Helium Bubble Growth at BCC Twist Grain Boundaries

Accepted Manuscript Helium Bubble Growth at BCC Twist Grain Boundaries J. Hetherly, E. Martinez, A. Caro, M. Nastasi PII: DOI: Reference:

S0022-3115(11)00785-9 10.1016/j.jnucmat.2011.08.009 NUMA 46044

To appear in:

Journal of Nuclear Materials

Received Date: Accepted Date:

29 March 2011 3 August 2011

Please cite this article as: J. Hetherly, E. Martinez, A. Caro, M. Nastasi, Helium Bubble Growth at BCC Twist Grain Boundaries, Journal of Nuclear Materials (2011), doi: 10.1016/j.jnucmat.2011.08.009

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Helium Bubble Growth at BCC Twist Grain Boundaries J. Hetherly, E. Martinez, A. Caro, M. Nastasi MST-8 Los Alamos National Laboratory, Los Alamos NM 87544

Abstract We study the growth of helium bubbles in α-Fe at low angle twist grain boundaries and in bulk using molecular dynamics and Metropolis Monte Carlo simulations. We describe the pressures and volumes of the helium bubbles and analyze the maximum pressure a bubble can sustain before emitting interstitial loops. We give a quantitative analysis of how these emitted loops behave differently in the bulk and at the grain boundary.

Introduction Ferritic steels, such as Fe-Cr steels, are good candidates for future fusion reactors due to their anticipated increase in radiation tolerance over current steels in the nuclear industry [1]. These prospective steels will experience especially high doses of irradiation along with the accompanying defects, dislocations, and He production. In particular, He together with vacancies tend to form He filled cavities within the material which can eventually embrittle the steel [2]. These bubbles in turn attract more He [3] and emit self–interstitials [4] and dislocation loops [5] to create room for the bubbles. Therefore, it is important to study the connection between the growth mechanisms and the pressure in the bubble as well as how bubble growth affects the surrounding lattice. In this paper we explore He bubble growth in pure bcc Fe as a representative base material for steels. Our studies focus on two particular situations, namely bubbles with density one (produced by substitution of a Fe atom by a He), and maximum density (He bubbles without incorporation of vacancies). While these conditions are not those of fusion environments, this study is a contribution towards the

1

development of an equation of state of He bubbles in metals, that would eventually cover all density values He has a well-known affinity for defect structures in Fe including grain boundaries [6], edge and screw dislocations ([7] and [8]), dislocation loops [9], point defects [10], and voids and He bubbles [11]. In this paper we study the interaction of He with screw dislocations as present in a twist grain boundary. The migration of He around a dislocation is in general highly dependant on how the He approaches the dislocation due to spatially varying stress fields ([12], [13], and [14]). However, the dislocations that form from low-angle twist boundaries are screw in character, which are axially symmetric sinks for He [15]. In particular, a network of intersecting screw dislocations forms, which appears to be a very effective sink for heterogeneous precipitation of He, in particular at its junctions. He bubbles could be characterized by a number of parameters describing the size of the bubble, the number of vacancies and He atoms, or the pressure. To fully characterize a bubble we need in fact two parameters, and here we choose the He density and the bubble radius. In this work we study two cuts on this r-He plane, namely He=1 He/Vac in the Metropolis simulations of heterogeneous precipitation, and He equal to the maximum possible value for a given material and microstructure (in this paper, Fe in bulk or twist GB’s) for the molecular dynamics (MD) simulations. By maximum possible value we mean the maximum amount of He a bubble can contain before it creates room by emission of host matrix atoms.

Computational Parameters Our simulations use the MD software LAMMPS with the concentrationdependent embedded atom method (CD-EAM) [16] for computing the forces. All simulations in this study use periodic boundary conditions (PBC) along all three axes. The potential we use was developed by Caro, et al. [5] for the Fe-Cr-He system. The Fe-Fe, Fe-He, and He-He interactions are taken from Ackland, et al. [17], Juslin, et al. [18], and Beck [19], respectively.

2

All helium-loading simulations are put through the following process. First, a position in the Fe sample is chosen to be the place where all He atoms will sequentially be created. For bulk Fe, this position is the center of a vacancy, created with the purpose to act as a nucleation site for the bubble; for the GB cases, the position of this initial vacancy is chosen close to the center of a dislocation junction. Then a cycle is set up as follows: i- a helium atom is introduced at the chosen location in the sample; ii- minimization is then performed using conjugate gradients to assure that the helium atom is not overlapping another atom; iii- MD is run at 0 bar and 300 K for 2 ps to allow relaxation of the sample; iv- another minimization is performed in order to obtain a clear understanding of the processes taking place in the material. This cycle is repeated until the desired number of He atoms has been created, typically between 500-800 atoms. All analysis is done on these last minimization frames. This scheme is meant to represent a He rich – vacancy poor irradiation condition, found at the end of the He profile in an implantation irradiation. The creation of He atoms inside the bubble is a way to simulate He interstitials diffusing towards a bubble and joining it. This represents the most extreme case of bubble growth since no vacancies are introduced, giving the pressure (or He density) as an outcome of the simulation in contrast to manually setting the density as in Caro, et al. [5], Schaublin, et al. [20], and Haghighat, et al. [21]. In order to make a fare comparison between the different samples, all bubbles start with roughly the same volume of one iron vacancy (.0117 nm3) with one helium atom inside. We employ the Voro++ [22] software to calculate the volume from the Voronoi cell for each atom. This allows us to accurately compute the volume and pressure for each individual atom as well as volumes and pressures of irregularly shaped bubbles.

Twist Sample Structure Each sample with a twist grain boundary (GB) has approximately 1.1×105 Fe atoms with the screw dislocation network in the xy plane. They are twice as large in the z direction (17.1 nm) than in x or y (8.6 nm). This is done to minimize

3

the interaction between the bubble and junctions above and below (due to PBC, two identical twist GB’s are created in each sample) so we can safely focus on interactions between the bubble and its immediate surroundings. The two samples tested in this paper are 2° and 6°grain boundaries. Samples are made by rotating the upper half of a crystal by +1° or +3° and the lower half by -1° or 3°. The rotation axis is (001), therefore a -screw dislocation network forms at each grain boundary. To analyze the dislocation network we used a dislocation detection algorithm (DXA)[23], which identifies dislocation lines and Burger vectors (Figure 1). The two samples have a different areal density of screw dislocations with four and sixteen junctions for 2° and 6° rotations respectively. The samples are loaded with various amounts of helium according to how many junctions it contains. The samples are loaded at one, two, four, and sixteen (6° sample only) junctions. The helium-loading rate for each simulation is 1 He / 2 ps × (number of junctions loaded). This loading scheme is chosen to observe what interactions the bubbles may have with other, nearby bubbles. Except for the case when all junctions are loaded, no two bubbles are loaded in adjacent screw junctions. This is to ensure any possible helium migration along the screw dislocation core (pipe diffusion) into other bubbles is held off for as long as possible. Results from Heinisch, et al. ([7] and [15]) suggest that while He is strongly attracted to the screw dislocation cores, He pipe diffusion along screw dislocations has an activation energy higher than in the bulk by a factor of 4 or 5 (although they used ½ screw dislocations and we use ). This result is consistent with our observation that no He is seen to diffuse from the bubbles.

4

Figure 1: View of the two screw dislocation networks formed at the middle of the 2° (left) and 6° (middle) sample and at the bottom. The second network appears at both the top and bottom due to periodic boundary conditions. The left and middle picture shows an atomistic view and the right is the analysis done by the DXA program on the 2° sample. The arrows represent burgers vectors (the sign is arbitrary).

He Placement Via MC Simulations To reveal that helium is thermodynamically attracted to the junctions and heterogeneously precipitate there rather than homogeneously in the bulk we ran a semigrand-canonical Metropolis Monte Carlo (MC) simulation with variance constraint ([24]) on the Fe 2° twist samples. We set the desired He composition to be loaded in a ramp linear in the MC steps, and allowed the sample to evolve, letting the helium go to locations it finds thermodynamically favorable (Figure 2). In these MC simulations Fe atoms are swapped for He atoms so the He is substitutional. The result shows that helium has a strong affinity for the screw dislocations and, in particular, the screw dislocation junctions. This forms the basis for choosing to load He at the screw junctions in our MD simulations, as done by others as well [15].

5

Figure 2: Placement of He (white) in the 2° screw dislocation network after the Metropolis Monte Carlo simulations in Fe (blue). This picture is taken looking down along the [001] direction at the dislocation network. Atoms are shown based on a potential energy threshold.

Results He-Fe Interactions and the He EOS We aim at describing the relation between pressure, size, and density of He bubbles in Fe. It is beneficial to discuss first how the pressure of the bubble differs from that expected by the helium equation of state (EOS) reported previously in [5]. The discrepancy in pressures that we find is due to the ironhelium interaction near the surface of the bubble. To demonstrate this effect we ran several simulations where helium was substituted for iron in a bulk sample. The substitutions were made in a spherical region of varying radii. This results in several samples with different bubble volumes but the same (initial) helium density of 1 He / Vac. The samples were then relaxed and pressures computed from the trace of the stress tensor calculated by LAMMPS and the atomic volume determined by Voro++. Figure 3 shows radial pressure profiles for three different bubble radius; we observe a well-defined region near the bubble-matrix interface that exhibits a sudden drop in pressure due to a combination of an increase in per-atom volume and iron-helium interaction. The increase in per-atom volume at the boundary is clearly seen in Figure 3, which shows a gap between He and the Fe matrix at 0 K as a consequence of a much softer He-He interaction compared

6

to the He-Fe interaction. Figure 3 shows that the drop in pressure for He atoms close to the interface for the actual bubble is less than the equation of state predicts based on the volume per atom alone. This implies that the He-Fe interaction acts in two ways: first it increases the volume per atom, thus decreasing the density and the corresponding pressure from the EOS for pure He; second it increases the overall pressure in those boundary He atoms with respect to the value of the EOS, creating a core-shell structure in the bubble whose size is determine by the range of the He-Fe interaction. Despite this increase, the actual pressure is still less than the value inside the bubble far from the interface, implying the bubble is not a homogeneous fluid. As the bubble gets smaller this shell effect dominates and results in bubbles that have, on average over all He atoms, much higher pressures that than predicted by the EOS at that He density. From this analysis it is clear that the pressure would be even higher in bubbles that are irregular in shape due to a higher number of He atom interacting with Fe. These observations have implications for coarser models of He bubble evolution, such as rate equations or phase field models, with respect to the equilibrium He concentration near the bubbles. Figure 4 shows a graph of average bubble pressure and average volume per atom versus inverse bubble radius to further illustrate the effect of size on the bubble pressure. The pressure data is fitted to the equation P(R)  P0 

2 with R

P0 1.05 GPa and   2.62 GPa  nm. For comparison, this potential gives   2.41 GPa  nm as the surface energy of an Fe(100)–He interface at 0 K. As the

 

 radius increases the helium EOS becomes a better approximation for the  description of the helium in the bubble, as it should. For our simulation the He

equation of state gives PEOS ( 1,T  0) 1.36 GPa . At this density, the radius where the pressure of the He matches the surface tension of the bubble is at

R 13.4 nm.



 7

Figure 3: View of He bubble and Fe lattice atoms colored by atomic volume (left). The redder colors 3

represent a larger volume (up to .02 nm ) and the cooler color represent smaller volume (minimum around 3

.0009 nm ). Pressure profiles for .75 nm, 2.1 nm, and 3.6 nm radius bubbles in bulk Fe are given on the right. Each of these profiles shows general agreement with the EOS until about .4 nm from the bubble’s surface at which point the He-Fe interaction alters the pressure significantly. (curves are smoothed for clarity)

We now comment on the properties of He at 0 K. The potential we are using for the He-He interactions gives excellent agreement with the experiments for the EOS at densities about 1 He/Vac and temperatures around room temperature and above. In Figure 3 and Figure 4 however, we choose to report data at 0 K where the potential is not accurate. The purpose behind this choice is just to give data with minimal thermal noise. To obtain the accurate data at room temperature and above, the EOS should be used. The EOS for the fluid phase of He for this empirical potential is given by [5], P(,T)  a0 ()  a1()T  a2 ()T 2

where

(1)



8

a0 ( )  6.347 10 1  5  3.446 10 0  4 1.820 10 0  3  3.84310 1  2 1.114 10 2  a1( )  1.08110 3  5  4.4310 3  4  4.812 10 3  3  2.24110 3  2 1.078 10 3  a2 ( )  6.847 10 7  5  2.499 10 6  4  2.165 10 6  3  9.776 10 7  2 1.21110 7 



Figure 4: Pressure and atomic volume versus inverse bubble radius for bubbles of radii ranging from .21 nm-3.6 nm. The pressure given by the He equation of state (1.36 GPa) is shown for reference. The linear fit ignores the case of a single He inside a vacancy (rightmost point) as this can hardly be called a “bubble.”

He Bubble Growth in Bulk Bubble growth in bulk Fe is briefly discussed here to give a comparison for growth at twist grain boundaries. The bulk sample is an 11.4 nm sided cube of 1.28×105 Fe atoms. The bubble seed is a He substitutional made by replacing one Fe atom with He. As already described, all subsequent He atoms are introduced at the same location without removing additional Fe atoms. The seed is chosen to anchor the location of the bubble since the binding energy of Hevacancy is large and diffusion is slow[25]. The helium loading rate is 1 He / 2 ps.

9

The growth of the helium bubble is characterized by a sharp initial spike in pressure and subsequent emission of ½ interstitial loops (Figure 5) that are roughly the size of the bubble at the time of emission. The emission of these loops significantly relieves the pressure in the bubble. Closer inspection of the pressure data shows that the pressure continues to drop inside the bubble immediately after the interstitial loop is emitted. The bubbles show faceting in agreement with Haghighat, et al. [21], yet never reach the He densities reported in that paper, the Fe matrix yields at lower pressure values. Figure 6 shows an inverse relationship between the pressure inside the bubble and the average atomic volume. Even though the atomic volume appears to vary inversely with the pressure, the total bubble volume (Figure 6) displays approximately linear growth for each additional He atom. Since the total volume is linear, the average atomic volume must be roughly constant. This implies that the small fluctuations about this constant value account for the relatively large fluctuations in pressure. This process is similar at the twist boundary with a few important differences discussed later.

Figure 5: Early bubble evolution in bulk Fe. The large initial spike in pressure is peaked at 10 He followed immediately by a deformation of the surrounding lattice and a drop in pressure (ΔP 10He-11He = 6.3 GPa). At 99 He the bubble reaches a local minimum in pressure of 27.2 GPa through the formation of a ½ interstitial loop and begins the pressure build-up and release cycle.

10

Figure 6: Pressure and average atomic volume inside a He bubble in pure Fe (left) and approximately linear volume growth of a single He bubble in pure Fe (right). Notice that after about 100 He atoms the pressure and average volume per atom curves are mirror images of each other, as one would expect.

The high pressure values shown in Fig. 6 for small bubbles (in excess of 50 GPa) produce substantial distortion of the lattice, as can be seen in the lattice atoms shown around the 10 He atom bubble in Figure 5. At these small bubble sizes, the high pressure produces only distortions, while at larger sizes we observe the emission of loops. These He pressures should not be compared with lattice strength; a single He interstitial has a pressure of ~ 60 GPa, producing a displacements of atoms around it. Larger bubble sizes are needed for the lattice to yield by emission of interstitial loops. Another point to highlight is that the high pressure within the bubble will have consequences on He diffusion in the matrix, creating a diffusion barrier that may prevent further precipitation of He atoms. While we do not address this point here, preliminary MD simulations of interstitial He migrationg towards bubbles show that this barrier exists but is not strong enough to prevent He precipitation. Analysis of the dislocations in bulk Fe using the DXA software reveals that the ½ dislocation loops may react to give dislocations. These dislocations are immobile and pin the dislocation loops. Furthermore, loops may interact as they are being formed at the bubble’s surface (Figure 7), in which case the loops cease to be ejected into the bulk and have a diminished 11

role in helping relieve the bubble pressure. The fact that no loops are emitted after 413 He is most likely due to nearby loops pushing back on the new loop trying to form and finite size effects preventing new dislocations from leaving the bubble. This could also explain how a dislocation forms at the bubble’s surface which acts to pin newly created loops at the surface. Therefore, the pressure inside an actual bubble may exhibit an even further decrease in pressure after 400 He if the formation and emission of loops is uninhibited. It is important to note that the observed emission of loops corresponds to an extremely high loading rate, 1 He atom every 2 ps, and that for lower rates other mechanisms such as emission of individual interstitials may be observed ([26] and [27]).

Figure 7: He bubble (red and blue surface) in bulk Fe emitting ½ dislocation loops (dark blue). A dislocation (red) formed at the surface that effectively pins all new loops to the surface and reacts with any new dislocations being emitted. One such reaction is the formation of a dislocation segment (light blue).

He Bubble Growth at Grain Boundaries Our simulations on twist GB samples show similar pressure and volume behavior to that of the bulk sample, but markedly different micro-structural evolution. First, we note that bubble pressures seem insensitive to the degree of rotation. There is virtually no difference in pressure among the different samples

12

and He loading schemes, which indicates no junction-junction interactions. The two twist samples diverge in behavior when all four junctions are loaded in the 2° sample and four junctions from a total of sixteen are loaded in the 6° sample (Figure 8). This is due the distortion of the screw dislocation network in the 2° sample (Figure 9). When all junctions are loaded the screw dislocation network is so severely distorted that the bubbles no longer go through the cycle of building up and sharp drop in pressure (Figure 8). This is especially apparent in the 6° sample when all sixteen junctions are loaded. After most of the bubbles coalesce, the pressure drops sharply with no build-up cycle afterward. This is mainly due to the bubbles effectively “cutting” the sample in half after the bubbles coalesce so there is much less He-Fe interaction. Similar, but less dramatic, effects were observed in the simulations where four junctions were loaded in both samples. Volume vs. number of He atoms curves (Figure 10) for most samples tested show an approximate linear behavior throughout the entire simulation. The samples that deviate from linearity are the samples where all screw dislocation junctions are loaded and where GB restructuring plays a significant role.

Figure 8: Pressure curves for the 2° twist sample for various He loading schemes (left). The 4-bubble simulation shows different behavior because all the junctions were loaded which caused severe distortion of the screw dislocation network. Pressure curves for the 6° twist sample for various He loading schemes (right). The 4-bubble simulation shows slightly different behavior because of bubbles migration within the

13

dislocation network while the 16-bubble simulation experienced extreme bubble coalescence, which dramatically altered the sample. (curves are smoothed for clarity)

Figure 9: View of the 2° twist boundary before and after He loading (all four junctions). Notice that the screw dislocation network is completely disrupted by the formation of the bubbles and dislocation loops.

Figure 10: Total bubble volume versus number of He atoms for most of the simulations in this paper. The linear fit is given for reference.

Dislocations formed at the grain boundary near the bubble are not seen to enter the bulk due to interactions with the screw dislocation network (Figure 11).

14

The formation of these dislocations still acts to relieve the pressure in the bubble, but not as effectively as in bulk.

Figure 11: View of a bubble (large red and blue surface) at the 2° twist boundary emitting ½ loops (dark blue). The bubble and screw dislocation network are centered for clarity. A dislocation (red) forms due to thermal interactions between the screw dislocations (light blue) and is not related to bubble activity.

Comparison to Bulk Now we compare helium bubble growth in the bulk to growth at twist grain boundaries. The pressure curves show a stark contrast below about 100 He atoms (Figure 12), after which they all exhibit virtually the same behavior. The pressure of the bubble in bulk increases to enormous pressures but then falls drastically before reaching 100 He atoms. Figure 13 shows the pressure profile for a bubble with 490 He atoms grown in bulk and a bubble grown at the grain boundary. The bubbles are of comparable size and density but show a slight difference in pressure. This is likely due to both the difference in volume per atom and different shape of the bubbles in bulk versus grain boundary. The bubbles at the grain boundary deform more easily with additional He than in bulk Fe. Also, the bubbles at the grain boundary are less spherical than within bulk, which causes more Fe-He interactions and alters the pressures of the bubbles. These

15

observations suggest that kinetics of growth for small bubbles may differ substantially between bulk and GB’s.

Figure 12: Comparison of pressures of a single bubble for 2°, 6°, and bulk samples within the domain that they exhibit the most contrast.

Figure 13: Comparison of pressure profiles for a bulk bubble and a bubble at a 2° twist boundary. (curves are smoothed for clarity)

16

Conclusions We studied He precipitation in Fe bulk and at twist grain boundaries. Using Metropolis Monte Carlo we find the preferred sites for precipitation to be the screw dislocation junctions. Using molecular dynamics we load the sample with He and analyze the growth mechanisms by which the Fe matrix creates room to accommodate the bubble. The preferred mechanism at a He loading rate of 1 He/ps is the emission of interstitial dislocation loops. The loops in bulk simulations leave the bubble while those emitted by bubbles at the grain boundary remain attached to the dislocation network. We find that bubbles of all sizes grown under conditions of He-rich vacancy poor environments never reach densities above 2 He/Vac, where the matrix reaches its limit strength and yields, to create room for the growing bubble. The pressure increases significantly with bubble size for small sizes, more so in bulk than at GB’s. We find a structure inside the bubble that reflects the influence on the energetics and stresses of the He-Fe interaction, creating a core-shell-gap structure with the shell and gap dimensions related to the strength of the He-Fe interaction. This structure affects the pressure-density relation in a way that for small bubbles it departs significantly from the EOS of He. We note that if He loading rate was reduced, we may observe the lattice accommodating additional He through other mechanism (possibly emission of Fe-interstitials). Our results from the two Fe samples rotated in the (001) plane show that the initial bubble pressure is relatively insensitive to small differences in twist angle with a slightly lower initial pressure in the 2° sample. The comparison between bulk Fe and twist boundaries show that the initial bubble pressure in bulk is substantially higher than at the twist grain boundaries, making bubble formation at the twist boundary more likely (Figure 12). The deviation of bubble pressure from the equation of state is explained by the He-Fe interaction. For instance, when the 6° sample is loaded with helium at all sixteen junctions, the bubbles coalesce and reduce its collective surface area and, hence, minimize the Fe-He interaction. This leads in a collective bubble pressure very close to that of the equation of state for helium at that density. All samples that do not 17

have all their junctions loaded show almost identical behavior after about 100 He atoms are introduced into the bubble. In all cases where the screw dislocation network stays relatively intact, bubble volume grows approximately linearly with the number of helium atoms regardless of bubble pressure or shape at a rate of about .007 nm3/He (Figure 10). The interstitial loops emitted by the bubbles behave differently in bulk and at grain boundaries. Loops emitted in bulk try to maximize their distance from the bubble while loops emitted at twist boundaries immediately interact with the screw dislocation network and never enter the bulk. This interaction between the dislocation loops and screw dislocation network severely distorts the grain boundary. Furthermore, if the bulk has a high density of bubbles and dislocation loops, dislocations may be prevented from leaving the bubbles surface and could react to form dislocation locks.

Acknowledgments Work performed with support from the Center for Materials at Irradiation and Mechanical Extremes, an Energy Frontier Research Center funded by the U.S. Department of Energy (Award Number 2008LANL1026) at Los Alamos National Laboratory, and with support from the Laboratory Directed Research and Development Program.

References 1 Bloom EE. The challenge of developing structural materials for fusion power systems. Journal of Nuclear Materials . 1998 OCT;258:7-17. 2 Braski DN, Schroeder H, Ullmaier H. The effects of tensile stress on the growth of helium bubbles in an austenitic stainless steel. Journal of Nuclear Materials. 1979;83:265-277. 3 Mansur LK, Coghlan WA. Mechanisms of helium interaction with radiation

18

effects in metals and alloys: A review. Journal of Nuclear Materials. 1983;119:1-25. 4 Morishita K, Wirth BD, de la Rubia TD, Kimura A. Effects of Helium on Radiation Damage Processes in Iron. 4th Pacific Rim International Conference on Advanced Materials and Processing. 2001 1-4. 5 Caro A, Hetherly J, Stukowski A, Caro M, Martinez E, Srivilliputhur S, ZepedaRuiz L, Nastasi M. Growth mechanisms of He bubbles in Fe and FeCr alloys (submitted). 2011. 6 Zhang L, Shu X, Jin S, Zhang Y, Lu GH. First-principles study of He effects in a bcc Fe grain boundary: site preference, segregation and theoretical tensile strength. Journal of Physics: Condensed Matter. 2010;22:375401. 7 Heinisch HL, Gao F, Kurtz RJ. Atomistic modeling of helium interacting with screw dislocations in alpha-Fe. Journal of Nuclear Materials. 2007;367370:311-315. 8 Heinisch HL, Gao F, Kurtz RJ, Le EA. Interaction of helium atoms with edge dislocations in a-Fe. Journal of Nuclear Materials. 2006;351:141-148. 9 Shim JH, Kwon SC, Kim WW, Wirth BD. Atomistic modeling of the interaction between self-interstitial dislocation loops and He in bcc Fe. Journal of Nuclear Materials. 2007;367-370:292-297. 10 Fu CC, Willaime F. Interaction between helium and self-defects in a-iron from first principles. Journal of Nuclear Materials. 2007;367-370:244-250. 11 Farrell K, Maziasz PJ, Lee EH, Mansur LK. Modification of radiation damage microstructure by helium. Radiation Effects. 1983;78:277-295. 12 Wang YX, Xu Q, Yoshiie T, Pan ZY. Effects of edge dislocations on interstitial helium and helium cluster behavior in a-Fe. Journal of Nuclear Materials. 2008;376:133-138. 13 Yang L, Zu XT, Wang ZG, Gao F, Wang XY, Heinisch HL, Kurtz RJ. Interaction of helium–vacancy clusters with edge dislocations in a-Fe. Nuclear Instruments and Methods in Physics Research B. 2007;265:541-546.

19

14 Kurtz RJ, Heinisch HL, Gao F. Modeling of He–defect interactions in ferritic alloys for fusion. Journal of Nuclear Materials. 2008;382:134-142. 15 Heinisch HL, Gao F, Kurtz RJ. Atomic-scale modeling of interactions of helium, vacancies and helium-vacancy clusters with screw dislocations in alpha-iron. Philosophical Magazine. 2010;90:885-895. 16 Caro A, Crowson DA, Caro M. Classical Many-Body Potential for Concentrated Alloys and the Inversion of Order in Iron-Chromium Alloys. Physical Review Letters. 2005;95:075702. 17 Ackland GJ, Mendelev MI, Srolovitz DJ, Han S, Barashev AV. Development of an interatomic potential for phosphorus impurities in α-iron. Journal of Physics: Condensed Matter. 2004;16:S2629. 18 Juslin N, Nordlund K. Pair potential for Fe–He. Journal of Nuclear Materials. 2008;382:143-146. 19 Beck DE. A new interatomic potential function for helium. Molecular Physics. 1968;14(4):311-315. 20 Schaublin R, Chiu YL. Effect of helium on irradiation-induced hardening of iron: A simulation point of view. Journal of Nuclear Materials. 2007;362:152160. 21 Haghighat SM, Lucas G, Schaublin R. State of a pressurized helium bubble in iron. EPL. 2009;85:60008. 22 Rycroft CH, Grest GS, Landry JW, Bazant MZ. Analysis of granular flow in a pebble-bed nuclear reactor. Physical Review E. 2006;74:021306. 23 Stukowski A, Albe K. Extracting dislocations and non-dislocation crystal defects from atomistic simulation data. Modelling and Simulation in Materials Science and Engineering. 2010;18:085001. 24 Sadigh B, Erhart P, Stukowski A, Caro A, Martinez E, Zepeda-Ruiz L. A scalable parallel Monte Carlo algorithm for atomistic simulations of precipitation in alloys (submitted). 2010. 25 Terentyev D, Juslin N, Nordlund K, Sandberg N. Fast three dimensional

20

migration of He clusters in bcc Fe and Fe-Cr alloys. Journal of Applied Physics. 2009;105:103509-1-12. 26 Trinkaus H, Wolfer WG. Conditions for dislocation loop punching by helium bubbles. Journal of Nuclear Materials. 1984;122 & 123:552-557. 27 Trinkaus H. Energetics and formation kinetics of helium bubbles in metals. Radiation Effects. 1983;78:189-211.

21