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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
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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
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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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