Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling
Fe-Cr alloys are the basis of high-Cr ferritic steels, which are the candidates for the structural materials for near future power plants. Recently, a significant effort has been put in the development of theoretical models dealing with the response of Fe-Cr alloys to irradiation. Here, we give a...
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irk-123456789-963382016-03-16T03:02:06Z Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling Terentyev, D. Olsson, P. Fe-Cr alloys are the basis of high-Cr ferritic steels, which are the candidates for the structural materials for near future power plants. Recently, a significant effort has been put in the development of theoretical models dealing with the response of Fe-Cr alloys to irradiation. Here, we give a brief overview of the current level of understanding of radiation damage in Fe-Cr alloys, based on the most recent results. In particular, we review and summarize data obtained using different atomistic modelling techniques in order to refine the most important findings achieved over the past few years. Сплави Fe-Cr становлять основу феритних сталей с високим вмістом Cr, що являються кандидатами у конструкційні матеріали для атомних електростанцій найближчого майбутнього. В останній час значні зусилля спрямовані на розробку теоретичних моделей, які розглядають реакцію Fe-Cr-сплавів на опромінення. В даній роботі наводиться короткий огляд сучасного рівня розуміння радіаційних пошкоджень у сплавах Fe-Cr, при цьому за основу беруться останні на даний момент результати. Зокрема, наводиться огляд та підводиться підсумок даних, що отримані за допомогою різних методів атомістичного моделювання з метою визначення найбільш важливих результатів, які досягнуті за останні кілька років. Сплавы Fe-Cr составляют основу ферритных сталей с высоким содержанием Cr, являющихся кандидатами в конструкционные материалы для атомных электростанций ближайшего будущего. В последнее время значительные усилия направлены на разработку теоретических моделей, рассматривающих реакцию сплавов Fe-Cr на облучение. В настоящей работе дается краткий обзор современного уровня понимания радиационных повреждений в сплавах Fe-Cr, при этом в основу берутся последние на данный момент результаты. В частности, дается обзор и подводится итог данных, полученных с помощью различных методов атомистического моделирования с целью определения наиболее важных результатов, достигнутых за последние несколько лет. 2009 Article Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling / D. Terentyev, P. Olsson // Вопросы атомной науки и техники. — 2009. — № 4. — С. 68-79. — Бібліогр.: 35 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/96338 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Fe-Cr alloys are the basis of high-Cr ferritic steels, which are the candidates for the structural
materials for near future power plants. Recently, a significant effort has been put in the
development of theoretical models dealing with the response of Fe-Cr alloys to irradiation. Here,
we give a brief overview of the current level of understanding of radiation damage in Fe-Cr
alloys, based on the most recent results. In particular, we review and summarize data obtained
using different atomistic modelling techniques in order to refine the most important findings
achieved over the past few years. |
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Article |
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Terentyev, D. Olsson, P. |
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Terentyev, D. Olsson, P. Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling Вопросы атомной науки и техники |
| author_facet |
Terentyev, D. Olsson, P. |
| author_sort |
Terentyev, D. |
| title |
Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling |
| title_short |
Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling |
| title_full |
Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling |
| title_fullStr |
Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling |
| title_full_unstemmed |
Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling |
| title_sort |
aspects of radiation damage effects in fe-cr alloys from the point of view of atomistic modeling |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/96338 |
| citation_txt |
Aspects of radiation damage effects in Fe-Cr alloys from the point of view of atomistic modeling / D. Terentyev, P. Olsson // Вопросы атомной науки и техники. — 2009. — № 4. — С. 68-79. — Бібліогр.: 35 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT terentyevd aspectsofradiationdamageeffectsinfecralloysfromthepointofviewofatomisticmodeling AT olssonp aspectsofradiationdamageeffectsinfecralloysfromthepointofviewofatomisticmodeling |
| first_indexed |
2025-07-07T03:31:58Z |
| last_indexed |
2025-07-07T03:31:58Z |
| _version_ |
1836957427920011264 |
| fulltext |
ASPECTS OF RADIATION DAMAGE EFFECTS
IN Fe-Cr ALLOYS FROM THE POINT OF VIEW
OF ATOMISTIC MODELING
D. Terentyev
Institute of Nuclear Materials Science, SCK CEN, Boeretang 200, B2400, Mol.;
P. Olsson
Département Matériaux et Mécanique des Composants, EDF R&D,
Les Renardières, F-77250 Moret-sur-Loing, France
E-mail: dterenty@sckcen.be
Fe-Cr alloys are the basis of high-Cr ferritic steels, which are the candidates for the structural
materials for near future power plants. Recently, a significant effort has been put in the
development of theoretical models dealing with the response of Fe-Cr alloys to irradiation. Here,
we give a brief overview of the current level of understanding of radiation damage in Fe-Cr
alloys, based on the most recent results. In particular, we review and summarize data obtained
using different atomistic modelling techniques in order to refine the most important findings
achieved over the past few years.
1. INTRODUCTION
High-Cr ferritic/martensitic steels are
candidates for structural materials for a large
number of future nuclear applications, from
fusion to fission accelerator-driven systems
and Gen IV reactors (e.g. [1] and references
sited therein). Independently of the specific
reactor concept, a reliable guide for the
assessment of the behaviour of these steels
in operation is the quantitative understanding
of the basic physical mechanisms acting
from the atomic to the macroscopic level and
determining the response of the material to
the applied mechanical, thermal and
environmental loads, under neutron
irradiation. The establishment of this
knowledge is beneficial for the safe
operation and design of most future nuclear
installations. Fe-Cr alloys are the basis of
high-Cr ferritic steels, so significant effort
has been put, in the development of multi-
scale models dealing with the response of
Fe-Cr alloys to irradiation aimed at
providing physical insight, rather than
parameters of design interest. In this paper,
we give a brief overview of the current level
of understanding of radiation damage in Fe-
Cr alloys, based on the most recent results
obtained by different modelling techniques.
2. EXPERIMENTAL OBSERVATIONS
SHOWING EFFECTS OF CR ON THE
PROPERTIES OF IRRADIATED
Fe-Cr ALLOYS
Experiments performed over the past thirty
years clearly show that the addition of Cr to Fe
influences significantly the response of the
alloy to irradiation, non-monotonically with
respect to the Cr content. One of the most
pronounced effects of Cr is the suppression of
void swelling, known since the late 1970 s [2].
This was explained by the trapping of point
defects by Cr atoms, which consequently
enhances the point defect recombination [2].
Another explanation proposed is the reduction
of the dislocation network mobility due to the
presence of <100> loops, whose population
depends on CCr [3]. Indeed, the ratio ½<111>-
to-<100> loops was found to increase with
CCr, so that the microstructure of pure Fe and
low Cr alloys is mainly composed of <100>
loops [4]. In addition, in the alloys with Cr
content large enough for the α-α' phase
separation to occur, further reduction of
swelling was observed, tentatively attributed to
the role of Cr-rich precipitates acting as
perfect sinks for point defects [5]. The
mechanism of swelling reduction based on
68 Серия: Физика радиационных повреждений и радиационное материаловедение (94), с. 68-79.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2009. №4-1.
trapping of point defects imply that swelling
should decrease with Cr content
monotonically, whereas experimental
measurements show a more complicated
dependence. Swelling in neutron irradiated (up
to dose ~30 dpa) Fe-Cr binary alloys is
presented in Fig. 1,a. At 380 °C, swelling
decreases with the addition of Cr and remains
low for concentrations between 1 and 10% and
then increases again in alloys with higher CCr.
At higher temperatures (400-460 °C), the
minimum of swelling shifts towards lower CCr
and is located around 3...6% Cr. But at the
same time, swelling is also reduced in alloys
containing more than 10% Cr, resulting in a
local maximum at ~10% Cr. Further increase
of temperature up to 615 °C leads to the
restoration of the minimum around 10%.
A remarkable effect of Cr is also seen in the
reduction of the shift in the ductile-to-brittle
transition temperature (ΔDBTT) in both high-
Cr steels and binary alloys. ΔDBTT was found
to reach a minimum around 9% Cr [6], in a
range of irradiation temperatures 300…410 °C
and doses 7-36 dpa. Recent experiments on
Fe-Cr binary alloys, irradiated in the BR2
reactor at 300 °C up to 1.5 dpa, also
demonstrate that radiation-induced hardening
(Δσy), shown in Fig.1,b, is higher in Fe-Cr
alloys and saturates at higher dose, than in
pure Fe [7]. Again, the dependence of Δσy on
CCr is not regular: it is larger than in Fe,
remains almost independent of Cr content with
a minimum at ~9% Cr, and then increases
again [7]. The increase of Δσy above 9% Cr is
qualitatively understandable in terms of α-α’
phase separation [8], however, no explanation
why there is an increase of the ΔDBTT and
Δσy below 9% Cr so far exists. Both swelling
and hardening are closely related with the
microstructure formed during irradiation. In
the case of moderate doses (i.e. few dpa, so
relevant for ΔσY shown in Fig.1,c), the main
defects detected using transmission electron
microscopy (TEM) are interstitial dislocation
loops. Experiments clearly show that the
presence of even small percentages (0.1%) of
Cr in ultra-pure Fe enhances the nucleation of
interstitial type dislocation loops, with a
subsequently higher radiation-induced
hardening [9]. Enhancement of ½<111> loop
density in Fe10%Cr irradiated by electrons at
25 °C has been also reported [10]. Finally, in-
situ TEM studies show higher thermal stability
of ½<111> loops in ultra pure Fe9%Cr than in
Fe [11]. The density of <100> loops formed
after severe neutron irradiation also depends
on CCr as shown in Fig.1,c [5].
0 5 10 15 20
0.01
0.1
1
10
S
w
el
lin
g
%
Cr content %
380°C
400°C(*)
460°C
615°C
3 6 9 12 15 18
0.01
0.1
1
10
0
10
20
30
40
50
N
um
be
r d
en
si
ty
(m
-3
) *
1
021
Cr content %
Number density
Mean radius
M
ea
n
ra
di
us
(n
m
)dose 140 dpa
TIRR=425°C
0 2 4 6 8 10 12
100
150
200
250
300 1.5 dpa
0.6 dpa
0.06 dpa
Δ
σ Y
(M
Pa
)
Cr content %
TIRR=300°C
(a) (b) (c)
Fig. 1. Swelling in the Fe-Cr alloys irradiated up to 30 dpa in the DFR fast reactor [2], (*)
refers to the alloys irradiated in BR-10 fast reactor up to 26 dpa [12] (a). Change in the yield
stress in the alloys irradiated in BR-2 at 300 °C up to dose 0.06-1.6 dpa [7] (b). Density and
mean size of <100> dislocation loops observed in the alloys irradiated in FFTF/MOTA at
425 °C up to 140 dpa [5] (c)
Understanding of such an interplay of the
radiation-induced swelling, hardening,
embrittlement and microstructure features
significantly varying with Cr content is
required. The role of atomistic modelling is to
envisage mechanisms involved in radiation
phenomenon which otherwise can not be
rationalized directly (or even indirectly) from
experiments. These are, for example, primary
damage, migration of point defects and their
69
nano-clusters, interaction of dislocations with
dislocation loops, etc. In other words,
processes occurring at the space and time scale
inaccessible on the experimental level. Here,
we summarize recent findings achieved using
different modelling techniques and discuss
implications of these findings.
3. RECENT PROGRESS IN
ATOMISTIC SIMULATIONS
3.1. Mixing enthalpy of Fe-Cr alloys
Experimental measurements of short range
order (SRO) parameter, performed after
thermal ageing and under irradiation [13],
revealed that it changes sign from negative to
positive at ~10% Cr. This implies that Cr
atoms repel each other in the alloys with
CCr<10%, thus providing a large solubility
limit, and tend to segregate only if CCr exceeds
~10%. The physical origin of this phenomenon
was recently explained based on results from
density functional theory (DFT) calculations,
showed that the mixing enthalpy (ME) of
random Fe-Cr alloys also changes sign from
negative to positive at ~8% Cr, with a
minimum around 5% Cr [14]. The origin of
the negative ME and repulsion between Cr
atoms is a two fold band structure effect [35]
and the so-called 'magnetic frustration' effect
[15]. The former explains the fact that the
substitution of Cr in Fe has a negative heat of
mixing by the effective lowering of the
occupation at the Fermi level as Cr is
introduced. The latter is because each Cr atom
favours its magnetic moment antiparallel to the
host Fe atoms as well as to Cr first nearest
neighbours. In dilute alloys, the most
energetically favourable situation is obtained
when the Cr atoms are distributed sufficiently
far from each other to avoid magnetic
frustration, which induces SRO and even
probably long range order in the alloys
containing 6.25 and 3.7 % Cr as proposed in
[16, 17]. By increasing Cr concentration, Cr-
Cr interactions cannot be avoided anymore and
this leads to a positive heat of mixing,
resulting in the Cr precipitation or α-α' phase
separation.
Based on the data obtained with DFT, for
the first time, the central-force many body
potential allowing to reproduce correct heat of
mixing of Fe-Cr system was developed [18].
The cross-potential describing Fe-Cr
interaction allows for the effect of not only d-
band, as standard EAM potential does, but also
s-band electrons, thus successfully
reproducing the heat of mixing from DFT [18].
The potential therefore was called the 'two-
band' model, so we will refer to as the 2BM
potential. The Fe-Fe part of the potential is
from [19], which is a significant improvement
over previous models (see [19] for details)
from the point of view of properties of self-
interstitial atoms (SIA) and their small clusters
as compared to DFT [20].
3.2. Primary damage in Fe-Cr alloys
and pure Fe
A complete comparative study of
displacement cascades in random Fe-Cr alloys
(with CCr=5,10 and 15%) and pure Fe
simulated with 2BM potential can be found in
[21]. It has been shown that no significant
difference in the primary damage state
between Fe and Fe-Cr alloys exists in terms of
number of defects produced, clustered
fractions and cluster size distributions. In Fe-
5%Cr and Fe-10%Cr alloys, SIAs and SIA-
clusters were found to be bound to Cr atoms.
Most of single SIAs appear as mixed <110>
dumbbells. Small <110> SIA clusters,
containing up to 5 defects, were not 'enriched'
by Cr. Concentration of Cr inside larger <111>
SIA clusters, is greater by about 3 and
1.5 times than in Fe-5%Cr and Fe-10%Cr
matrices, respectively. Enrichment of SIA
defects by Cr was not detected in Fe-15%Cr.
Formation of Cr-Cr dumbbells was almost
never observed in all studied alloys. A very
small but statistically meaningful change of
the SRO parameter in Fe-5%Cr and Fe-15%Cr
was detected, whereas no change occurred in
Fe-10%Cr. The change to negative SRO
parameter in Fe-5%Cr was due to the break-up
of the Cr-Cr first nearest neighbour (1st nn)
pairs, while changes to positive SRO
parameter in Fe-15%Cr was due to the
formation of Cr clusters during solidification
of the molten region.
The details of the different arrangements of
Cr atoms near interstitial defects are, however,
of minor importance for the post-cascade long-
term evolution. The main message received
from the comparative study of cascades in Fe
and Fe-Cr alloys is that the number density
70
and size distribution of defects produced are
not affected by the presence of Cr.
Observation of the cascade-induced Cr
ordering and clustering is potentially important
but it is not clear for the moment if this effect
could be as effective as phase changes driven
by cascade-induced defects.
3.3. Mobility of point defects and their
clusters in Fe and Fe-Cr alloys
The mobility of point defects and their
clusters is one of the most important
parameters determining the microstructure
evolution in irradiated metals. Thus, the
precise knowledge of migration properties of
vacancy and interstitial defects versus defect
size is a fundamental prerequisite for any
model intended to describe microstructure
evolution under irradiation. One of the
conventional ways to study migration
properties of elementary point defects and
their small clusters in metals is to perform
measurements of resistivity by during
isochronal annealing of pre-irradiated samples.
Resistivity recovery studies performed in
dilute and concentrated alloys (up to 15% Cr)
have shown that the recovery indeed depends
on the Cr content (for review and most recent
result see [22]). The recovery stages associated
with the onset of a single SIA and Di-SIA
cluster (in pure Fe) were found to be
significantly affected by Cr in concentrated
alloys. Whereas, the stage attributed to the
vacancy long-range migration remains
unchanged. This was interpreted in terms of no
influence of Cr atoms on the vacancy
migration. Similar conclusion was drawn
based also on results of positron annihilation
spectroscopy. At the same time, DFT studies
have also shown that the interaction of Cr
atoms with a single vacancy in Fe is negligibly
small (for review and the most recent results
see [23]). Comparative MD study of vacancy
migration in pure Fe and Fe-12%Cr have
shown that the difference between migration
energy estimated in pure metal and in the
alloys is only 0.07 eV (lower in alloy) [24].
We therefore focus on the study of the
mobility of SIA defects, since these are mainly
seen to be affected by Cr in experiments.
3.3.1. Classification of SIA clusters
Prior to considering the effect of Cr on the
mobility of SIA defects, it is useful to give a
brief overview of the recent DFT and MD
findings with respect to the properties of SIA
defects in pure Fe. DFT studies suggest that a
single SIA and SIA clusters with size up to
five defects occupy a <110> configuration
(preferred to <111>) and therefore while
migrating follow a 3D path by performing
translation-rotation jumps via movement of
individual <110> dumbbells [20, 25]. The
ground state for larger SIA clusters is a platelet
of <111> crowdions which consequently
exhibit 1D-motion [26]. Using accurate DFT
calculations, the migration energy of a single
SIA was found to be 0.33 eV, and this was
also well reproduced by the 2BM potential
[27]. The migration energies of 3D moving
SIA clusters were estimated to vary from 0.15
up to 0.4 eV using both static DFT
calculations [20] and MD simulations with the
2BM potential [25]. MD study of the mobility
of <111> SIA clusters with the 2BM potential
has shown that the migration energy weakly
depends on the cluster size and it was
estimated to be ~0.05 eV [25]. It is therefore
important to keep in mind that even relatively
large (up to few tens of defects) <111> SIA
clusters exhibit fast 1D motion in contrast to
small 3D-migrating SIA clusters.
3.3.2. Interaction of SIA with Cr atoms
in dilute solution
A thorough study of the interaction of a
single SIA and Cr atoms by DFT and with the
2BM potential in pure Fe was performed in
[23, 27]. It has been found that single SIAs
interact strongly and mainly attractively with
Cr atoms. The strength and sign of the
interaction depends on the local Cr
distribution, as well as on the configuration of
the SIA, which could be a <110> dumbbell or
<111> crowdion, as shown in Fig.2. A
dumbbell is bound to an isolated Cr atom
therefore forming a mixed (i.e. Fe-Cr)
dumbbell and the energy required to separate a
Cr and dumbbell is about 0.1 eV (Fig.2, #1).
The presence of a second Cr nearby an Fe-Cr
dumbbell may increase the binding energy up
71
to 0.15 eV (Fig.2, #2) or results in a repulsive
interaction (Fig.2, #5), depending on the
particular arrangement of Cr atoms. The
highest repulsion (0.42 eV) occurs when the
two Cr atoms form a dumbbell (Fig.2, #5).
This repulsion stems from the effect of
magnetic frustration determining Cr-Cr
repulsion in Fe. The binding energy of a
<111> crowdion to an isolated Cr (Fig.2, #4)
was found to be much higher than that with a
dumbbell. However, the formation energy of
an Fe-Cr crowdion is still higher than that of
an Fe-Cr dumbbell (the excess formation
energy is 0.4 eV), so the most stable
configuration of an SIA is the mixed dumbbell
[23, 27].
Interstitial – Cr
Vacancy – Cr
0.10.21#6
0.0070.057#7
-0.00530.014#8
0.330.42#5
-0.4-0.37#4
#3
#2
#1
-0.19-0.15
-0.05-0.15
-0.14-0.08
PotentialDFT
Interstitial – Cr
Vacancy – Cr
0.10.21#6
0.0070.057#7
-0.00530.014#8
0.330.42#5
-0.4-0.37#4
#3
#2
#1
-0.19-0.15
-0.05-0.15
-0.14-0.08
PotentialDFT
#1 #4
#2
#3
#5
#6
#7 #8
SIA-Cr Interaction
Vacancy-Cr Interaction
Interaction energy (eV)
Fig. 2. Interaction between Cr atoms
and point defects [27]. Filled circle
denotes a Cr atom, empty cube denotes
a vacancy
The mobility of a single SIA in random Fe-
Cr alloys (with CCr=1-15%) was studied by
MD simulations [27]. The results obtained
have shown that an SIA migrates via
translation-rotation jumps as in pure Fe and
this mechanism is shown in Fig.3,a. The SIA
migration energy, Em(SIA), estimated from the
Arrhenius slope was seen to slightly decrease
with Cr content (e.g. in Fe-15%Cr was
reduced by 0.1 eV), as compared to the pure
Fe, where Em(SIA)=0.31 eV. As a
consequence of this, at low temperature the
SIA appears to migrate faster in Fe-Cr alloys.
At the same time, a reduction of the SIA
diffusivity at high temperature was expressed
in the prefactor, found to decrease
monotonically with Cr content. The migration
of the SIA in concentrated alloys was found to
be irregular i.e. the time delay before a jump
differed significantly, sometimes reaching few
nano-seconds in simulations performed at
300 °C. This was explained by the presence of
some low energy states for the SIA, producing
a trapping effect [27]. The migration of the
SIA in the alloys with CCr not exceeding 10%
preferentially occurred via movement of the
mixed dumbbell, thus mainly contributing to
the Cr mass transport. The preferential motion
of the mixed dumbbell is not only determined
by the binding energy, but also due to the
lower migration barrier, as can be seen from
Fig. 3 [28]. The formation of the Cr-Cr
dumbbell was never observed in alloys with
CCr up to 10%. Very rarely, the formation of
Cr-Cr dumbbells was seen in Fe-15%Cr [27].
The results of MD simulations suggested
the presence of some deep traps for an SIA,
whose concentration seemed to increase with
Cr content [27]. As has been already
mentioned the Cr-SIA interaction depends
significantly on the local atomic environment,
so that in concentrated alloys the free energy
of the formation of an SIA constantly changes
while the latter is migrating in the lattice.
Static calculations were performed to provide
the characterization of possible traps for SIAs
in the concentrated alloys [27].
72
Em=0.23 Em=0.33 Em=0.42
[001]
[110]
(a) (b) (b)
Em=0.23 Em=0.33 Em=0.42
[001]
[110]
(a) (b) (b)
(c
Fig. 3. Migration energy and path of the mixed [110] Fe-Cr dumbbell into: (a) [101] Fe-Cr
dumbbell, (b) [101] Fe-Fe dumbbell and (c) [110] Fe-Cr dumbbell, as obtained by DFT
calculations [28]. A filled circle denotes a Cr atom
2 3 4 5 6 7 8 9 10
1E-5
1E-4
1E-3
0.01
0.1
1
1E-5
1E-4
1E-3
0.01
0.1
1
EFe=6.23 eV EFe=8.87 eVEFe=3.55 eV
Tri - SIADi - SIA
Fe-1Cr
Fe-5Cr
Fe-10Cr
Fe-15Cr
P
(E
i)
Formation energy (eV)
Single SIA
Fig. 4. Distribution of the formation energy of single SIA, di- and tri-SIA clusters in Fe-Cr alloys
of different content, estimated by static calculations with the 2BM potential in [29]
The distribution of the formation energy
probability, P(Ef), was calculated for the
random alloys with CCr=1-15%. P(Ef) for
single SIA, di-SIA and tri-SIA in Fe-Cr alloys
is plotted in Fig. 4 [29]. The formation
energies of the same defects, in their lowest
energy configurations in pure Fe, are shown as
vertical lines in the figure. With increasing Cr
content P(Ef) broadens showing the presence
of the low-energy states with probability ~10-3
and lower. Based on this data, the strength and
concentration of the strongest traps was
estimated for the SIA defects in the alloys, as
proposed in [27] and summarized in Tabl. 1.
The latter shows that the trapping energy
varies from 0.2 to 0.8 eV, depending on a
defect size, generally increasing with CCr.
Table 1
Trap concentrations (atomic fraction) and energies (eV) for 3d-migrating sia defects
in Fe-Cr alloys [29]
Defect Quantity Fe-1%Cr Fe-5%Cr Fe-10%Cr Fe-15%Cr
Ctrap 2.8·10-3 8.6·10-4 8.6·10-5 1.6·10-3
Single SIA
Etrap 0.2 0.3 0.5 0.3
Ctrap 5.7·10-3 6.9·10-4 1.3·10-3 1.1·10-2
Di-SIA
Etrap 0.2 0.4 0.4 0.4
Ctrap 1.5·10-3 7.7·10-4 1.9·10-4 6.25·10-4
Tri-SIA
Etrap 0.4 0.6 0.8 0.8
73
3.4. Effect of Cr on the diffusivity
of 1D-migrating SIA defects
The strong crowdion-Cr binding energy,
found by DFT and confirmed by calculations
using the 2BM potential [27], transfers to the
SIA clusters as well. Crowdions in <111> SIA
clusters exhibit binding energy which varies
from 0.4 to 0.15 eV depending on the position
of a crowdion in the cluster habit plane and the
size of a cluster. The strongest interaction
occurs at the cluster edge and the weakest in
the center. The Cr-SIA cluster binding energy,
estimated for crowdions at the edge of a
cluster, decreases with cluster size down to
~0.15 eV, which is close to the binding energy
of the Cr and the perfect edge dislocation.
MD studies of the mobility of <111> SIA
clusters were performed in random Fe-Cr
alloys with different Cr content using the 2BM
potential [30]. A significant reduction in the
diffusivity of <111> SIA clusters was
observed in Fe-Cr alloys as compared to pure
Fe. The actual value of the diffusivity
reduction depends on three parameters:
temperature, cluster size and Cr content. The
most convenient way to represent obtained
MD results is to give the ratio between the
diffusion coefficients in Fe-Cr and in Fe
(Rn=Dn
Fe-Cr/Dn
Fe, where n is the size of the
cluster). The Cr concentration dependence of
Rn, estimated at 350 °C, is shown in Fig 5,a.
The left-hand axis of Fig. 5,a shows the
change in the SIA cluster free energy, ΔFMD,
as defined in [31] as:
)/ln(k FeFeCr
B
MD
nnn DDTF =Δ . (1)
The reduction of the diffusivity is strong
and non-linear: R7 reaches a minimum 10-2 at
10% Cr, meaning that the diffusion coefficient
is a hundred times smaller in Fe-10%Cr than
in pure Fe. With increasing the cluster size, the
minimum diffusivity shifts towards smaller Cr
concentration and Rn increases, e.g. R91=0.1 at
2%Cr.
0 5 10 15 20 25 30
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
Δ
F n0 (e
V
)
Cr concentration (at%)
Cluster size
7 SIAs
19 SIAs
37 SIAs
61 SIAs
91 SIAs
0 5 10 15 20
-0.4
-0.3
-0.2
-0.1
0.0
0.1
10-3
10-2
10-1
100
Δ
F nM
D
(e
V
)
Cr concentration (at.%)
Cluster size
7 SIAs; 37 SIAs; 91 SIAs
D
nFe
C
r /D
nFe
(a) (b)
0 5 10 15 20 25 30
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
Δ
F n0 (e
V
)
Cr concentration (at%)
Cluster size
7 SIAs
19 SIAs
37 SIAs
61 SIAs
91 SIAs
0 5 10 15 20
-0.4
-0.3
-0.2
-0.1
0.0
0.1
10-3
10-2
10-1
100
Δ
F nM
D
(e
V
)
Cr concentration (at.%)
Cluster size
7 SIAs; 37 SIAs; 91 SIAs
D
nFe
C
r /D
nFe
(a) (b)
Fig. 5. Reduction in the diffusion coefficient and variation of the free energy of <111> SIA clusters in
Fe-Cr alloys (at T=350 °C) as compared to pure Fe, estimated using (a) MD and (b)
static calculations [30]
In order to rationalise obtained MD results
the physical process of SIA cluster migration
in Fe-Cr was modelled by expressing the
reduction of the diffusion coefficient, Rn, in
terms of the average binding energy bE seen
by the cluster due to the presence of Cr for a
given solute concentration [31]. The model
implies that the cluster moves in a field of
energy hills and valleys, determined by the
changing local atomic environment. The
decrease of the cluster diffusivity comes from
the increase of the time the cluster spends in
the different positions, as compared with the
pure crystal, which is proportional to
)/exp( Bb TkE , where Bk is Boltzmann's
constant and T is the absolute temperature.
The diffusion coefficient of the cluster in Fe-
Cr, DFe-Cr, then could be expressed knowing
the corresponding in pure Fe, DFe, as:
TkF
n
TkE
nn
BB eDDD /Fe1/FeFeCr be Δ− =〉〈= , (2)
74
where the averaging is performed over
different cluster positions in the system. ΔF
defined in Eqs. (1) and (2) and estimated using
static simulations is presented in Fig. 5, b. It
has been therefore shown that the model
developed in [31] clearly succeeds in capturing
the basic cause for SIA cluster slowing down
in Fe-Cr alloys, compared to pure Fe, not only
qualitatively, but also quantitatively. These
results are however, only valid for the random
alloys. It can be seen from Fig. 5, b that the
saturation of the Cr-crowdion interaction
occurs for CCr>12%. In this range of
concentrations the α-α' phase separation may
occur.
3.5. Interaction of SIA defects with Cr
precipitates
Fe-Cr alloys as well as high-Cr ferritic
steels containing more 10% Cr exhibit α-α’
phase separation, which results in the
formation of finely dispersed Cr-rich
precipitates. These are coherent with the
matrix and contain about 95% Cr.
Experimental measurements show that the
precipitates may exhibit extremely high
density (up to 1024 m-3) [8] at the onset of the
phase separation and therefore may play an
important role in the microstructure evolution
if they somehow affect mobility of radiation
defects.
Table 2
Formation energies (eV) of sia defects in pure chromium for 〈110〉 and 〈111〉 configurations.
Data for pure fe is given in brackets [20]
Method EIP calculations [29] Ab initio calculations [28]
Defect 〈110〉 config. 〈111〉 config. 〈110〉 config. 〈111〉 config.
Ef (SIA) 5.59 (3.59) 5.62 (4.01) 5.68 (3.75) 5.76 (4.45)
Ef (di-SIA) 10.31 (6.23) 9.55 (6.68) 10.24 (6.68) 10.11 (7.43)
Ef (tri-SIA) 14.78 (8.87) 13.91 (9.46) 14.72 (9.49) 14.08 (10.01)
Ef (vacancy) 2.56 (1.71) 2.71(2.15*) [23]
The interaction of SIA defects with Cr-rich
coherent precipitates was recently studied in
[29]. Where the corresponding formation
energies of a single SIA and di- and tri-SIAs in
Fe-Cr, estimated using DFT and 2BM were
reported and are summarized in Tabl. 2.
Comparison of these results with the data
presented in Fig. 2, reveals that the formation
or migration of SIA-type defects in the Cr
precipitates is strongly unfavourable, because
of the large excess of the formation energy.
The excess of the formation energy of a
vacancy is about 0.5 eV. However, a vacancy
was found to be slightly bound to the Cr
precipitate (the latter is oversized), with the
energy ~0.1 eV. This is, however, much less
than the variation of the formation energy of a
vacancy in the concentrated Fe-Cr alloys,
estimated to vary from 0.2 to 0.4 eV in Fe-
5%Cr and Fe-10%Cr, respectively. Based on
these results, one expects that the Cr
precipitates will act as 'scattering' centers for
3D-migrating SIA defects and not affect their
mobility. Weak attractive vacancy-precipitate
interaction will be effectively 'screened' due to
the presence of dispersed Cr atoms in the
depleted matrix around Cr precipitates. The
formation and accumulation of both
interstitials and vacancies inside the Cr
precipitates is excluded.
The interaction of <111> SIA defects with
the Cr precipitates in Fe10Cr matrix was
studied using static and MD simulations in
[30]. MS calculations revealed the presence of
the strong repulsive interaction occurring
when the <111> SIA cluster approaches the
precipitate. The maximum repulsive energy
was found when the <111> SIA cluster is
located at the precipitate-matrix interface. The
origin of the observed strong repulsion is the
large formation energy excess of <111> SIAs,
as has been already discussed in the previous
section. By increasing the precipitate size, the
energy barrier increases as well. The typical
values of the repulsive energy of <111> SIA
clusters (with sized varied from 7 to 331
75
defects) reported in [30] for precipitates with
size of 1.5-3 nm, vary from 5 up 30 eV. Such
energy barriers are not expected be
surmounted with the help of thermal
fluctuations. Thus large 1D migrating clusters
are expected to be pinned (or immobilized)
between Cr precipitates.
4. MOBILITY OF RADIATION
INDUCED SELF-INTERSTITIAL
DEFECTS IN Fe-Cr ALLOYS:
SUMMARY
Gathering results and discussion presented
in sections 3.3; 3.4 and 3.5, we provide a brief
summary of the most important effects of Cr
on the mobility of SIA defects and their
clusters in dilute and concentrated Fe-Cr
alloys.
3D-migrating (i.e. single SIAs and small
SIA clusters containing up to defects):
• in dilute Fe-Cr alloys, can be trapped by
single Cr and pairs of Cr atoms, with binding
energies not exceeding 0.2 eV. In addition, the
migration energy of the mixed Fe-Cr dumbbell
is about 0.1 eV lower than that of the <110>
Fe-Fe dumbbell;
• in concentrated Fe-Cr alloys, may
encounter configurational traps involving a
few Cr atoms, with binding energies in the
range of about 0.2-0.8 eV. The concentration
and the strength of these traps strongly depend
on the Cr content;
• are not bound to the Cr precipitates and
are not expected to accumulate inside them.
The Cr precipitates therefore do not affect
mobility of these defects.
1D-migrating interstitial clusters:
• with increasing Cr concentration, their
diffusivity first decreases and then increases
again. The diffusivity reduction exhibits a
minimum at a certain Cr content. The depth
and critical concentration at which the
minimum occurs depends on the cluster size.
• with size of about 7-30 SIAs exhibit the
minimum diffusivity at 6-10% Cr;
• with increasing cluster size, the
minimum diffusivity shifts towards lower
concentrations and stays at ~1 %Cr for small
dislocation loops (i.e. 2-3 nm);
• are not bound to Cr precipitates and are
not expected to penetrate inside them. The Cr
precipitates therefore cause an effective
pinning of these clusters and hence
dramatically reduce their mobility.
5. IMPLICATIONS OF THE
RESULTS PROVIDED BY
ATOMISTIC SIMULATIONS AND
CONCLUSIONS
As shown, in Fe-Cr alloys the
microstructure evolution under irradiation is
the consequence of strongly coupled,
concomitant phase changes and defect
migration processes that inextricably influence
each other. The strong reduction of the
mobility of <111> SIA clusters and small
dislocation loops (in Fe-Cr as compared to Fe)
should lead to a pronounced decrease of
annihilation of small 1D-migrating SIA
clusters, created in collision cascades initiated
by fast neutrons, at dislocations, grain
boundaries and other sinks. Meaning that
additional recombination sites for freely
migrating vacancies and small vacancy
clusters will be provided. This should lead to a
lower equilibrium vacancy concentration and a
decrease in swelling. In addition, the
nucleation rate of the dislocation loops is
expected to be higher in Fe-Cr alloys, but their
growth rate correspondingly lower, because of
the reduced mobility of 3D-migrating SIAs
(coupled with the unaffected vacancy
migration). This agrees well with the
experimental results from [10], mentioned
already in Section 2.
The decrease of <111> SIA cluster
diffusivity is non-monotonic in the random
solution limit. Small clusters with size up to 20
SIAs exhibit a diffusivity minimum at 9-
12% Cr, which shifts towards lower Cr
concentrations with the cluster size, as
schematically shown in Fig. 6,a. At elevated
temperatures and/or under irradiation either Cr
ordering or α-α' phase separation is expected
to occur, depending on the Cr content in the
alloy. Strong repulsive interaction between Cr
precipitates and <111> SIA clusters will result
in further suppression of the mobility of these
defects. The cross section of the <111> SIA
cluster - precipitate interaction depends on the
density of precipitates and size of a cluster and
precipitate. The onset of the α-α'-phase
separation results in the formation of
nanometric Cr precipitates with enormously
76
high density (up to 1024 m-3) [8]. Coalescence
of Cr precipitates with the corresponding
decrease of the precipitate density should
'soften' the blockage of 1D-migrating clusters
by α' particles.
In the low-concentration alloys, the
enhanced Cr ordering should lead to a strong
decrease of the value of ΔF (defined in Section
3.4). The mobility of <111> SIA clusters will
be governed by the interaction with isolated Cr
atoms and therefore the migration energy
should have a weak dependence on the cluster
size and will stay around 0.4-0.3 eV for the
small (in-cascade created) SIA clusters. The
degree of ordering, in turn, depends on the
heat of mixing and temperature, thus the alloys
containing around 5% Cr should have the
lowest SRO parameter. The change in the
diffusivity reduction due to the effect of Cr
ordering/precipitation is schematically shown
in Fig. 6,b.
Rn
1
10 20
Cluster size ↑
Rn
1
10 205 15
Ordering/precipitation
(a) (b)
Rn
1
10 20
Cluster size ↑
Rn
1
10 205 15
Ordering/precipitation
(a) (b)
1
10 20
Cluster size ↑
Rn
1
10 205 15
Ordering/precipitation
(a) (b)
Fig. 6. Variation of the diffusivity
reduction (Rn=DFe-Cr/DFe) of 1D-
migrating SIA clusters (a) due to
the increase of a cluster size and
(b) in the presence of the long-
range order or Cr precipitation.
Dashed curves show the trend
The pinning of <111> SIA clusters by α'
particles should lead to a steep increase of the
nucleation rate of ½<111> loops. The
formation of <100> loops was proposed to
occur via unfaulting of small ½<111> loops
[33] or in the reaction between the two
½<111> loops [34]. So far, there is no
experimental confirmation for the proposed
mechanisms. If, however, these mechanisms
are relevant, the nucleation rate and the density
of <100> loops should increase in the presence
of α-α' phase separation. The experimental
data shown in Fig.1,c, reveals a steep increase
in the density of <100> loops in alloys with
CCr=12-18%Cr, irradiated at 425 °C, which
potentially should exhibit α-α'-phase
separation [32].
By combining these observations, the
general picture of void swelling behaviour as a
function of Cr content can be built. The
swelling minimum experimentally observed at
relatively low doses (1-10 dpa) around 10% Cr
(see Fig. 1,a) can be mainly attributed to the
reduction of the mobility of in-cascade created
<111> SIA clusters. Consequently, the
minimum of the hardening also stays around
9-10% Cr (see Fig.1,a), since the mean size of
defects (mainly of SIA type) obstructing the
motion of dislocations is minimal in these
alloys. With increasing dose (and irradiation
temperature) the α' precipitation or Cr ordering
occurs, in the alloys with CCr=10-20%
(spinodal decomposition occurs at larger CCr)
and around 5%, respectively. As a
consequence of these effects, <111> SIA
clusters will be more effectively reduced either
in high-Cr or low-Cr alloys, providing a local
maximum at ~10% Cr. Indeed, the swelling,
measured at higher temperature or dose
usually shows two separated minima (see Fig.
1,a). In the temperature range where Cr
ordering/precipitation is no longer possible the
swelling minimum is restored at ~10% Cr.
Concluding, we provided evidence from
atomistic modelling that the reduction of
½<111> dislocation loops and small 3D-
migrating SIA clusters, the formation of
<100> loops, induced swelling and hardening
are interconnected, and all strongly depend on
the Cr content. Two main characteristics of the
Fe-Cr system seem to provide in many cases
the key for the interpretation of the behaviour
of these alloys under irradiation as observed
experimentally. These characteristics are: (i)
the change of sign of the mixing enthalpy from
negative to positive in the low Cr
concentration region; (ii) the existence of a
strong interaction between self-interstitials and
Cr atoms, which dramatically reduces the
diffusivity of 1D-migrating self-interstitial
atom clusters in Fe-Cr compared to Fe.
77
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НЕКОТОРЫЕ СТОРОНЫ ЭФФЕКТОВ РАДИАЦИОННЫХ
ПОВРЕЖДЕНИЙ В СПЛАВАХ Fe-Cr С ТОЧКИ ЗРЕНИЯ
АТОМИСТИЧЕСКОГО МОДЕЛИРОВАНИЯ
Д. Терентьев, П. Олсон
Сплавы Fe-Cr составляют основу ферритных сталей с высоким содержанием Cr,
являющихся кандидатами в конструкционные материалы для атомных электростанций
ближайшего будущего. В последнее время значительные усилия направлены на
разработку теоретических моделей, рассматривающих реакцию сплавов Fe-Cr на
облучение. В настоящей работе дается краткий обзор современного уровня понимания
радиационных повреждений в сплавах Fe-Cr, при этом в основу берутся последние на
данный момент результаты. В частности, дается обзор и подводится итог данных,
полученных с помощью различных методов атомистического моделирования с целью
определения наиболее важных результатов, достигнутых за последние несколько лет.
ДЕЯКІ СТОРОНИ ЕФЕКТІВ РАДІАЦІЙНИХ
ПОШКОДЖЕНЬ У СПЛАВАХ Fe-Cr З ТОЧКИ ЗОРУ
АТОМІСТИЧНОГО МОДЕЛЮВАННЯ
Д. Терентьєв, П. Олсон
Сплави Fe-Cr становлять основу феритних сталей с високим вмістом Cr, що являються
кандидатами у конструкційні матеріали для атомних електростанцій найближчого
майбутнього. В останній час значні зусилля спрямовані на розробку теоретичних моделей,
які розглядають реакцію Fe-Cr-сплавів на опромінення. В даній роботі наводиться
короткий огляд сучасного рівня розуміння радіаційних пошкоджень у сплавах Fe-Cr, при
цьому за основу беруться останні на даний момент результати. Зокрема, наводиться огляд
та підводиться підсумок даних, що отримані за допомогою різних методів атомістичного
моделювання з метою визначення найбільш важливих результатів, які досягнуті за останні
кілька років.
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