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http://www.diva-portal.org Preprint This is the submitted version of a paper published in Journal of Nuclear Materials. Citation for the original published paper (version of record): Chang, Z., Dmitry, T., Sandberg, N., Samuelsson, K., Bonny, G. et al. [Year unknown!] Assessment of the dislocation bias in fcc metals and extrapolation to austenitic steels. Journal of Nuclear Materials Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Permanent link to this version: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-163299 Assessment of the dislocation bias in fcc metals and extrapolation to austenitic steels Zhongwen Changa , Nils Sandberga,b , Dmitry Terentyevc , Karl Samuelssona , Giovanni Bonnyc , Pär Olssona a KTH Royal Institute of Technology, Reactor Physics, Roslagstullsbacken 21, SE-106 91 Stockholm, Sweden b Swedish Radiation Safety Authority, Solna Strandväg 96, SE-171 16 Stockholm, Sweden c SCK-CEN, Nuclear Materials Science Institute, Boeretang 200, B-2400 Mol, Belgium Abstract A systematic study of dislocation bias has been performed using a method that combines atomistic and elastic dislocation-point defect interaction models with a numerical solution of the diffusion equation with a drift term. Copper, nickel and aluminium model lattices are used in this study, covering a wide range of shear moduli and stacking fault energies. It is found that the dominant parameter for the dislocation bias in fcc metals is the width of the stacking fault ribbon. The variation in elastic constants does not strongly impact the dislocation bias value. As a result of this analysis and its extrapolation, the dislocation bias of the widely applied austenitic stainless steels of 316 type is predicted to be about 0.1 at temperature close to the swelling peak (815 K) and typical dislocation density of 1014 m−2 . This is in line with the bias calculated using the elastic interaction model, which Email addresses: [email protected] (Zhongwen Chang), [email protected] (Pär Olsson) Preprint submitted to Journal of Nuclear Materials February 13, 2015 implies that the prediction method can be used readily in other fcc systems even without EAM potentials. By comparing the bias values obtained using atomistic- and elastic interaction energies, about 20% discrepancy is found, therefore a more realistic bias value for the 316 type alloy is 0.08 in these conditions. Keywords: Dislocation bias, Atomistic calculation, Interaction energy, fcc 1 1. Introduction 2 Irradiation of metals can significantly alter their properties such as di- 3 mensional stability. Since void swelling was first discovered under neutron 4 irradiation in 1967 [1], intensive efforts have been applied to characterize and 5 understand the mechanisms behind its emergence. The preferential absorp- 6 tion of self interstitials (SIA) at dislocations, first suggested by Greenwood, 7 Foreman and Rimmer [2], has been incorporated in rate theory models for 8 swelling as a possible driving force for radiation induced dimensional change. 9 As the SIAs are absorbed more efficiently at dislocations than vacancies, a 10 net excess number of vacancies is accumulated in the bulk and either con- 11 dense as new voids or increase the volume of existing voids by flowing into 12 them. The higher absorption rate is caused by the stronger attraction of 13 an SIA to a dislocation as compared with a vacancy. The parameter that 14 characterizes the difference in the absorption efficiency is the bias factor. 15 Various swelling models have been constructed based on the micro struc- 16 ture evolution under irradiation. The first and probably still the most pop- 2 17 ular model is so-called the standard rate theory (SRT) model, based on the 18 concept of sink bias [3, 4, 5]. It is formulated within the framework of the 19 mean field type chemical reaction rate theory. The model implies that the 20 irradiation produces only Frenkel pairs created evenly in space and time, and 21 voids are neutral sinks absorbing both vacancies and SIAs equally. The main 22 driving force for swelling, therefore, is the dislocation bias. A more system- 23 atic and detailed model, the Bullough, Eyre and Krishan (BEK) model [6], 24 was formulated on the extension of the SRT model. It took into account the 25 vacancy loops produced by vacancy emission and biased interstitial absorp- 26 tions. In this model, the dislocation bias (Bd ) is still the dominant driving 27 force for the irradiation-induced void swelling [4]. To model the effects of 28 high energetic neutron irradiation, the more sophisticated production bias 29 model [7, 8, 9] has been proposed. It characterises the damage production 30 and annihilation more accurately than the previous two models because it 31 incorporates generation of mobile SIA clusters known to be produced directly 32 in displacement cascades. The 1D migrating SIA clusters play an important 33 role in this model, and the dislocation are biased in absorption of both SIAs 34 and mobile SIA clusters. 35 The dislocation bias factor is thus an essential parameter for the present 36 computational models for void swelling. The study of the bias factor is mo- 37 tivated by both the fundamental scientific interest and technological need 38 to tailor candidate materials for the high swelling resistance as required for 39 the next generation of nuclear reactors. To assess the propensity in regards 3 40 to void swelling of those candidate structural materials, the computational 41 evaluation of dislocation bias is one of the first steps to be done before the 42 actual irradiation testing. However, the dislocation bias factors vary signifi- 43 cantly with crystalline structure, material composition, irradiation tempera- 44 ture and dislocation density. Therefore an efficient and repeatable approach 45 is required to evaluate the dislocation bias since the conditions change con- 46 tinuously under irradiation. Various analytical studies based on elasticity 47 theory have been carried out [10, 11, 12, 13] to evaluate the dislocation bias, 48 however as Wolfer pointed out [14], the frequently used isotropic elastic the- 49 ory is not enough to describe the elastic interactions between dislocation and 50 migrating defects, and the near-dislocation-core interaction from continuum 51 elasticity theory is insufficient. Furthermore, the dislocation bias of an alloy 52 is not available from a purely analytical approach. 53 In our previous work [15], a method that combines the interaction en- 54 ergy from atomistic calculations and the bias calculation from a numerical 55 finite element method (FEM) was shown to be an improvement of the an- 56 alytical method, and it gives a reasonable prediction of the dislocation bias 57 in fcc Cu. A similar method has been also used recently to study the effect 58 of anisotropy, SIA orientation, and one-dimensional migration mechanism 59 on the bias of edge dislocations in bcc Fe and fcc Cu [16]. However, the 60 calculation of atomistic interaction energy in an alloy are hindered by the 61 development of the alloy semi-empirical embedded atom method (EAM) po- 62 tential and by the complexity of local chemical composition. In the present 4 63 work, three representative face centered cubic (fcc) model lattices are chosen 64 for a systematic study of the dislocation bias in fcc crystals. Atomistic sim- 65 ulations with empirical potentials are applied to map the dislocation-point 66 defect (PD) interaction energy and a numerical solution using the finite el- 67 ement method is obtained for the diffusion equation in order to estimate 68 capture efficiencies and the dislocation biases in Cu, Ni and Al. By ma- 69 nipulating the anisotropic elastic interaction models (The elastic interaction 70 models are always assumed to be anisotropic in this paper, unless otherwise 71 stated), a systematic study is performed in order to evaluate the impacts of 72 the stacking fault energy (SFE) and elastic constants on the dislocation bias. 73 The dislocation bias of a typical austenitic steel is then predicted by extrapo- 74 lating the results obtained for pure fcc metals and taking the experimentally 75 known elastic properties and SFE. 76 2. Theory and Methods 77 A detailed description of the methods employed in this work is presented 78 in [15]. The main idea is to solve the diffusion equation with a drift term 79 numerically using an interaction map which describes the interaction profile 80 of dislocation and PDs. The diffusion equations are solved by applying the 81 finite element method. The capture efficiency is defined as the ratio of PD 82 fluxes with and without interaction with the dislocation, i.e. Z = 83 flux of PDs including the interaction with the dislocation and J0 is the flux 84 excluding the interaction. The dislocation bias is defined as Bd = 5 J . J0 J is the ZSIA Zvac − 1. 85 By using atomistic simulations we obtained the interaction energy be- 86 tween a dislocation and PDs in an fcc metal without suffering from the lim- 87 itations imposed by elasticity theory. Comparison of atomistic and elastic 88 interactions shows about the reliability of the elastic description. The elastic 89 interaction model is built following the detailed description in our previous 90 work [15]. All elastic constants used in the models are determined by the 91 EAM potentials via molecular static calculations. The elastic interaction en- 92 ergies are then used to obtain the dislocation bias following the same method- 93 ology as used for the atomistic interaction energies. It worth noticing that 94 for the analytical solution in the framework of linear elasticity theory, a dis- 95 location in a fcc crystal is usually treated as a single core line with a0 /2h110i 96 Burgers vector, even though the dislocations in fcc metals are characterized 97 by splitting into two partials. To mitigate this problem, a two partial dislo- 98 cation model is constructed by superimposing the interaction energy of two 99 individual dislocation cores each with a Burgers vector of a0 /4h110i, sepa- 100 rated by a distance which is the same as in the atomistic interaction energy 101 profiles. This ignores the screw component of each partial [17], but since that 102 component mainly induces shear deformations in the lattice, it is expected 103 to interact only weakly with the PDs. 104 In the atomistic simulations, the computational model containing an edge 105 dislocation is set up by misfitting two half crystal lattices so that the upper 106 one contains an extra half plane, as described in detail in [18]. They join along 107 the dislocation slip plane and thus an edge dislocation with b = a0 /2h110i is 6 108 generated in the center. The simulation boxes in Cu, Ni and Al are about 109 70a0 ∗ 7a0 ∗ 76a0 in the [110], [-11-2] and [-111] directions respectively. In 110 order to model an infinite straight dislocation, periodic boundary conditions 111 were applied in the direction of the Burgers vector and in the direction of 112 the dislocation line, while a fixed boundary conditions were applied in the 113 direction that is normal to the glide plane. A typical dislocation density in the 114 simulation cell in this case is 1.5 · 1015 m−2 . A combination of the conjugate 115 gradient and quasi static relaxation with constant volume was applied to 116 relax the crystal and obtain the equilibrium structure of the dislocation. 117 A vacancy is created by removing one atom from the lattice. An SIA is 118 inserted as a dumbbell containing two atoms aligned along {100} directions 119 and placed at a distance of 0.2 a0 from each other, centred on a lattice site. 120 Given the three different orientation of the dumbbells, we performed calcu- 121 lations on each configuration, and used the average of these three obtained 122 energies as input for the bias calculation. The interaction energy is defined 123 as the difference between the formation energy of a PD with and without a 124 dislocation. The interaction maps are calculated by positioning PDs on each 125 lattice site. 126 Large scale molecular statics calculations were performed using the DYMOKA 127 code [19]. Full interaction energy landscapes around the dislocation core for 128 PDs were obtained using EAM potentials for Cu [20], Ni [21] and Al [22]. 129 The potentials reproduce the properties of defects in the bulk crystal in good 130 agreement with reference data obtained from experiments and ab initio cal7 131 culations as shown in Tab.1. 132 Due to the splitting of the dislocation in an fcc lattice, the dislocation-PD 133 interaction range is relatively large, hence the fixed and periodic boundary 134 conditions should be carefully treated. The artificial contribution to the 135 interaction energy originated from the strain induced by the fixed boundary 136 conditions has been removed from the atomistic interaction energy maps as 137 described below. 138 Atoms located near the fixed atomic layers can not fully relax thus in- 139 troducing non-physical strain, which in turn affects the interaction energy. 140 Several positions were chosen along the direction normal to the dislocation 141 glide plane to compute the PD formation energy with the strained lattice 142 constants. Later, these data are used for the correction that removes the 143 impact of the fixed boundary conditions. To eliminate the contribution to 144 the interaction energy from the image dislocations, the isotropic elastic in- 145 teraction model is applied to create the two neighbouring image dislocations 146 whose contribution is correspondingly subtracted. 147 To obtain the bias numerically from FEM, it is unavoidable to deal with 148 the integration area which is denoted as the core region of the dislocation. 149 Inside the core boundary, the PDs are assumed to be absorbed and therefore 150 the PD concentrations are zero. A dislocation is usually seen as a cylinder for 151 simplification and the core radius is regarded as a variable in the previous 152 bias calculations [23]. We have studied the impacts of the choice of the 153 core geometry in a previous work [15]. In this work, considering that one 8 154 integration circle around both partial cores may not be representative for 155 a large partial dislocation splitting such as in the austenitic alloy, we use 156 two circles to represent the two partial dislocation core regions separately. 157 To assign some physical meaning to the dislocation core radius we used an 158 interaction energy gradient threshold [16]: bO|E| ≥ kB T (1) 159 The radii determined by this criterion are different for different defect species 160 and different interaction profiles. In our calculations for Cu, Ni and Al, 161 the radii in atomistic interaction energies are 12 Å and 6 Å for SIAs and 162 vacancies, respectively, while 8 Å and 4 Å are used in elastic interaction 163 maps for SIAs and vacancies, respectively. 164 In order to study the influence of dislocation densities on the bias calcu- 165 lation, different dislocation densities were generated by expanding the region 166 described by the atomistic interaction and matching it to the anisotropic elas- 167 tic solution in the outskirts. In this manner the near core region is described 168 as accurately as possible while at the same time one can obtain dislocation 169 densities on the same order of magnitude as in technological materials. 9 170 3. Results 171 3.1. Interaction energies 172 The interaction energies of PDs with an edge dislocation have been cal- 173 culated in Cu, Ni and Al. The comparison between anisotropic elastic and 174 atomistically obtained interaction energy map reveals that the elastic descrip- 175 tions of the atomistic features in the dislocation core region is insufficient, as 176 shown in Fig.1, Fig.2 and Fig.3 for Cu, Ni and Al, respectively. In these fig- 177 ures, sub-plots A and B are, respectively, atomistic- and elastic interaction of 178 dislocation and SIAs. C and D represent the vacancy-dislocation interaction 179 in the atomistic- and elastic models. The difference between A and B, and 180 between C and D are shown in E and F, respectively, in order to have a more 181 detailed view of where the divergence emerge. The difference attributed to 182 the insufficient description of the elastic core model. 183 In the atomistic calculations, the dislocation splits into two partials fol- 184 lowing the energy minimization in accordance with Frank’s rule. In copper 185 the stacking fault energy is ESF =44.4 mJ/m2 . The splitting distance result- 186 ing from the stacking fault is calculated to be 30 Å according to elasticity 187 theory [24] d = 188 is the Poisson ratio and Esf is the stacking fault energy. In our atomistic 189 calculations, the positions of the two partials are determined by identifying 190 atoms with maximal energies, which would occur in the dislocation core. 191 This gives a distance of 35 Å between the two partials. In the case of nickel, 192 the ESF =113 mJ/m2 which corresponds to a splitting distance of 19 Å from Gb2 (2+ν) 8π(1−ν)Esf where G is shear modulus, b is Burgers vector, ν 10 193 theoretical calculation while 22 Å is found from the atomistic calculations. 194 For Al, the ESF =129.4 mJ/m2 which leads to a partial distance of 9 Å while 195 the calculated distance is 14 Å. We consider these results to be in acceptable 196 agreement and the regular underestimation of the stacking fault ribbon is 197 due to the insufficiency of the isotropic elasticity theory. 198 3.2. Bd calculations and predictions 199 The bias factors computed using the atomistic interaction energies, ac- 200 counting for the boundary conditions and image dislocations, are shown in 201 Fig.4. The results corresponding to the dislocation density of 1014 m−2 are 202 obtained for the temperature range 603 – 1000 K. At the same temperature 203 and dislocation density, BdAl >BdNi >BdCu is observed. The dislocation bias, 204 meanwhile, is proportional to the swelling rate of the material according to 205 the SRT model. Under this presumption, these results suggest that copper 206 should exhibit a lower swelling rate than nickel and aluminium under the 207 same irradiation conditions. This is in agreement with neutron irradiation 208 experiments described in [25] that shows that nickel is more prone to irradi- 209 ation induced swelling as compared to copper. An analysis based on electron 210 irradiation data [26] also indirectly suggests a larger bias for nickel than for 211 copper. The Bd calculated using atomistic interactions are about 20% lower 212 than these using the elastic interaction energies, which shows the inaccuracy 213 of the elastic interactions used to obtain the dislocation bias. This shows, 214 however, the opposite trend comparing to our previous work [15], where the 11 215 atomistic interaction energies result in higher dislocation bias compared to 216 that using the elastic interaction energies. The reason stems from the choice 217 of the dislocation core radius. In the previous work, the same dislocation 218 radii are used for the integration while in the present work, the criterion 219 of Eq.1 is used and the radii are thus different for atomistic- and elastic 220 interaction energies. The criterion in the present work is better motivated 221 comparing to the arbitrary choice in the previous work. 222 To assess the impacts of elastic constants and the partial splitting dis- 223 tances on the bias calculations, the elastic constants of Cu, Ni and Al, as well 224 as variable partial core distances are used in the elastic model to simulate ma- 225 terials with different SFEs, considering that the SFE is the major component 226 in determining the partial splitting distance. As shown in the inset figures in 227 Fig.5, the elastic constants of Cu are used to generate the elastic interaction 228 model with partial distances of 14 Å, 22 Å and 35 Å, respectively. The bias 229 factors are calculated correspondingly at the temperature of 815 K and 1000 230 K with the dislocation density of 1014 m−2 . At both temperatures, the bias 231 decreases as the partial distances increase. The same trend is observed when 232 the elastic constants of Ni and Al are used. Comparing the Bd calculated 233 with interactions that are generated using Cu, Ni and Al elastic constants 234 at the same partial distance, it is seen that BdCu >BdNi >BdAl in the defined 235 temperature range. At d = 35 Å, the Bd calculated using elastic constants 236 of Cu is about 7% larger than that using elastic constants of Ni while the Bd 237 calculated using elastic constants of Ni is about 6% larger than that of Al. In 12 238 these constructed interaction models, all calculation parameters are the same 239 except the elastic constants used to describe the interaction. Therefore, in 240 this case, the difference in Bd originates only from the variation of the elastic 241 constants. To identify the most important elastic properties in determining 242 the Bd , an empirical parameter [B/G] is selected in order to obtain an ap- 243 proximately linear relation related to the dislocation bias factors, where B 244 is the bulk modulus and G is the shear modulus. Given that the interaction 245 energy of a dislocation and a SIA is determinant in the bias calculation, the 246 capture efficiency ZSIA is further studied as a function of the unitless param- 247 eter [B/G]. This parameter takes into account that the interaction energy is 248 decided by the relaxation volume in the isotropic elastic interaction model, 249 and the relaxation volume can be seen as a balance between compressing 250 the two central atoms and their neighbours, and shearing of the surrounding 251 crystal. As shown in Fig.6, by constructing elastic interactions with 14, 22 252 and 35 Å partial distances, and using the elastic constants from Ni, Al and 253 Cu, the ZSIA s are proportional to the empirical parameter [B/G]. The values 254 of this parameter for Ni, Al and Cu are marked on the x-axis. For typical 255 austenitic alloys, the [B/G] value is in the range of 2 and 2.3 as shown in 256 the Tab.2. Therefore the capture efficiency ZSIA for the alloy is supposed 257 to located about 1% below Ni in Fig.6. The typical range is marked on the 258 x-axis in the figure. 259 As it is seen from Fig.5, the difference in Bd induced by employing dif- 260 ferent elastic constants are much less pronounced compared to that induced 13 261 by the variance of the partial distance. Further studies are made by extend- 262 ing the partial distances and calculating the corresponding Bd with elastic 263 interactions constructed using the elastic constants from Ni since Ni locates 264 closest to the austenitic alloy in Fig.6 and it is the austenitizer in austenitic 265 alloys. When the partial distance is small enough, the two partials collapse 266 back to a single core dislocation. The single core dislocation bias is relatively 267 large: around 35% as shown in Fig.7. As the partial distance increases, 268 the calculated Bd decreases and converges to a certain level. The reason is 269 probably that as the partial distance increases, the overlap effect on lattice 270 sites decreases, therefore the interaction energies around the partial cores 271 decrease. Since the diffusion potential is an exponential function of the in- 272 teraction energy, it gets weaker as a consequence of the partial separation. 273 When the two partial cores are far enough from each other, they are seen 274 as twice the dislocation densities with half the Burgers vector on each par- 275 tial dislocation, comparing to the case when partial distance is zero with 276 one strong dislocation core. At a large partial distance, the decrease of Bd 277 due to SFE decrease is eliminated by the increase of Bd due to the disloca- 278 tion density increase. From Fig.7 the Bd converges to about 0.1 after the 279 partial distance reaches 100 Å. The converged value 0.1, compared to the 280 non-splitting full core value 0.35, has been lowered with a factor of about 3. 281 This implies that the stacking fault distances plays a more significant role 282 than the elastic properties when assessing dislocation bias factors. 283 With the above analysis, it is possible to predict the dislocation bias on 14 Gb2 4πEsf [24], an approximate value for 284 the austenitic alloys. Using equation d = 285 equilibrium separation is calculated to be 127 Å by inserting G=77.5 GPa, 286 b=2.5 Å and Esf = 30 mJ/m2 . The parameters are shown in Tab.2. As seen 287 in Fig.7, an equilibrium distance of 127 Å corresponds to a bias factor about 288 0.1 when using the elastic interaction model with the elastic constants of Ni. 289 When we substitute the elastic constants of Ni with that of an austenitic 290 alloy, as discussed already, the Bd might be about 1% lower than it is using 291 Ni elastic constants, which is negligible. Therefore, the bias factor calculated 292 using the elastic interactions for austenitic alloy is estimated to be 0.1. The 293 tolerance of this prediction is relatively high since the separation distances of 294 austenitic alloys lie on the plateau region in Fig.7. To benchmark the predic- 295 tion, a FeNiCr alloy EAM potential [21] is used to construct a Fe-10Ni-15Cr 296 alloy. The elastic constants calculated from the EAM potential are used in 297 the elastic interaction model to obtain the interaction energies of this alloy. 298 This is used to estimate the bias factor and we obtain a value of 0.093 with an 299 equilibrium partial distance of 104 Å, which is in agreement with the above 300 prediction value of 0.1. This implies that even without a proper EAM poten- 301 tial, by applying the elastic constants and SFE obtained from experiments, 302 it is possible to predict the dislocation bias for a fcc alloy. However, the 303 predicted values are obtained by using only the elastic interaction energies. 304 From the comparison of Bd values using atomistic- and elastic interaction 305 models in Cu, Ni and Al, the dislocation bias are about 20 % higher using 306 the elastic interaction energies than using the atomistic ones. Therefore, the 15 307 more realistic dislocation bias in the alloy should be about 20 % lower than 308 what we predicted using the elastic interaction energies. This results in an 309 estimation of Bd ≈ 0.08 in the actual austenitic alloy at the temperature of 310 815 K and a dislocation density of 1014 m−2 . 311 4. Conclusions 312 In this work, an efficient and easily reproducible approach is proposed 313 to perform a systematic study of the dislocation bias factors in fcc Cu, Ni 314 and Al model lattices. The atomistic interaction energies between an edge 315 dislocation and point defects are calculated and applied to obtain the dis- 316 location bias factor for the three model lattices. The results are compared 317 with the bias calculated using anisotropic interaction models. It is found 318 that BdAl >BdNi >BdCu at the same temperature and dislocation density, which 319 is in agreement with experiments. 320 The elastic models are applied to study the fundamental parameters that 321 influence the Bd values by changing the elastic constants and the partial dis- 322 location distances in the anisotropic model. The results show that the Bd 323 is more sensitive to the change of equilibrium partial dislocation separation 324 distances than to the change of elastic constants, regardless of temperatures. 325 As the separation distance gets larger, the bias tends to converge. When the 326 two partial cores are far enough from each other, they will act as indepen- 327 dent dislocations with half the Burgers vector, but in a system with twice 328 the dislocation density, compared to the case when partial distance is zero. 16 329 Therefore the bias factor tends to increase. However, the trend is balanced 330 by the tendency of decrease induced by larger separation distance. 331 By estimating the partial dislocation separation of the austenitic alloy, a 332 prototype of 316 stainless steel, we predict the dislocation bias to be about 333 0.1 at temperature close to the swelling peak (815 K) and typical disloca- 334 tion density of 1014 m−2 . This value is in agreement with the dislocation 335 bias calculated from numerical FEM using the elastic interaction model. By 336 taking into account the overestimation of the bias induced by using elastic 337 interaction energies, a more realistic bias value of 0.08 is predicted under the 338 same conditions. 339 In this study, we have shown that the SFE has an important effect on 340 dislocation bias because it is related to the equilibrium splitting of partials. 341 However, we have not considered the effect of the SF interface itself on defect 342 capture. There is a possibility that the SF surface either facilitates of hinders 343 defect diffusion to the partial cores, and it may also serve as a recombination 344 area for PDs, thereby influencing the resulting bias of the dislocation. These 345 aspects are left to future studies. 346 Acknowledgements 347 This work is supported by the national project on Generation IV reactor 348 research and development (GENIUS) in Sweden, by the Göran Gustafsson 349 Stiftelse and by the European Atomic Energy Community’s (Euratom) Sev- 350 enth Framework Programme FP7/2007-2013 under grant agreement No.604862 17 351 (MatISSE project). This work contributes to the Joint Program on Nuclear 352 Materials (JPNM) of the European Energy Research Alliance (EERA). The 353 Swedish National Infrastructure for Computing (SNIC) sources have been 354 used for part of this work. 355 [1] C. Cawthorne, E. Fulton, Nature 216 (1967) 575–576. 356 [2] G. Greenwood, A. Foreman, D. Rimmer, J. Nucl. Mater. 1 (1959) 305– 357 324. 358 [3] A. Brailsford, R. Bullough, J. Nucl. Mater. 44 (1972) 121–135. 359 [4] C. Woo, B. Singh, A. Semenov, J. Nucl. Mater. 239 (1996) 7–23. 360 [5] S. Golubov, B. Singh, H. Trinkaus, Philos. Mag. 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Mater. 328 (2004) 107–114. 21 Figure 1: Edge dislocation – point defect interaction energies for the different approaches in Cu model lattice, A) Atomistic SIA; B) Elastic SIA; C) Atomistic vacancy; D) Elastic vacancy; E) Difference between A and B; F) Difference between C and D. 22 Figure 2: Edge dislocation – point defect interaction energy maps for the different approaches in Ni model lattice, A) Atomistic SIA; B) Elastic SIA; C) Atomistic vacancy; D) Elastic vacancy; E) Difference between A and B; F) Difference between C and D. 23 Figure 3: Edge dislocation – point defect interaction energy maps for the different approaches in Al model lattice, A) Atomistic SIA; B) Elastic SIA; C) Atomistic vacancy; D) Elastic vacancy; E) Difference between A and B; F) Difference between C and D. 24 Al Atomistic Al Elastic Cu Atomistic Cu Elastic Ni Atomistic Ni Elastic Dislocation bias factor Bd 0.3 0.25 ρd=10 14 -2 (m ) 0.2 0.15 0.1 600 700 800 900 1000 Temperature (K) Figure 4: Temperature dependence of Bd for the atomistic and elastic cases at the dislocation density of 1014 m−2 . 25 Figure 5: Bd varies with partial dislocation separation distances. The Bd are calculated from anisotropic elastic interaction models using elastic constants from Cu, Ni and Al. The insets depict the interactions calculated using elastic constants from Cu with a separation of 14 Å, 22 Å and 35 Å respectively. 26 1.32 d=3.5 nm d=2.2 nm d=1.4 nm 1.28 ZSIA ρd=10 1.24 14 -2 (m ) T=815 K 1.2 1.16 2 Austenitic alloy Ni 2.5 Al 3 Cu 3.5 B/G Figure 6: ZSIA as an approximate linear function of an empirical parameter B/G at different partial separation distances. Typical B/G values for austenitic alloy is marked on the x-axis. 27 0.35 815 K Cij=Cij(Cu) 815 K Cij=Cij(Ni) 815 K Cij=Cij(Al) 0.3 1000 K Cij=Cij(Cu) 1000 K Cij=Cij(Ni) 1000 K Cij=Cij(Al) Bd 0.25 0.2 14 ρd=10 -2 (m ) 0.15 0.1 0 50 100 150 200 Partial distance (Å) Figure 7: Bd calculated from constructed elastic model as a function of partial distances at 815 K and 1000 K and the dislocation density of 1014 m−2 . 28 Table 1: Fundamental parameters of Cu, Ni, Al from EAM potentials. Cu Ni Al EAM Ref EAM Ref EAM Ref c a a0 (Å) 3.615 3.615 3.519 3.519 4.032 4.032c for Evac (eV) 1.27 1.27f 1.48 1.79a 0.68 0.66m for g k E<100>SIA (eV) 3.063 2.8-4.2 4.08 4.08 2.68 2.59l Ecoh (eV/at.) 3.54 3.54d 4.45 4.45a 3.36 3.36j ESF (mJm−2 ) 44.4 45e 113 128b 129.4 144i Notes: The Ref values are from experimental measurements (marked as bold) and other calculations. a is from [27]; b is from [28]; c is from [29]; d is from [30]; e is from [31]; f is from [32]; g is from [33]; h is from [34]; i is from [35]; j is from [36]; k is from [37]; l is from [38]. 29 Table 2: Fundamental parameters of Cu, Ni, Al and typical austenitic alloy. Cu Ni Al Austenitic alloy ∆V volume) -0.3 -0.07 -0.4 -0.2a ∆V volume) 1.8 1.2 2.1 1.4a 1.5e G (GPa) 41 75 29 70a 75b 77c B (GPa) 138 180 79 160a 157b 159c Tm /2 (K) 815 1000 603 973 Esf (mJ/m2 ) 44 113 129 18a 30d Notes: 316 type alloy is used as a representative for the austenitic alloy here. a is from the EAM potential; b is from [39], c is from [40], d is from [41]. e is from [42]. Note that the compositions are slightly different in the different references. Relax (atomic vac Relax SIA (atomic effective 30

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