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ScienceDirectActa Materialia 84 (2015) 208–214

www.elsevier.com/locate/actamat

Helium segregation to screw and edge dislocations in a-ironand their yield strength

Enrique Martınez,⇑

Daniel Schwen and Alfredo Caro

Material Science and Technology Division, MST-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 21 July 2014; revised 21 October 2014; accepted 23 October 2014

Abstract—Helium (He) presents one of the mayor concerns in the nuclear materials community as it modifies the mechanical properties of the systemwithstanding fast neutron spectra, promoting swelling and embrittlement. Ferritic/martensitic steels are one of the main candidates as structuralmaterials for future nuclear applications. Experimentally the bubble distribution is observed to vary depending on irradiation conditions (temper-ature, dose rate and total dose). However, traditional atomistic models decouple the role of temperature in the mechanical properties from its effect onthe bubble distribution. In this paper we study substitutional He segregation to screw and edge dislocations in a-Fe at different temperatures. We use anobject kinetic Monte Carlo methodology to obtain general trends in bubble distribution and a canonical Monte Carlo algorithm, with full atomistic fidel-ity, to find the He distribution at the dislocation cores. Molecular dynamics has subsequently been applied to study the yield strength, which increasessignificantly in the presence of He, more remarkably for the edge dislocation. The total stress fits a Kocks relation. However, if the lattice resistance issubtracted, the relation between the critical shear stress and the temperature is non-monotonic for the screw character. To reproduce this effect, we pro-pose to modify the Kocks relation, adding a second-order term in temperature that extends the range of applicability of the model.� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Helium; Segregation; Dislocation mobility; Modeling; Monte Carlo

1. Introduction

The presence of helium (He) in next-generation nuclearenvironments [1–3] poses great material challenges as it pre-cipitates in the form of bubbles, and usually segregates intodislocations and grain boundaries, promoting swelling,hardening and embrittlement, which is ultimately the mainfailure mechanism occurring in these materials [4–7]. Thedistribution of bubble diameters and spacing between themis controlled by the irradiation conditions (temperature,dose rate and total dose) and the material properties(mobilities and energetics).

He effects have been extensively studied both experimen-tally and theoretically. On the experimental front, Krameret al. [4] studied the effect of He on the mechanical proper-ties of 304 stainless steels. They observed a severe loss ofductility rationalized in terms of He segregation to carbideparticles at grain boundaries that enhanced void formationand cracking. More recent results on He implantation showa strong embrittlement effect in different steels [8–15] evenfor small He contents. This deterioration of the mechanicalproperties is associated with He segregation to extendeddefects. Insight into this matter has been provided by theo-retical models. Rate theory has been used to study bubbleformation and bubble to void transition that promotes

http://dx.doi.org/10.1016/j.actamat.2014.10.0661359-6462/� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights

⇑Corresponding author.; e-mail: enriquem@lanl.gov

swelling [16–18]. Although useful in several respects, ratetheory needs an extended list of parameters, usuallyobtained either from experiments or lower-scale computa-tional models. This is one of the main reasons why classicalmolecular dynamics (MD) simulations have become soprominent.

Various atomistic studies have focused on the interac-tion of dislocations with preexisting He bubbles and theenergetics of He at dislocations or grain boundaries in Fe[19–25]. However, the combined study of both He segrega-tion into dislocations, and how this segregation modifiesthe yield strength of the system is not available in the liter-ature, even though this scenario is physically more mean-ingful. That is, for a given material and irradiation doserate, temperature determines the bubble distribution on dis-locations, varying in diameter and spacing. This effectmeans, overall, that the yield stress for a dislocation toovercome a fixed distribution of obstacles also dependson temperature, and therefore, both processes are convo-luted. Understanding this coevolution is the purpose of thiswork, in which we use both object kinetic Monte Carlo(OKMC) and canonical metropolis Monte Carlo (CMC)in the Kawasaki dynamics formalism, to obtain differentbubble distributions as a function of temperature, doserate and dislocation character. We then study the mechan-ical response of these decorated dislocations and obtainan analytical expression for the increase of yield stressinduced by He.

reserved.

E. Martınez et al. / Acta Materialia 84 (2015) 208–214 209

2. Results

The study of the real dynamics of He segregation toscrew and edge dislocations in a-Fe and the resultant dislo-cation mobility is challenging. MD calculations are limitedin simulated time scale and therefore it is difficult to extractmeaningful physics from them. Kinetic MC algorithmshave been proposed to try to overcome the MD time-scaleproblem [26–30]; however, in the case of He hopping inthe presence of dislocations, the existence of complexmechanisms and low barriers decreases dramatically theirefficiency. This study focuses on the bubble distributionat a given temperature and how the temperature modifiesthe spacing between bubbles and the bubble diametersrather than in the actual dynamics of the process. Conse-quently, we use two different MC algorithms, an OKMCto study general trends in the distribution of bubbles, anda CMC method [31] with a fixed rate of transmutation ofFe into He that mimics the behavior of He in the presenceof a dislocation field. This algorithm has been implementedin the MD code LAMMPSLAMMPS [32], obtaining a very powerfulhybrid MC–MD scheme that takes into account chemicalmixing, structural relaxations and thermal vibrations.

2.1. General trends for bubble diameter and spacingdepending on temperature and dose rate

To obtain general trends of the bubble distributions on alinear sink, the OKMC algorithm [33] has been used. Themain goal is to convince the reader that the bubble distribu-

Fig. 1. OKMC results for the evolution depending on temperature anddose rate of (a) He bubble diameter and (b) He bubble spacing.

tion depends on both temperature and dose rate. For that,a quasi-1-D model on an underlying body-centered cubic(bcc) lattice with 8192 sites (2 � 2 � 1024) has been setupwith He atoms diffusing to first nearest neighbors with amigration barrier of Em ¼ 0:8 eV and a prefactor ofm ¼ 1012 s�1. He atoms are created at random at a pre-scribed dose rate. Defects interact to form clusters withan infinite binding energy, i.e. when a cluster forms it willnot dissolve (mimicking the large He–He binding energy).Runs have been carried out at different temperatures anddose rates until the fraction of defects in the sample was0.25. The results are shown in Fig. 1, where the evolutionof the bubble diameter (D) and spacing (L) between bub-bles is plotted. Both D and L are monotonically increasingfunctions of the temperature. Temperature increases thedefect mobility for a given dose rate, which increases thereaction probability and therefore the propensity for bub-ble nucleation. We also note that the lower the dose rate,the larger the bubble diameter and therefore the largerthe spacing between bubbles. For a given temperature,increasing the dose rate decreases the bubble nucleationprobability.

The evolution of the bubble diameter was fitted withfunctions of the simple form: D ¼ D0 þ a � ðT � T 0Þb. Thediameter has been obtained, assuming sphericity, usingthe relation hNiX ¼ 1

6pD3, where hNi is the average number

of particles in a cluster and X is the atomic volume. Thetotal number of defects in the sample is known, and there-fore, a relation between the diameter and the spacingbetween bubbles can be derived: L ¼ D3pLT

6XNT, with LT the total

length of the sample and NT the total number of defects.General trends are therefore obtained following this meth-odology (see Table 1).

2.2. He segregation to screw and edge dislocations asdetermined by canonical Monte Carlo

Two samples containing a screw and an edge disloca-tion, respectively, were generated using linear elasticity[34]. Both samples are oriented in the ½1�10�; ½111� and½�1�12� directions and fixed boundary conditions in the½1�10� direction and periodic boundary conditions in theother two were applied. The Burgers vector of both disloca-tions is the a0

2½111�, where a0 is the lattice parameter of

a-Fe[35]. The dislocation line for the screw case is alongthe ½111� direction, while for the edge the dislocation linelies along the ½�1�12� direction. Two layers of 12 A each werefixed in the upper and lower region of the simulation box.The Fe–Fe interatomic potential given by Ackland et al.[36], the He–He interaction by Beck [37] and the Fe–Heby Juslin and Nordlund [38] were used in both MD andCMC simulations. Three temperatures have been consid-ered: 100, 300 and 500 K. The He atoms were inserted inthe system at the expense of Fe atoms, i.e. transmutingFe into He at a prescribed CMC rate. In this way, substitu-tional He atoms were introduced every 8000 CMC steps.To sample the vibrational degrees of freedom, 500 MDsteps were performed between CMC steps. This methodol-ogy leads to bubbles with a He-to-vacancy ratio close toone, similar to the calculated equilibrium value [35,39].The CMC stopped when a total of 100 He atoms wereinserted in the sample, which represents a concentrationper dislocation unit length of cscrew

He ¼ 3:68 nm�1 andcedge

He ¼ 3:57 nm�1 for the screw and edge dislocations,respectively.

Table 1. Fitting parameters for the bubble diameter(D ¼ D0 þ a � ðT � T 0Þb). T 0 ¼ 300 K in all cases.

D0ðnmÞ a b

/ ¼ 1 dpa s�1 0.55 0.018 1.2/ ¼ 10 dpa s�1 0.50 0.021 1.1/ ¼ 100 dpa s�1 0.44 0.029 1.0

210 E. Martınez et al. / Acta Materialia 84 (2015) 208–214

2.2.1. He segregation to screw dislocationsDislocations are preferential sites for heterogeneous

nucleation of He bubbles. The thermodynamic driving

Fig. 2. He bubble distribution at a screw dislocation core in a-Fe. Light brownanalysis while He atoms are depicted in blue: (a) T ¼ 100 K; (b) T ¼ 300 K; (clegend, the reader is referred to the web version of this article.)

Fig. 3. He bubble distribution at an edge dislocation core in a-Fe. Lightneighbor analysis while He atoms are depicted in blue: (a) T ¼ 100 K; (b) T ¼this figure legend, the reader is referred to the web version of this article.)

force for a substitutional He atom to be attracted towardsthe dislocation is given by the gradient of chemical poten-tial. At 0 K this gradient can be estimated by calculatingthe interaction energy. This interaction energy is definedas [40]:

E intHe�D ¼ E½He�D� þ E½Fe� � E½D� þ E½He�f g ð1Þ

where E½He�D� is the energy of the system with the substi-tutional He at the dislocation core, E½Fe� is the energy ofthe pure Fe system taken as reference, E½D� is the energyof the system with the dislocation and E½He� is the energy

atoms represent non-bcc Fe atoms according to the common neighbor) T ¼ 500 K. (For interpretation of the references to color in this figure

brown atoms represent non-bcc Fe atoms according to the common300 K; (c) T ¼ 500 K. (For interpretation of the references to color in

Fig. 4. Canonical MC results for He bubbles: (a) diameter; (b) spacing.Points are fitted to the expression obtained from OKMC calculations,i.e. D ¼ D0 þ a � ðT � T 0Þb and L / D3 (solid lines).

Fig. 5. Stress–strain curves for a screw dislocation in a-Fe as itovercomes a distribution of He bubbles at different temperatures: (a)T ¼ 100 K; (b) T ¼ 300 K; (c) T ¼ 500 K. The He bubble distributionsare those obtained using the CMC algorithm.

E. Martınez et al. / Acta Materialia 84 (2015) 208–214 211

of the Fe system with one substitutional He. We haveobtained a value of Eint

He�D ¼ �0:28 eV for the screw dislo-cation. MC simulations were performed to obtain the Hesegregation distribution in a system with a screw disloca-tion for the three temperatures of interest (see Fig. 2).The sample dimensions were 16.2 � 27.2 � 14 nm. At100 K, the average number of He atoms per bubble is 10,which leads to an average diameter of around 0.59 nm fora cHe ¼ 3:68 nm�1 and an average spacing between bubblesof 2.72 nm. The average diameter at 300 K is about 0.62 nmwith a bubble spacing of 3.02 nm and at 500 K the averagediameter is about 0.67 nm, which leads to a bubble spacingof 3.88 nm. The bubbles can be considered spherical atevery temperature since surface tension becomes the domi-nant term in the energetics of the system.

2.2.2. He segregation to edge dislocationsThe interaction energy of a substitutional He atom at

the core of the edge dislocation is EintHe�D ¼ �0:5 eV, which

results in a strong thermodynamic driving force for the Heto segregate at the dislocation core. This interaction energyis more attractive than for the screw dislocation and thuswe expect a stronger pinning effect for the edge character.

As was done for the screw dislocation, we have studiedsubstitutional He segregation into the edge dislocation coreusing the CMC methodology. Fig. 3 shows the bubble con-figuration at the dislocation core for the temperatures stud-ied. The sample dimensions were 16.3 � 9.78 � 28 nm. AtT = 100 K the average bubble diameter was 0.51 nm withan average spacing between bubbles of 2 nm. When thetemperature was increased to 300 K, the average bubblediameter became 0.64 nm and the bubble spacing 3.5 nm.Finally, for 500 K we found an average diameter of0.77 nm and an average spacing of 6.29 nm. For all temper-atures again the bubble structure can be consideredspherical.

Fig. 4 shows the segregation behavior using both theCMC and OKMC approaches. The resulting trends aresimilar and can be described by the relation D ¼ D0 þ a �ðT � T 0Þb, with T 0 ¼ 100, and D0 ¼ 0:59 nm, a¼ 6:05 � 10�5

and b¼ 1:2 for the screw dislocation and D0 ¼ 0:51 nm,a¼ 6:5� 10�4 and b¼ 1:0 for the edge dislocation. Thefunction is close to linear, with a crossover that correlatesto the relation of the substitutional He binding energy tothe dislocation cores and the He binding energies to thebubbles themselves. The spacing between bubbles asobtained from CMC is shown in Fig. 4(b). For the screwdislocation we see that the points follow accurately thefitting function. On the other hand, for the edge dislocationthere is a slight deviation from the predicted values, prob-ably due to lack of better statistics and the fact that thebinding energies between He were not proper in the OKMCmodel.

2.3. Mobility of screw and edge dislocations

Once the He configurations have been obtained, thesample yield strength (s) for both dislocation types has beenstudied with a constant 109 s�1 shear strain rate applied tothe upper layer of the simulation box in the direction of theBurgers vector.

We analyze the case of the screw dislocation first. Theresults are plotted in Fig. 5. At 100 K the stress requiredfor the dislocation to overcome the He bubbles was around900 MPa. The lattice resistance at this temperature as given

by the empirical potential used in this study is about540 MPa. The stress to overcome the He bubbles is largerthan the lattice resistance stress and therefore the disloca-tion, once released from the obstacles, will glide in the pho-non drag regime. For 300 K, the yield stress was calculatedto be around 650 MPa, which compares to the lattice resis-tance of about 180 MPa, while at 500 K the stress requiredfor the dislocation to overcome the He bubbles was about580 MPa with the lattice resistance regime occurring below96 MPa. Therefore, the required stress for the dislocations

Fig. 6. Stress–strain curves for an edge dislocation in a-Fe as itovercomes a distribution of He bubbles at different temperatures: (a)T ¼ 100 K; (b) T ¼ 300 K; (c) T ¼ 500 K. The He bubble distributionsare those obtained using the CMC algorithm.

Fig. 7. (a) Total yield strength for a screw and an edge dislocations vs.temperature for a He concentration of cscrew

He ¼ 3:68 nm�1 andcedge

He ¼ 3:57 nm�1. The red circles show the lattice resistance for ascrew dislocation, fitted with an exponential function. (b) Yieldstrength vs. temperature for screw and edge dislocations subtractingthe lattice resistance for the same He concentration as in (a). (Forinterpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

212 E. Martınez et al. / Acta Materialia 84 (2015) 208–214

to move increases considerably in the presence of He, evenat elevated temperatures, resulting in strong hardening.

For the edge dislocations in a-Fe gliding on a f110gplane it is well known that the Peierls stress is extremelylow [41,42] and the mobility of these dislocations is oftenconsidered athermal. Fig. 6 shows the stress–strain curvesupon simple shear loading. We observe that He modifiesenormously the stress required for the dislocation to moveat all temperatures. At 100 K the yield strength increases upto around 800 MPa before the dislocation overcomes theHe bubbles and starts gliding. Increasing the temperaturedecreases the number of obstacles and increases the proba-bility the dislocation will move. At 300 K the yield stressbecomes 500 MPa and at 500 K the stress needed for thedislocation to glide is of the order of 400 MPa.

We highlight here that these results reflect not only theeffect of temperature on the bubble–dislocation interaction,but also include the effects of temperature on the bubblepatterning (size and distance between bubbles) that charac-terizes the He heterogeneous precipitation process.

3. Discussion

Traditional models for the yield strength dependence ontemperature for a random distribution of obstacles followthe phenomenological relation developed by Kocks andArgon [43]:

s ¼ s0 1� TT 0

� �p� �q

ð2Þ

where s0 is the stress at which the migration barrier for thedislocation becomes zero and T 0 is the critical temperatureat which the thermal energy is high enough for the disloca-tion to overcome the barrier with no stress applied. In gen-eral, 0 6 p 6 1 and 1 6 q 6 2, although their values aresomehow arbitrary [44].

Eq. (2) applies to a random distribution of precipi-tates—a distribution that may result from a homogeneousnucleation process. In the case of He precipitating heteroge-neously to dislocations, we observe that temperature intro-duces two effects. On the one hand, as temperatureincreases, the bubble diameter increases and the bubble

spacing along the dislocation core also increases and, aswe shall see, this effect will lead to a non-monotonic rela-tion between stress and temperature. On the other hand,temperature increases the probability of overcoming thePeierls barrier imposed by the lattice resistance. This effectcan be clearly seen in Fig. 7, where the total yield strengthfor both screw and edge dislocations is shown in Fig. 7(a)while the yield strength increment with respect to the latticeresistance limit is plotted in Fig. 7(b). In the first case, Eq.(2) reproduces accurately the values obtained from atomis-tic calculations with the parameter values shown in Table 2.We observe that the values for p and q are indeed within thegeneral range. However, s0 for the edge dislocation is some-what large. This value is related to the binding energy of theHe to the dislocation which is larger for the edge segmentthan for the screw, and therefore the incremental stress tobe applied at 0 K to overcome the barrier is substantiallyhigher.

The total yield stress includes both the lattice resistancecomponent and the strengthening due to the obstacles. Forthe edge dislocation the lattice resistance is negligible, whilefor the screw dislocation there is an important componentdue to the Peierls barrier in the temperature regime studiedin this work [45]. If the lattice resistance for the differenttemperatures is subtracted from the total stress [45], theobstacle strengthening is then obtained. The results areshown in Fig. 7(b). For the edge component the curve isvirtually identical to the total stress. On the other hand,for the screw dislocation, we observe a non-monotonicbehavior with a maximum at about 350 K. This maximumreflects the fact that the relation between the He bubble dis-tribution, which depends on temperature, and the criticalshear stress is not linear. To model this non-monotonicbehavior a new term might be added to Eq. (2), renderingthe form:

s� ¼ s�0 1� TT 0

� �p

þ aTT 0

� �r� �q

ð3Þ

Table 3. Fitting parameters for the obstacle strengthening s�ðT Þ.T cðKÞ s�0 (MPa) a p q r

Screw 500 240 1.38 2.0 1.95 1.0Edge 2350 7600.03 0.0 0.03 0.96 –

Table 2. Fitting parameters for the total yield stress sðT Þ.T cðKÞ s0 (MPa) p q

Screw 1850 1142.88 0.53 1.04Edge 2350 7600.03 0.04 1.06

E. Martınez et al. / Acta Materialia 84 (2015) 208–214 213

where the asterisk indicates the component due to obstaclestrengthening. A least-squares fit to the MD data is shownin Table 3 and Fig. 7(b).

Quantitative analysis of the results presented heredepends on the accuracy of the interatomic potential. Morerecent Fe interatomic potentials, such as those by Gordonet al. [46] or Marinica et al. [47] that reproduce moreaccurately the dislocation migration barrier, would pro-vide better estimates for the critical shear stress. However,the qualitative description of the various effects to beaccounted for under irradiation depending on temperaturewould not change. The fact that the binding energy of Heatoms to the edge dislocation is larger than for the screwwould most probably remain the same since the atomicdensity is lower in the edge case (note that this result agreeswith previously published trends [48]). The effect of doserate might be more subtle. In a diffusion-controlled regimethe He would have enough time to diffuse and form largebubbles at the dislocation cores. On the other hand, in areaction-controlled regime He atoms will accumulatequasi-statically at the dislocations without time to migrateand react, therefore remaining dispersed at the cores. Inan intermediate regime, the relation between the bindingenergy, the He mobility, the dose rate and the temperaturewould prescribe the bubble distribution, as shown before,which would depend on the interatomic potential used.This, in turn, would result in a critical shear stress that willdiffer for each case. Again, the goal of this study is not toobtain perfectly accurate values for the stresses, but toprovide a methodology that is able to feed large-scalecalculations, such as crystal plasticity models, to studythe macroscopic effect of the formation of bubbles indiverse irradiation environments in an Fe matrix.

Note also that if one is to use the Bacon–Scattergoodmodel [49,50] that was developed for “strong” obstaclesand that relates the critical shear stress with the harmonicmean of the bubble diameter (D) and spacing (L) 1 the tem-perature effect should be taken into account. As a first-order approximation, substituting the order of magnitudefound above for D and L, we would obtain:

s� � 1

T 3ln T�1 þ T�3� ��1 þ DðT Þ

n oð4Þ

which in general decays faster than the results shown inFig. 7(a) and would present a minimum instead of amaximum if the lattice resistance is subtracted (Fig. 7(b)).

1s� ¼ lb2pL ln D�1 þ L�1

� ��1 þ Dn o

where l is the shear modulus, b is

the Burgers vector and D is a phenomenological parameter.

4. Conclusions

In summary, we have studied in this work the role oftemperature on the yield stress of screw and edge disloca-tions in the presence of He bubbles. We have shown, usingan OKMC methodology, that the distribution of bubbles ata dislocation core depends on both dose rate and tempera-ture, and from these calculations, laws for the diameter andspacing between bubbles were derived, showing the compe-tition between diffusional properties and the particle crea-tion rates. We have studied He segregation to screw andedge straight dislocations in a-Fe using an equilibriumMC algorithm, including both vibrational and chemicalcontributions. We observe the formation of bubbles prefer-entially at the dislocation core in both cases. For a givendose rate the distribution of He atoms at the dislocationcores depends significantly on the temperature. Bothapproaches, OKMC and CMC, give similar results, sup-porting the picture that emerges from them in the form ofan equation for the size of, and distance between, bubbles.

The study of the dislocation mobility shows that Hestrongly pins dislocations, increasing the system yieldstrength. Traditional phenomenological Kocks modelsmanage to follow the total critical shear stress given bythe atomistic results if the right set of parameters is chosen.We further show that, if the lattice resistance term is sub-tracted from the total yield stress, the obstacle strengthen-ing for the screw case results in a non-monotonicrelation, which invalidates these traditional approaches.We have added a new term to the Kocks relation of theshear stress with the temperature to be able to reproducethe MD results. The parameters obtained in this studymight be used in crystal plasticity calculations to analyzethe macroscopic effect of He in the deformation modes offerritic systems.

Acknowledgements

The authors want to thank J. Marian and J.P. Hirth for usefuldiscussions. Work performed with support from the Center forMaterials at Irradiation and Mechanical Extremes, an EnergyFrontier Research Center funded by the US Department ofEnergy (Award Number 2008LANL1026) at Los AlamosNational Laboratory. This research used resources provided bythe LANL Institutional Computing Program. LANL, an affirma-tive action/equal opportunity employer, is operated by Los Ala-mos National Security, LLC, for the National Nuclear SecurityAdministration of the US DOE under contract DE-AC52-06NA25396.

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