Journal of Nuclear Materials 425 (2012) 41–47

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Journal of Nuclear Materials

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Derivation of kinetic coefficients by atomistic methods for studying defectbehavior in Mo

Z. Insepov a,⇑, J. Rest a, A.M. Yacout a, A.Yu. Kuksin b, G.E. Norman b, V.V. Stegailov b, S.V. Starikov b,A.V. Yanilkin b

a Argonne National Laboratory, Argonne, IL, USAb Joint Institute for High Temperatures, Moscow, Russia

a r t i c l e i n f o

Article history:Available online 24 August 2011

0022-3115/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jnucmat.2011.08.019

⇑ Corresponding author.E-mail address: insepov@anl.gov (Z. Insepov).

a b s t r a c t

A multiscale concept for irradiated materials simulation is formulated based on coupling moleculardynamics simulations (MD) where the potential was obtained from ab initio data of energies of the basicdefect structures, with kinetic mesoscale models. The evolution of a system containing self-interstitialatoms (SIAs) and vacancies in crystalline molybdenum is investigated by means of MD. The kinetics offormation of di-SIA clusters and SIA–vacancy recombination is analyzed via approaches used in thekinetic theory of radiation ageing. The effects of 1D diffusion of SIAs, temperature, and defect concentra-tions on the reaction rates are also studied. This approach can validate both the kinetic mechanisms andthe appropriate kinetic coefficients, offering the potential to significantly reduce the uncertainty of thekinetic methodology and providing a powerful predictive tool for simulating irradiation behavior ofnuclear materials.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

A vast literature exists on the properties of body-centered cubic(bcc) metals, including Mo and Mo alloys irradiated with neutrons,electrons, and ions [1–18]. Interstitials, vacancies, and rare-gas fis-sion product atoms generated during these different types of irra-diations contribute to lattice damage, though to different degrees.The evolution of self-interstitial atoms (SIAs) and vacancies (VACs)is the first stage of defective structure relaxation during irradiationand contributes to relaxation after radiation cascade formation (aFrenkel pair mechanism). Along with that are the processes involv-ing large cluster formation directly generated in radiation cascades(an in-cascade mechanism). This stage plays an important role inthe nucleation process of the clusters of SIAs, dislocations, andvoids. Specifically, sessile clusters were considered as a nuclei forinterstitial loops (see, e.g., [19] and references therein).

Conventional radiation damage cascade simulations utilize theprimary knock-on atom approach, in which a high-energy particledisturbs the crystalline system by transferring its kinetic energy tothe lattice atoms, in the process creating Frenkel pairs (FPs). Thesesimulations are computationally expensive, in particular due tovery small time steps required to accurately capture the forceson the atoms during collision events. As a result, while the ‘‘ballis-tic phase,’’ during which FPs are created, is well captured in the

ll rights reserved.

simulations, the longer-lived ‘‘kinetic phase,’’ during which mostof the defects evolve into clusters or annihilate, remains less acces-sible. Since the input parameters for the steady-state rate equa-tions are based on this kinetic phase, we use an alternativemethod that focuses more on the diffusion-based long time evolu-tion of the defects. In this method, a high concentration of defects(both vacancies and interstitials) were introduced into the systemand are allowed to interact among themselves and with the micro-structural sinks. This scenario resembles the situation at the end ofthe ballistic phase of a radiation-damage simulation.

An in-cascade mechanism of large cluster formation was exper-imentally observed in several face-centered cubic (fcc) metals Ni,Cu, Ag, Pt, and Au and in the bcc metal W. These experimentalobservations were confirmed by molecular dynamics simulations[19–21]. However, this mechanism was not observed in fcc Aland in bcc Fe and V (see, e.g., [21] and references therein).

As for the material of interest to this work, Mo, in a recent studyof bcc Mo [22] it was found that the dose dependence of the elec-trical resistivity and the density of defect clusters, as well as sev-eral other findings, cannot be explained by in-cascade defectcluster formation. Therefore, we assume that the FP mechanismis viable for describing cluster formation in irradiation cascadesin Mo.

Modern kinetic theory of the radiation damage based on the dif-fusion-controlled reaction kinetics and mean field theory describesthe kinetics of these processes in terms of diffusion coefficients andinteraction radius of defects [7–14]. However, such a traditional

42 Z. Insepov et al. / Journal of Nuclear Materials 425 (2012) 41–47

approach cannot provide accurate values of both the kinetic coeffi-cients and the defect interaction radii, which can depend on tem-perature. These parameters should specifically be addressedbecause of the complexity of mechanisms of SIA + SIA interactionsand VAC + SIA interactions [15]. The mean field theory, for exam-ple, does not take into account the probability of binding of two de-fects into a bigger cluster, which in nucleation and chemicalreaction theories is called a steric factor. This factor can be muchless than unity, which is assumed in the existing radiation damagetheory [7,16–18,23,24].

A one-dimensional diffusion of SIAs and clusters of SIAs in Moshould be taken into account at low temperatures [25,26]. A kineticMonte Carlo (KMC) study has shown that 1D diffusion can changethe kinetics of the interaction between SIAs and sink strength dra-matically [27]. A similar effect can be observed with SIAs andvacancies.

This paper describes molecular dynamics (MD) simulations ofthe evolution of a pure Mo system containing SIAs and vacancies.Included are results from investigating the influence of 1D diffu-sion on kinetics, as well as a description of the kinetics obtainedby using an accurate, newly developed Mo interatomic potentialthat correctly describes thermodynamic properties and energiesof the main defect structures and applies this new potential tothe radiation cascade studies [28].

Basic properties (such as formation energies and structure) ofSIAs in Mo were studied using density functional theory (DFT)[25,29]. It was found that the difference in formation energies ofcrowdion and dumbbell h111i configuration is negligibly small,which implies that the SIA mobility at very low temperatures isin accordance with the data from the resistivity recovery experi-ments. The Finnis–Sinclair EAM interatomic potential [30] doesnot reproduce correctly the hierarchy of formation energies of SIAsthat results in very low SIA diffusivity. A recent EAM potential forMo proposed in Ref. [26] takes into account this hierarchy but pre-dicts anomalous thermal expansion and gives a large overestima-tion of the melting temperature.

This paper presents a multiscale approach that couples kineticrate theory with ab initio and atomistic MD simulations. This ap-proach allows one to bridge time and spatial scales from the quan-tum mechanical to the macroscopic world. A similar multiscalemethod was suggested by the present authors for studying clusterformation in rarefied gases and thin film formation processes in1991 [31] and by Stoller and Greenwood in 1998 [32]. Stollerand Greenwood briefly explored using MD-simulated cross sec-tions and proposed two approaches: a hybrid model that bridgesdifferent time and length scales (‘‘handshaking’’) and an approachthat involves passing information between models (‘‘handing-off’’)[32]. In related multiscale approaches [8–10] the results of ab initioand MD calculations are used in a KMC model.

In the present paper, an atomistic MD model of defects cluster-ing and annihilation is developed that addresses the evolution ofthe microstructure of Mo after the initial stage of point defects pro-duction and continuing until the stage where annealing of the de-fects becomes plausible.

2. Simulation method

The point defects in an ideal bcc Mo crystalline structure wereintroduced into the system at the beginning of the simulation.The rate of damage generation during the defect evolution wasset to zero.

Two types of initial defects were considered: SIAs in the form ofcrowdions and vacancies.1 The initial space distributions of defects

1 We discuss these defects in detail in [28].

were uniform for both defect types. Crowdions have four equivalentdirections in the basic crystal cell: [111], [�111], [1 �11], and [11 �1].These equivalent orientations were assumed to be distributed uni-formly. Such a simple MD model is applicable to simulation of theevolution of point defects produced by electron irradiation or inthe collision cascades. The developed model is applicable to a qua-si-equilibrium state where the temperature and defect concentra-tions gradients were not too high.

Two types of initial microstructures were considered: a crystal-line system containing only crowdions and a system with bothcrowdions and vacancies. Such an initial setup enables separatestudy of the kinetics of the cluster formation and recombination.The typical size of a simulation cell for the simulations of SIA + SIAreactions was 100 � 100 � 100 lattice units (31 � 31 � 31 nm)containing about 2 � 106 atoms. The initial numbers of crowdionswere 300, 400, and 600. For evaluation of vacancy and SIA recom-bination rates, a 100 � 100 � 100 system with 200 crowdions and800, 1000, 1500, and 2000 vacancies was used, and a150 � 150 � 150 system with 200 crowdions and 3000, 4000,5000, and 6000 vacancies was applied. The vacancies were as-sumed to be immobile during the calculation. In all cases three-dimensional periodic boundary conditions were used.

First, the structure of the defects was optimized by minimizingthe total potential energy. Then, the system was equilibrated at agiven temperature and zero pressure. Equilibrium was reached inseveral picoseconds. A range of temperatures T = 300 K, 400 K,600 K, 800 K, and 1000 K was considered.

The interaction between atoms was described by the embed-ded atom method potential [33]. This potential was parameter-ized by an ab initio force-matching method [34] that adjustsper atomic forces, stresses and energies of a vast set of atomicconfigurations to those obtained via ab initio DFT calculations,including a set of configurations containing defects, and there-fore this method was capable of reproducing a correct potentialenergy map and hierarchy of the formation and migration ener-gies of the defects [27]. The MD calculations were carried out byusing the LAMMPS code [35]. Obtaining accurate kinetic rateswith an MD method requires accurate diagnostic proceduresfor calculating the number of the defects in the studied systems.For this paper, the simulation focused on several simple kineticprocesses, such as defect recombination and di-SIA formation.Therefore, diagnostic methods were developed to enable map-ping interstitials, vacancies, and di-SIAs. In order to increasethe accuracy of the structure identification, certain processeswere developed and then optimized. For example, preliminaryrelaxation of the atomic positions was performed by minimizingthe total potential energy. This approach allowed the eliminationof thermal fluctuations that could have obstructed the visualobservation and overall accuracy of the method.

The types of defects are recognized taking into account a num-ber of per atom parameters that characterize the local environ-ment of atom: coordination number, centro-symmetryparameter, and potential energy. The coordination number (CN),is calculated as the number of neighboring atoms within thesphere of a radius rc. A typical value of rc is r2 + r3, where r2 andr3 are the radii of the second and third coordination spheres,respectively. This local parameter was also applicable for studyingthe vacancies. The centro-symmetry (CS) parameter was used as asecond local variable; it characterizes the deviation from the sym-metry of the local environment. The local potential energy (PE) ofthe atom was used as a third local parameter. The values of theseparameters are shown in Table 1.

Vacancies are identified by PE and CS. As an additional conditionthe CN parameter is used, with CN < 13.5, because the ideal latticehas 14 neighbors in a sphere of radius (r2 + r3). Eight atoms of thevacancy environment correspond to one vacancy.

Table 1Diagnostics parameters for different defect structures.

Type Crowdion Dumbbell di-SIAmob1 di-SIAmob2 di-SIAsessile Vacancy

PE, eV >�5.4 (�5.8..–5.4) (�6.6..-6.0) (�5.9..-5.5) >�6.0 <�6.6CS <2 <3 (8.5..9.1) <2 (7.6..8.5) (6..8)CN >14.5 >14.5 >14.5 >14.5 >14.5 <13.5

Z. Insepov et al. / Journal of Nuclear Materials 425 (2012) 41–47 43

3. Results

The following is a discussion of the behavior of the varioustypes of defects, including crowdions, dumbbells, di-SIAs, andvacancies, that were observed in our simulations.

2 Although this model should be applicable to di-SIA formation, we did not try touse it for this purpose.

3.1. Point defects in Mo

A crowdion and a dumbbell along a h111i crystallographicdirection are two stable structural forms of SIAs in Mo with thelowest formation energies and therefore are the most stable SIAdefects. Because they have almost identical potential energies,these defects are able to transform into each other by a one-dimen-sional motion. Such motion provides the possibility for a one-dimensional diffusion already at low temperatures. The change ofthe propagation direction is possible via change of structure fromh111i to h110i and finally to an equivalent h111i dumbbell. Wenote that a dumbbell defect along a h110i direction has a largerformation energy, ED110–ED111 � 0.27 eV [28], and SIAs stay in thisstructure for only a very short time. Hence this structure is not spe-cially tracked in our analysis of the simulations.

The structure of the crowdion consists of one additional Moatom in the bcc cell along the h111i diagonal. The energy minimi-zation of this structure results in expanding the lattice along theh111i direction.

The h111i dumbbell is characterized by two atoms symmetri-cally located around the center of the basic cell. Since 1D motionof SIA takes place as a rearrangement of a h111i chain from crow-dion to dumbbell structure, the direction of the SIA chain is parallelto the direction of its motion.

Two stable types of di-SIAs were observed. The first and moststable one is the mobile configuration, which can be representedby two parallel interstitials. The mobile configuration can changeduring the motion to a sessile cluster configuration, which formswhen the two additional atoms appear in the same elementarybcc cell. The atoms are redistributed in symmetrical positions onthe extension of three equivalent atomic chains, for example,[111], [11 �1] and [1 �11]. The equilibrated sessile configurationlooks like a star formed by three atomic chains.

The main objective of our investigation of the mechanisms ofdefect recombination and clusterization was to obtain favorableconfigurations of defects that result in the reaction and the reac-tion distances (interaction radii). The main reaction path of themobile di-SIA formation is the interaction between the two parallelcrowdions or dumbbells close to each other. The distance of inter-action R0 is about

ffiffiffiffiffiffi3ap

0 (a0 is the lattice parameter). The sessile di-SIAs are formed mainly by the interaction of two crowdions corre-sponding to nonparallel directions. But the formation from twoparallel crowdions is observed also.

The mechanisms of the formation of the two types of di-SIAs areclosely interlaced. Two parallel crowdions can form a sessile di-SIA,and two nonparallel crowdions can form a mobile one. The interac-tion distance between the nonparallel SIA configurations is largerthan that of the parallel one. The sessile di-SIA can rearrange tothe mobile form during the simulation. The reason is that the mo-bile form has a lower formation energy (the difference is about0.4 eV) than does the sessile form [28].

If the system contains both vacancies and SIAs, recombination(annihilation) between these two defect types is observed. Themain conclusion from the observation of SIA + SIA and VAC + SIAreactions is that the reaction distance is very high: it can reach10 Å.

3.2. Kinetic rate theory

The following basic rate equations describe the kinetics of thesystem of defects with the number densities (concentrations) ofinterstitials ci and vacancies cv:

dcidt ¼ G� k2

vDici � k2i Dvcv � k2

isDici;

dcvdt ¼ G� k2

vDici � k2i Dvcv � k2

vsDvcv: ð1Þ

where G is the defect production rate in dpa/s, Da (a = v, i) is the dif-fusion coefficients of vacancy or interstitial, and ka are the sinkstrengths. The second and third terms in the right-hand side ofthe equations describe recombination through a diffusion of inter-stitials to vacancies and vacancies to interstitials (hence k2

v and k2i

are the strength of vacancy as a sink for interstitials and vice versa).The last terms correspond to clusterization of interstitials andvacancies (k2

is and k2vs are the strength of corresponding clusters as

a sink for interstitials and vacancies).The rate equations can be solved by direct numerical integra-

tion or, in limited cases, by a simple analytical solution. Our MDmodel was constructed by simplifying the rate equations and bysetting the rate G to zero (since all the point defects are presentin the system from the beginning of the simulation and are notintroduced later). The diffusion coefficient of vacancies Dv is muchlower than that for interstitials, Di, within the whole temperaturerange studied, so all the terms with Dv are negligible.

The values of sink strength are substantially different for 1D and3D diffusion [20]. For recombination in the case of 3D diffusion k2

vis equal to 4pRc, while for 1D diffusion k2

v is equal [20] to 6p2R4c2,where c = cv is the concentration of vacancies. In our simulation wehave 1D diffusion at low temperatures and nearly 3D diffusion athigh (1000 K) temperatures. At 300 K the typical time of the 1D dif-fusion without reorientations is several nanoseconds, and the cor-responding mean free path length is larger than about 10 nm [36].

A ‘‘hedgehog’’ model is proposed to explain the complex aniso-tropic character of interactions between self-interstitials andvacancies in Mo. The model was applied to study the annihilationof self-interstitials with vacancies.2 According to this model, SIAsmove almost freely along four equivalent directions h111i. There-fore, if a Frenkel pair of SIA + VAC defects occupies the same h111iaxis and if the defects are closer than a certain recombination dis-tance, they spontaneously annihilate. Our MD calculations haveshown that the number of such spontaneous sites in a bcc Mo latticenear the vacancy ranges from 42 to 200, located along the fourequivalent h111i directions around the vacancy. An effective isotro-pic radius of a sphere around the vacancy comprising the same num-ber of lattice nodes would be 2a0.

44 Z. Insepov et al. / Journal of Nuclear Materials 425 (2012) 41–47

To simplify the analysis, we describe the kinetics in terms of 3Ddiffusion, but we take into account the dependence of the constantrate on temperature. This approach allows us to investigate thetransition from 3D to 1D diffusion. Eq. (1) then can be simplified:

dcidt ¼ �arcicv � aiicici;

dcvdt ¼ �arcicv :

ð2Þ

Here ar and aii are the rate constants in terms of the binary reac-tions; aii ¼

ffiffiffi2p

Di=X2=3 is the rate constant for association of two SIAs

into one di-SIA. Eq. (2) can be solved analytically in order to com-pare with MD results.

Fig. 2. Dependence of the SIA number on time for different initial concentrationsci0: 1–1.5 � 10�4, 2–2 � 10�4, 3–3 � 10�4. T = 300 K.

Fig. 3. Dependence of rate constant aii on the initial concentration for twotemperatures T: 1–300 K, 2–1000 K. The system size is 2 million atoms.

3.3. MD calculations of kinetic coefficients

Fig. 1 illustrates the evolution of the number of interstitialatoms and di-SIAs in a system without vacancies. During the first10 ps the system reaches quasi-equilibrium state, where the tem-perature and pressure are equilibrated and the very close intersti-tials interact by forming clusters. The diagnostics is verified by thetotal number of recognized defects, Ntotal, equal to NSIA + Ndi-SIA. IfNtotal equals its initial value N0, the diagnostics accurately recog-nizes all defect structures. After a long calculation time the Ntotal

changes, indicating that three SIAs and larger clusters are formed.Integration of Eq. (2) with cv = 0 gives the following solution:

1=ci � 1=ci0 ¼ atii; ð3Þ

enabling comparison of our MD results with the kinetic theory to bemade in the coordinates 1/ci � 1/ci0 vs. t, in order to obtain the rateconstant.

Fig. 2 shows the evolution of defect concentrations obtained byMD at three initial concentrations of SIAs at 300 K. The dependen-cies are well approximated by the linear equation with the slopeaii, which turned out to be nearly independent of the initial con-centration. The resulting rates are presented in Fig. 3 for differentinitial concentrations and for two temperatures. One can concludethat the reaction is second order at both temperatures, althoughthe motion is strongly 1D at T = 300 K and mostly 3D at T = 1000 K.

Fig. 4a shows the time dependence of the concentrations of mo-bile and sessile clusters during clustering of SIAs. An interestingfeature of this plot is that the number of sessile clusters saturatesafter about 30 ps, whereas the number of mobile clusters is stillgrowing. This situation means that the sessile clusters most prob-

Fig. 1. Evolution of the number of SIAs and di-SIAs, and total number of recognizeddefects. The system size is 2 million atoms, and T = 1000 K.

ably transform to the mobile configuration and reach a quasi-equi-librium concentration.

Fig. 4b shows that the relative number of sessile cluster be-comes less at T = 1000 K, which characterizes thermally activatedtransformation of the sessile clusters into mobile ones. Based onthe temperature dependence of the quasi-equilibrium concentra-tion of sessile clusters, one can estimate activation energy of tran-sition into mobile clusters as 0.2 eV.

3.4. Determination of rate constants

The recombination rates of SIA and VAC were determined fromMD simulation of the systems, where the concentration of vacan-cies is much larger than the concentration of interstitials. It pre-sents the possibility of neglecting the change of vacanciesconcentration: cv(t) � cv0. In this case Eq. (2) results in the follow-ing simple solution:

ci ¼ ci0 expð�aivcv0tÞ; ð4Þ

The concentrations of interstitials were fitted according to Eq. (4) atfive temperatures. Fitting procedures were conducted at several ini-tial concentrations of vacancies, to ensure that the results do notdepend on the initial concentration of vacancies (Fig. 5). One can

Fig. 4. Time evolution of SIA concentrations: (a) dependence of the number of different types of di-SIAs on time T = 600 K; (b) dependence of the ratio of the immobileconfiguration of di-SIAs on time for three temperatures: T = 300 K, T = 600 K, T = 1000 K. The system size is 2 million atoms.

Fig. 5. Calculation of the rate constants ar of annihilation between interstitials and vacancies at five temperatures obtained by MD in this work: (a) T = 300 K, (b) T = 400 K, (c)T = 600 K, (d) T = 800 K, and (e) T = 1000 K.

Z. Insepov et al. / Journal of Nuclear Materials 425 (2012) 41–47 45

conclude that the reaction is second order at both temperatures,although the motion is strongly 1D at T = 300 K and the average dis-

tance between the reorientations of crowdion chain is much largerthan the average distance between SIA and VAC. This conclusion

46 Z. Insepov et al. / Journal of Nuclear Materials 425 (2012) 41–47

contradicts the results of the theory [20] and KMC [27]. The discrep-ancy arises from the assumption accepted in the theory and KMCthat vacancies (and voids) are sinks of unlimited capacity.

3.5. Comparison of MD results with kinetic theory

In conventional diffusion theory based on the stationary con-centration profile, the recombination rate constants for the reac-tions between interstitials and vacancies can be obtained asfollows [7]:

ar ¼ 4pR0ðDi þ DvÞ=X � 4pR0Di=X; ð5Þ

where R0 is a temperature-independent recombination radius,which usually is set to a lattice parameter. In [39], the recombina-tion radius was obtained from resistivity experiments to be be-tween 0.7 and 2.5a0 for molybdenum. Our Frenkel pairrecombination model assumes that the recombination radius canbe a temperature-dependent value, R1(T):

ar ¼ 4privðDi þ DvÞ=X � 4privDi=X; ð6Þ

where riv = R0 + R1(T), R0 is the constant radius with a typical valueat low temperatures, and R1 is the temperature-dependent term.Our MD calculations have shown that the number of lattice sitesleading to a spontaneous recombination in a bcc Mo near the va-cancy is about 200, which are located along the four equivalenth111i directions around the vacancy. An effective isotropic radiusof a sphere around the vacancy comprising the same number of lat-tice nodes would be 3a0. The temperature-dependent term R1 is setas an Arrhenius expression with the activation energy barrier corre-sponding to 3D diffusion:

R1ðTÞ ¼ nR0 exp � E3da

kT

!: ð7Þ

Here, n = 0.125 is the steric factor that takes into account eight pos-sible ways for an interstitial hopping from one gliding direction toanother. The diffusion coefficient of interstitials, Di, is determinedby 1D diffusion,

Di ¼ D0 exp � E1da

kT

!; ð8Þ

and Fig. 6 plots various approximations that determine the temper-ature dependence of the Frenkel pair recombination rate constant ar

given by Eqs. (6)–(8).The conventional isotropic interaction radius theory of Frenkel

pairs recombination [37] also shows that the radius of interactionof a long-range isotropic elastic field near the vacancy can depend

Fig. 6. Dependence of the recombination rate constant of SIA + VAC reaction ontemperature: solid circle – our MD results; open triangle and squares – Eq. (5) withD = 1 � 10�4 cm2/s and R0 = 0.7 and 2.5a0, where a0 is the Mo lattice parameter [7];open circles and solid line – Eq. (6).

on temperature. The theory results in a decrease of the interactionradius with increasing temperature. The discrepancy between the-ory and MD results can be attributed to the 1D character of the dif-fusion, which is not taken into account in [37].

Several formulas are given in the literature for determining theSIA + SIA association rate constant based on the theory of diffusion-limited reactions [11,13,14]. A simple approximation of the rateconstant can be made by an Arrhenius formula:

aii ¼ aii0 expð�Ea=kTÞ; ð9Þ

where aii0 is the pre-exponential constant, Ea the activation energy,and T the temperature. The parameter a0 can be obtained from thefollowing requirement: a(300 K) = aMD (300 K). The best fit betweenthe kinetic coefficients approximated by (9) is obtained with theactivation energy Ea = 0.072 eV.

Theoretical values of the rate constants for the SIA + SIA interac-tion can also be obtained by the conventional formula (see, e.g., [7]for more detail).

aii ¼ffiffiffi2p� DiðTÞ=X2=3; ð10Þ

where Di (T) is the SIA diffusion coefficient depending on tempera-ture and X is the atomic volume. The dependence a(T) can be takeninto account just by the dependence D(T). This gives the followingapproximation formula:

aii ¼ffiffiffi2p

X2=3 � DMDðTÞ=Dð300KÞ; ð11Þ

which, together with Eq. (9) (with slight modification), has been ap-plied to analyze 3D point defect diffusion (see, e.g., [7]). However,previous studies have not taken into account the significantly aniso-tropic 1D behavior of defect migration in irradiated bcc Mo at lowand intermediate temperatures. When one does take into accountthe almost free movement of SIAs inside the ‘‘needles’’ of the hedge-hog model, one obtains the following expression, where the expo-nential dependence is enhanced by a stronger power dependenceon T:

a1dii ¼

2ffiffiffi2p

X2=3 D1di ¼ n

2ffiffiffi2p

X2=3 � ATs exp � E1da

kT

!; ð12Þ

where A, s are the adjustable parameters and n is the steric factor.Fig. 7 shows a comparison of the MD results with kinetic theory,

where the kinetic theory coefficients were obtained by using Eqs.(9)–(12). The figure shows that the expressions (9) and (10), appli-cable to the conventional 3D diffusion mechanism and widely used

Fig. 7. Dependence of the rate constant of SIA + SIA reaction on temperature:diamond symbols – our MD results; dashed line – Eq. (9) with Ea = 0.072 eV; solidline – Eq. (11). Our new model is shown by a dotted line drawn by Eq. (12) and hascorrect curvature fitting the MD results.

Fig. 8. Comparison of the concentrations of sessile clusters obtained by MD in thiswork with experimental dislocation loops density in [38].

Z. Insepov et al. / Journal of Nuclear Materials 425 (2012) 41–47 47

in kinetic rate theory, fail to satisfactorily describe the MD results.In particular, they give the wrong asymptotic behavior: the Arrhe-nius (dash) and the kinetic rate theory (solid) lines saturate athigher temperatures, whereas the MD data shows a much fastergrowth. Adding a power dependence to pre-exponential accordingto Eq. (11) significantly improves the comparison and changes thecharacter of curve to a growing type that fits the MD data.

3.6. Comparison of sessile cluster concentration with experiment

The concentration of the immobile sessile clusters saturateswith time (see Fig. 4a), whereas the concentration of mobile clus-ters continues to grow. We did a preliminary comparison of thecalculated concentration of sessile clusters with the experimentaldensity of small dislocation loops observed in Mo irradiated byelectron bombardment [38].

Our preliminary results, plotted in Fig. 8, show that the sessileclusters can become nuclei of the dislocation loops in irradiationexperiments, since they do not change their concentration afterreaching equilibrium in the system at any given temperature.

4. Summary

In the present paper, a multiscale concept is proposed for sim-ulating the evolution of radiation defects in Mo. The proposed con-cept consists of using atomistic simulation methods to studyradiation damage, defect formation, and growth processes and tocalculate the probabilities of elemental processes and reactions.All these types of simulated processes are also applicable to pro-cesses in irradiated nuclear materials. Since interaction potentialsare important for the new concept, a new Xe-Mo interatomic po-tential was developed by using an ab initio force-matching method[28].

MD simulations were conducted for SIA + SIA and VAC + SIAreactions, which play a crucial role in the initial stages of radiationdamage. Specifically, the influence of 1D diffusion on kinetics was

considered by using simulation at different temperatures.Although the diffusion of SIAs is strongly one dimensional atT = 300 K, the reaction of SIA + SIA is described as a second-orderreaction also at high temperatures. The same conclusion is drawnfor the recombination of vacancies and SIAs.

The rate constants of SIA + SIA and VAC + SIA reactions alsowere calculated based on the kinetics, and the temperature depen-dencies were obtained. Comparisons indicate that the values fromthe kinetic theory are typically smaller than the MD results.

Acknowledgments

This work was supported by the US Dept. of Energy under Con-tract DE-AC02-06CH11357, president Grant MK-3174.2011.8 andRFBR Grant 09-08-01116-a.

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