of the first assumption, where the formationof SIA-like defects above the larger hexagonal border is notpermitted. A similar situation is observed for the case ofNdefect � �2(l e 1) e 1. Since the probability of formation of defectpairs may be artificially reduced for these cases, the velocity may be

underestimated; this is the application limit for this model. As it isunlikely that clusters originally categorized in Groups #1 to #3would initiate their conservative climb motion until they absorbsome extra atoms and become categorized in Group #4 or #5, theconservative climb motion experimentally observed is mostly forclusters in those groups. Hence, for the subsequent calculations, wefocused on the data of Group #4 and set Ndefect to 2(l e 1) e 2.

Fig. 9 shows the Arrhenius plot to derive the diffusion coefficient(Dcc) of the conservative climb for clusters with l¼ 4, 5, and 6 basedon the Einstein equation. The value of Dcc is expressed by Eq. (4):

DccðlÞ ¼ Dcc0ðlÞ$exp�� EccðlÞ

kT

�; (4)

where Dcc0 and Ecc denote the pre-exponential factor and theactivation energy for the conservative climb, respectively. The valueof Ecc varies for different cluster sizes: Ecc ¼ 1.67, 1.85, and 1.98 eVfor l ¼ 4, 5, and 6, respectively. It can be seen that Ecc is quite highand increases with the cluster size. In previous studies, Ecc wasanticipated to be equivalent to Em [18,24], which was also thoughtto be smaller than Em

i . However, by analyzing the atomisticbehavior, we clarified that Em i≪ Em≪ Ecc. This result demonstratesthat Ecc is determined not by a single process, but severalcompeting processes and the conservative climb is a very slowmotion even for clusters categorized in the faster groups.

The plot of the velocity of the conservative climb at 873 and973 K as a function of the length of the edge (Fig. 10) shows that the

Fig. 10. Dependence of the velocity of the conservative climb on the length of the edge at (a) 873 K and (b) 973 K.

T. Okita et al. / Acta Materialia 118 (2016) 342e349348

velocity is proportional to the inverse eighth power of the clustersize. At lower temperatures of 673 and 773 K, even though it wasdifficult to calculate the velocity for a larger cluster of l ¼ 7 becausethe calculations would take too long, the same dependency on thecluster size was obtained for clusters with 4 � l � 6. The velocityshows much stronger dependency on the cluster size than a pre-diction based on the dislocation theory [18]. In the previous theory,the velocity was calculated to be proportional to the inverse thirdpower of the cluster size, whichwas derived by two size-dependentfactors: the inverse square dependence of the gradient in theinteraction energy with the stress field of the dislocation, multi-plied by the inverse dependence caused by more defects necessaryfor one conservative climb motion of a larger cluster. In this study,we found another stronger factor: the probability of a defect pair toescape from the recombination. Defect pairs, which form at thecorners in most cases, start to migrate along one of the adjacentedges. Because DEtot < 0, they prefer to make a reverse migration,resulting in recombination and annihilation. In order to escapefrom this strongly attractive process, they essentially requiremigration to the other corner of the edge and further advance to thenext edge. This process becomes much more difficult withincreasing the cluster size. Owing to the significant reduction in thenumber of defects contributing to the occurrence of the conserva-tive climbwith increasing the cluster size, the velocity shows muchstronger dependence on the cluster size. Such a strong dependencycan be obtained for the first time with this atomistic simulation, inwhich the change in total energy caused by the change in thecluster shape is incorporated as well as the change in interactionenergy based on the MD and MS simulations.

5. Conclusions

We have constructed a model based on atomistic behavior thatdescribes the absorption process of an SIA cluster to an edgedislocation by conservative climb. The activation energy for pipe-diffusion of an atom along the edge of the cluster (Em), the inter-action energy of an atom with stress field of dislocation (Eint), andthe change in the total energy (DEtot) caused by the change in thecluster shape through one jump of an atom, all of which arerequired for time-evolution model of the conservative climb, werequantified by MD and MS simulation of BCC-Fe with the inter-atomic potential [21]. A total of 40 repeated kMC simulations wereconducted at each condition that varies the temperature, clustersize, and the initial number of defects existing on the edges of thecluster (Ndefect).

The velocity of the conservative climb was found to be depen-dent on the inverse square of the distance from the dislocation,

which agrees with the Einstein equation. However, even in thesame cluster size, the velocity of the conservative climb stronglydiffers (by more than three orders of magnitude) with Ndefectbecause a defect on the cluster edge, especially an SIA-like defect,significantly enhances the formation of another defect pair. Theactivation energy for the conservative climb is not equal to Em; infact, it is much larger at 1.67e1.98 eV for a cluster with l ¼ 4e6compared to Em¼ 1.02 eV or the activation energy for bulk diffusionof a dumbbell SIA of 0.36 eV [22]. As the cluster size increases, asmaller number of defects escape from recombination via reversemigration of the defect formation process. Hence, the velocity ofthe conservative climb depends on the cluster size much morestrongly than the velocity predicted by dislocation theory.

This model enables simulation of the behavior of a larger SIAcluster that forms by collision cascade as it is absorbed by a dislo-cation without changing its Burgers vector. An SIA cluster with lowNdefect may increase Ndefect by absorbing SIAs from the bulk andstart the conservative climb, which can be detected by some ex-periments. Although even the fastest motion cannot be fully eval-uated byMD simulations because of time constraints, the validity ofthis model can be verified by comparing the results with in situTEM observations.

Acknowledgements

This paper includes results of the program “Development ofInnovative Resource-Renewable Boiling Water Reactor as HighPerformance Transuranium Burner”, which is supported by theMinistry of Education, Culture, Sports, Science and Technology(MEXT) in Japan. We would like to thank Editage (www.editage.jp)for English language editing.

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