Primary damage production in the presence of extended defects and growth of vacancy-type dislocation loops in hcp zirconium

Abstract

Production rates in long-term predictive radiation damage accumulation models are generally considered independent of the material's microstructure for reactor components. In this study, the effect of preexisting microstructural elements on primary damage production in α-Zr—and vice versa—is assessed by molecular dynamics (MD) simulations. a-type dislocation loops, c-component dislocation loops, and a tilt grain boundary (GB) were considered. Primary damage production is reduced in the presence of all these microstructural elements, and clustering behavior is dependent on the microstructure. Collision cascades do not cause a-type loop growth or shrinkage, but they cause c-component loop shrinkage. Cascades in the presence of the GBs produce more vacancies than interstitials. This result, as well as other theoretical, MD, and experimental evidence, confirms that vacancy loops will grow in the vacancy-supersaturated environment near GBs. Distinct temperature-dependent growth regimes are identified. Also, MD reveals cascade-induced events where a-type vacancy loops are absorbed by GBs. Fe segregation at the loops inhibits this cascade-induced absorption.

Full text

PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
Primary damage production in the presence of extended defects and growth of vacancy-type
dislocation loops in hcp zirconium
Cong Dai, Fei Long, Peyman Saidi, Laurent Karim Béland,*Zhongwen Yao, and Mark R. Daymond†
Department of Mechanical and Materials Engineering, Queen’s University, Kingston, Ontario, Canada K7L 3N6
(Received 1 August 2018; revised manuscript received 4 January 2019; published 9 April 2019)
Production rates in long-term predictive radiation damage accumulation models are generally considered
independent of the material’s microstructure for reactor components. In this study, the effect of preexisting
microstructural elements on primary damage production in α-Zr—and vice versa—is assessed by molecular
dynamics (MD) simulations. a-type dislocation loops, c-component dislocation loops, and a tilt grain boundary
(GB) were considered. Primary damage production is reduced in the presence of all these microstructural
elements, and clustering behavior is dependent on the microstructure. Collision cascades do not cause a-type
loop growth or shrinkage, but they cause c-component loop shrinkage. Cascades in the presence of the GBs
produce more vacancies than interstitials. This result, as well as other theoretical, MD, and experimental
evidence, confirms that vacancy loops will grow in the vacancy-supersaturated environment near GBs. Distinct
temperature-dependent growth regimes are identified. Also, MD reveals cascade-induced events where a-type
vacancy loops are absorbed by GBs. Fe segregation at the loops inhibits this cascade-induced absorption.
DOI: 10.1103/PhysRevMaterials.3.043602
I. INTRODUCTION
Long-term exposure to radiation causes structural damage
and limits the lifetimes of nuclear reactor components, many
made from Zr alloys. Irradiation-induced point defects re-
arrange to form point-defect clusters, dislocation loops, and
voids [1]. In particular, in Zr alloys c-component loops are
believed to have a significant impact on irradiation growth
and irradiation creep as well as the degradation of mechanical
behavior [2]. In addition, the formation and distribution of
irradiation-induced point defects are determined by the mate-
rial’s prior microstructure [3]. For instance, early experiments
by Griffiths et al. [4] on neutron-irradiated Zr show that
interstitial loops were estimated to make up about 25% of the
total dislocation loops observed near grain boundaries (GBs),
for a 2–3-μm grain size.
Irradiation damage of Zr alloys has been extensively stud-
ied using transmission electron microscopy (TEM) [5], x-ray
analysis [6], and positron annihilation spectroscopy [7]. How-
ever, these experimental instruments have limitations in pro-
viding detailed information in regards to the nanosize defects
that form during picosecond collision cascades. Thus, differ-
ent computational and modeling techniques such as molecular
dynamics (MD), kinetic Monte Carlo (kMC), and rate theory
have been applied to various size- and timescale problems.
Given the small MD time steps—typically ∼1×10−15 s,
and as small as 1 ×10−18 s in collision cascade simulations—
the evolution of radiation-induced damage that can be taken
into account by MD is of the order of a few tens of nanosec-
onds. There are three main stages to primary damage pro-
duction that take place over a few picoseconds: a supersonic
*laurent.beland@queensu.ca
†mark.daymond@queensu.ca
phase, a sonic phase, and a thermally enhanced recovery phase
[8]. MD is well suited to making physically valid predictions
over these timescales, within the limits of the choice of
interatomic potentials (see Refs. [9–13] for discussions about
designing potentials for cascade simulations); MD can also
uncover atomistic phenomena that take place on timescales
of a few nanoseconds, revealing relevant mechanisms that
should be included in kMC and rate-theory models, such as
one-dimensional (1D) and three-dimensional (3D) diffusion
of defects [14–16], or interaction of point defects, defect
clusters, and extended defects such as dislocation loops and
line dislocations [17–19].
To calculate damage accumulation and evolution over
longer timescales, the results of these MD simulations–
damage production rates and clustering behavior—are typi-
cally used as inputs in kMC and rate-theory based models. For
example, Arévalo et al. [20,21] used a database of displace-
ment cascades from MD to study the effect of temperature on
the accumulation of damage in pure Zr using kMC. Likewise,
using kMC calculations, the anisotropic diffusion of point
defects in Zr was analyzed by Fan et al. [22]. For even
longer-time treatments, rate-theory models are used, deter-
mined by the coupled reaction-diffusion equations [23–25].
Woo and Gösele [26] proposed a rate theory that considers
anisotropic diffusion of point defects. They found that the
difference in diffusional anisotropy (DAD) between vacancies
and interstitials could cause a large bias in their reaction rates
with sinks, even if the dislocation structure was isotropic.
The DAD between vacancies and interstitials was justified
by molecular dynamics simulations based on empirical po-
tentials. Later, Samolyuk et al. [68] showed, using atomistic
simulations based on electronic structure calculations, that
DAD was much lower than that predicted using empirical
potentials. The ab initio based DAD alone cannot explain
radiation-induced growth of Zr below 750 K. Interestingly,
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DAI, LONG, SAIDI, BÉLAND, YAO, AND DAYMOND PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
including one-dimensional diffusion of cascade-induced self-
interstitial atom (SIA) clusters in the rate theory—which
were revealed by MD—can lead to reasonable predictions
of radiation-induced growth [27,28]. A key takeaway here is
that the predictions—and physical validity—of rate theories
are largely determined by the mechanisms and rates that they
include, which are often provided by atomistic simulations.
Recently, Payant [29] used cluster dynamics models to pre-
dict the microstructure evolution as a function of dose and
temperature, and predicted significantly higher loop densities
than those experimentally observed. They emphasized the
importance of MD simulations in providing a database on the
effects of collision cascades with preexisting defects.
Such long-timescale models typically neglect the effect
of preexisting microstructure on primary damage production
rates. This is a good first-order approximation, but it may
break down as the densities of extended defects increase as
damage accumulates. It may also break down in the case of
nanograined materials, which are designed to have a very high
density of grain boundaries.
The effect of preexisting microstructure on primary dam-
age production has been assessed by a number of previous
MD-based studies. Ludy and Rupert [30] studied collision
cascades induced by primary knock-on atoms (PKAs) with
kinetic energies of up to 2.5 keV near high-angle GBs in Cu,
and found that GBs easily absorb SIAs while more residual
vacancies remain in the bulk. This suggests that high-angle
GBs in Cu capture SIAs in cascade. The cascade sink pref-
erence of GBs in α-Zr was also studied [31,32]. The cascade
sink is defined as the asymmetry between interstitial-type and
vacancy-type primary damage production when cascades oc-
cur in nonpristine materials. Performing cascade simulations
in the vicinity of five different GB structures, Hatami et al.
[32], observed that GBs in α-Zr are not necessarily biased
toward interstitials, and can preferentially absorb vacancies.
Hatami et al. considered PKAs with initial kinetic energy
less than 9 keV, and neglected the effect of PKA direction.
Jin et al. [33] analyzed radiation damage evolution in Cu
bicrystals by simulating overlapping cascades, and they re-
ported a mechanism for annihilation of defect clusters during
irradiation. Jin et al. also discussed how the short simulation
time between collision cascades yields a significantly higher
effective dose rate than that observed in experiments. This
timescale problem limits MD in the investigation of diffusion-
based microstructural evolution.
As mentioned above, realistic MD data in regards to defect
production are valuable input parameters for kMC and rate-
theory models [34]. MD can help us understand in detail the
accumulation of damage in irradiated materials, e.g., intra-
cascade recombination, the creation of primary clusters, and
cascade-induced growth or shrinkage of preexisting extended
defects. Augmenting cascade-induced event rates databases
will improve model-based predictions, and provide a more
comprehensive view of radiation damage [35]. Likewise, sim-
ulating the kinetics of defects over timescales of nanoseconds
may reveal mechanisms that the higher-scale models need
to take into account. Our study has thus focused on the
following three themes. First, primary cascade production
in the presence of dislocation loops. Second, the growth
of dislocation loops near GBs. Third, the stability of the
Y= <1120>
X: <1100>
X= <1120>
Y= <1100>
Z: <0002>
Z= <0002>
FIG. 1. The asymmetric tilt grain boundary (1¯
100)/(11¯
20)
0002(θ=30◦) is viewed along the 0002tilt axis; atoms on
consecutive {0002}planes are shown as black and white. The crystal
orientations are shown in the left-hand side for each for grain.
dislocation loop near GBs and the effect of alloying on this
stability.
MD is used to simulate collision cascades in α-Zr that over-
lap with extended defects: a-type or c-component dislocation
loops, and a tilt GB. A mixture of theoretical arguments, MD
simulations, and experimental evidence—in situ transmission
electron micrographs (TEMs) of Zircaloy-2 irradiated with
a 3-MeV proton beam—are combined to show that vacancy
dislocation loops near GBs grow in a vacancy-supersaturated
environment. MD simulations of a-type and c-component
loop growth in this environment are reported. The stability of
these loops near a grain boundary under irradiation is assessed
by MD. The effect of Fe solutes on this stability is assessed by
a hybrid molecular dynamics/Monte Carlo (MD/MC) scheme.
II. METHODS
A. Simulation description
The MD was simulated using the LAMMPS (large-scale
atomic/molecular massively parallel simulator) [36]. There
are a few [37–39] Zr potentials available. As reported in [40],
the MA07 [39] potential (No. 2) provides a better description
of vacancy binding energies than the MA07 [39] potential
(No. 3). However, the latter (No. 3) provides a better descrip-
tion of the stacking faults on prismatic, basal, and pyramidal
slip systems [40], which largely controls the energetics of
extended defects such as dislocation loops. We chose the
latter one (No. 3). Also, it should be noted that the short-
range interactions predicted by this potential are stiffened,
which is key when predicting primary damage production
[9,10,41]. This potential has been used in a number of recent
radiation damage simulation studies [42–46]. An asymmet-
ric (1¯
100)/(11¯
20)0002(θ=30◦) tilt grain boundary was
created. Its boundary plane orientations are the (1¯
100) and
(11¯
20) surfaces of the two grains. This GB is illustrated in
Fig. 1. The GB structure was constructed using a conjugate
gradient energy minimization [47] in conjunction with an
atom deletion criterion [48]. Periodic boundary conditions are
imposed along all directions (X,Y, and Zaxes). Typical sam-
ples contain 4 000 000 atoms with dimensions Lx=44.5nm,
Ly=56.0 nm, and Lz=20.6nm.
Cascade simulations were carried out at 573 K, which is
a common environmental temperature during irradiation in
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PRIMARY DAMAGE PRODUCTION IN THE PRESENCE OF … PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
nuclear reactors [1]. Within the time of collision cascade sim-
ulations using MD, only short-range diffusion is considered.
The purpose of the cascade simulations in this study is to in-
vestigate primary damage production in the presence of differ-
ent preexisting microstructures. In this context, temperature
effects should be relatively minor. For this reason, only one
cascade simulation temperature was considered. Long-range
diffusion—a process that crucially depends on temperature—
is largely ignored. Moreover, recent cascade simulations by
Nordlund et al. [13] suggest that the effect of temperature on
primary damage production is minor, an observation consis-
tent with several previous studies [34,49,50]. The energy of a
PKA was set to be 50 keV, and the directions of the PKA were
randomly selected. Ions and electrons both significantly con-
tribute to the stopping of 50-keV ions. In this study, we solely
considered ion stopping. In the future, it would be interesting
to consider the influence of electron stopping and electron-
phonon coupling on cascade-microstructure interactions, e.g.,
using a two-temperature model [51,52]. It should be noted,
however, that the present article focuses on differences be-
tween cascades in the bulk and cascades in the presence
of preexisting microstructure. We expect that the absolute
level of primary damage production is likely to be affected
by electron stopping and electron-phonon coupling to a mi-
crostructure much larger extent than microstructure-induced
differences in primary damage production. This assumption
and ensuing computational strategy are commonly used to
simulate radiation damage in this energy range [53–55]. Also,
future two-temperature model-based work should likely take
into account the effect of preexisting microstructure on the
coupling between the atomic and electronic subsystems. For
instance, the approach suggested by Tamm et al. [56,57] could
prove fruitful in such an endeavor.
At least ten independent cascade simulations were em-
ployed to obtain statistics. The cascade simulations were
performed with a constant number of atoms, volume, and
energy (NV E ensemble) with a variable time step.
The Zr-Fe interactions are described by the embedded-
atom method (EAM) developed by Saidi et al. [58]. Their
model can accurately reproduce the formation energy and
lattice parameter of all five stable and metastable intermetallic
compounds of the Zr-Fe phase diagram. To simulate alloying
segregation, the variance-constrained semi-grand-canonical
(VC-SGC) ensemble [59] was employed, which combines
MD for structural relaxation, and MC to push the atomic
chemical configuration toward equilibrium. The fraction of
swapped atoms is 0.2 and the MD runs between the MC
swaps comprising 100 integration steps of 1 fs each. The
chemical composition is controlled by the chemical potential
difference between the two species and the target concentra-
tion. Images of atomic configurations were produced with the
OPEN VISUALIZATION TOOL (OVI TO)[60]. A Wigner-Seitz cell
method [61–63] was used to identify the type of point defects,
where atom positions are analyzed with respect to a geometric
structure with perfect lattice atom positions.
B. EXPERIMENTS
The annealed alloy Zircaloy-2 was irradiated by a 3-MeV
proton beam at 180 °C and at a flux of 3.26 ×1018 ion/cm2
(∼0.11 displacement per atom) on a 15 mm ×15 mm area
with the Tandem accelerator at the Reactor Material Testing
Laboratory (RMTL), Queen’s University. TEM foils were
prepared by back-polishing and subsequent thinning using a
solution of 10% perchloric acid and 90% methanol at –40 ◦C
using a Struers TenuPol-5. TEM imaging was carried out
on an FEI Tecnai Osiris Scanning/TEM (S/TEM). In order
to observe the growth behavior of the irradiation-induced
loops and the effect of grain boundary as a defect sink, in
situ TEM annealing experiments were carried out with one
irradiated sample loaded in a Gatan 625 double tilt heating
stage at 673 K. Bright field (BF) images were taken from three
areas close to the grain boundary in one grain with the same
reflection prior, during, and at the end of heating, so that the
change of a given group of defects could be monitored.
III. RESULTS AND DISCUSSIONS
A. MD simulations: The primary cascade with different
extended defects
The interactions of cascades with a perfect crystal, a-type
loops, c-component loops, and a tilt GB are statistically stud-
ied and reported in Table I. An illustration of residual defects
identified using the Wigner-Seitz cell method is provided
in the Supplemental Material [64]. Cascades produce fewer
residual interstitial-type defects NIand residual vacancy-type
defects NVin the simulation boxes containing a preexisting
defect than in the perfect crystal. Vacancy and interstitial
TABLE I. The number of residual interstitials (NI), the number of residual vacancies (NV), the number of interstitial clusters (NI
C), the
number of vacancy clusters (NV
C), the proportion of interstitial defects contained in clusters (PI
C), and the proportion of vacancy defects
contained in clusters (PV
C) after cascade simulations with 50-keV PKA energies at 573 K. The simulation boxes contained either a perfect
crystal, an a-type interstitial or vacancy loop, a c-component interstitial or vacancy loop, or a tilt GB.
a-Type a-Type vacancy c-Component c-Component
Pristine interstitial loop loop interstitial loop vacancy loop Tilt GB
NI78 (6.1) 65 (5.2) 54 (5.0) 67 (8.6) 28 (5.2) 37 (6.0)
NV78 (6.1) 63 (5.4) 63 (4.5) 34 (9.1) 41 (8.9) 80 (11.5)
NI
C7 (1.3) 6 (0.5) 6 (0.5) 6 (1.1) 4 (0.9) 3 (0.7)
NV
C4 (0.8) 4 (0.4) 4 (0.3) 3 (0.6) 4 (0.5) 4 (0.5)
PI
C66.83% (5.23%) 66.62% (2.35%) 58.88% (2.9%) 79.64% (4.39) 70.21% (2.73%) 69.69% (5.08%)
PV
C81.8% (4.14%) 63.95% (3.61%) 60.83% (3.67%) 69.6% (4.91%) 74.02% (6.34%) 82.68% (3.65%)
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(a) Time: 1 ps (b) Time: 101 ps (c) Time: 330 ps (d) Time: 340 ps (e) Time: 400 ps
a-type interstitial loop
GBs
FIG. 2. A thermal relaxation simulation illustrating the absorption of an a-type interstitial dislocation loop by the GB at 573 K. The clock
starts when the a-type interstitial loop is formed. Only displaced atoms are shown based on common neighbor analysis. (f) is the potential
energy profile of the system as a function of time.
production is balanced in the presence of a-type loops, and
is imbalanced in the presence of c-component loops and a tilt
GB. As described in Table I,thevalueofNVis almost half that
of NIfor c-component interstitial loops, while the value of NI
is almost half of NVfor c-component vacancy loops and a tilt
GB. This imbalanced cascade production is associated with a
shrinkage of c-component loops and vacancy supersaturation
near GBs.
Clustering of cascade-induced defects is characterized by
calculating the number of interstitial clusters NI
Cand the
number of vacancy clusters NV
C, as well as the proportion of
interstitial/vacancy defects in clusters PI
Cand PV
C. The number
of clusters does not seem altered by a-type loops, but it is
decreased by c-component loops and a tilt GB. In particular,
NI
Cnear a tilt GB is less than half that of a perfect crystal,
which is consistent with the comparison between NIin the two
cases. Moreover, vacancies have a lesser propensity to cluster
in the presence of a-type and c-component loops, but have the
same propensity to cluster in the presence of a tilt GB as in a
perfect crystal.
The production rate of interstitials and vacancies is gen-
erally assumed to be identical in rate theory [23,25], and
the fraction of interstitial clusters and vacancy clusters is
only analyzed in a perfect crystal [28]. Also, cascade-induced
shrinkage of c-component loops is neglected. The results
presented in Table Icould be used to modify rate-theory and
kMC models.
B. Evidence for vacancy supersaturation near grain boundaries
In the previous section, it was observed that GBs promote
vacancy supersaturation by acting as a cascade sink. However,
this result might depend on the grain boundary orientation
[32,65,66]. Here, by combining classical diffusion sink ar-
guments, MD simulations of dislocation loop stability, and
experimental evidence, vacancy supersaturation near GB is
established in a definitive manner.
In a first approximation, a GB is a perfect sink for all point
defects and defect clusters during the long-time annealing
stage. It is well known that diffusion coefficients for SIAs and
their clusters are much higher than those of vacancies and their
clusters in α-Zr, respectively [67,68]. If reactions between
defects are neglected, a simple Fickian diffusion model will
predict a vacancy surplus near the GBs.
The stability of a-type dislocation loops near a tilt GB
GBs might affect the damage accumulation and evolution
not only by serving as a cascade sink and a diffusion sink,
but also by absorbing certain types of loops. We studied the
thermal stability of a-type dislocation loops near a GB. An
interstitial loop was created 10 nm below the tilted GB, and
its initial habit plane is (10¯
10) as displayed in Fig. 2(a).After
relaxation at 573 K for 101 ps, this loop began to tilt, and
it had a chair-shape structure after relaxation for 330 ps [see
Fig. 2(c)]. At 340 ps, half of the loop was absorbed by the GB
and it was then completely absorbed as illustrated in Fig. 2(e).
The potential energy profile as a function of time is provided
in Fig. 2(f), which decreases as the loop approaches the GB.
An a-type vacancy loop was also created below the GB at
the same separation distance. This a-type vacancy loop in
the plane (10¯
10) was stable during the relaxation at 573 K.
Thus, a-type vacancy loops are thermally more stable than the
a-type interstitial loops near the GB.
C. In situ TEM observation: Identifying the type
of loops near the GBs
The previous sections of this article bring forward three
theoretical results pertaining to defects near GBs. First, MD
revealed that cascades generate more vacancies than inter-
stitial near a GB. Second, we quickly highlighted that—
given that vacancies are less mobile than interstitials—simple
Fickian arguments predict that, near a sink such as a GB,
the concentration of vacancies should be higher than the
concentration of self-interstitials. Third, MD revealed that
a-type vacancy loops are more stable than a-type interstitial
loops near a GB. These three results suggest that dislocation
loops near grain boundaries will mainly be vacancy type. In
this section, we present experiments that were carried out to
analyze the type of dislocation loops near the GBs.
Most of the loops generated by 3-MeV proton irradiation
are smaller than 10 nm. Their vacancy or interstitial nature
cannot be determined because of their small size. The loops
were annealed for 3 h at 673 K to increase their size due to
absorption of point defects (mostly vacancy) and subsequent
coarsening, and hence let us identify their vacancy/interstitial
nature using BF images. According to the inside/outside con-
trast method proposed by Maher and Eyre [69], and Föll and
Wilkens [70], one needs to determine the loop Burgers vector
b, habit plane normal n, the angle between the diffraction
vector gand the habit plane normal n, and also the sign of
the Bragg deviation parameter sg. The BF images in Fig. 3
were taken with (10¯
1¯
1) and ( ¯
1011) reflection with deviation
parameter sg>0 along the [11¯
23] zone at the same area after
in situ annealing, as shown in Figs. 3(a) and 3(b), respectively.
As indicated by red arrows, almost all the loops have shown
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PRIMARY DAMAGE PRODUCTION IN THE PRESENCE OF … PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
(1 0 11)
--
(1011)
-
(a) (b)
FIG. 3. Bright field images taken with (10¯
1¯
1) and (¯
1011) reflec-
tion along the [11¯
23] zone axis from a sample annealed at 673 K for
3 h. Red arrows indicating vacancy loops near the grain boundary
have larger size distribution.
outside contrast with the (¯
1011) reflection. Given that loops
with ±a/3[1¯
210] Burgers vector are invisible with this re-
flection, the loops shown in Fig. 3have either ±a/3[11¯
20]
or ±a/3[¯
2110] Burgers vector. It is known that loops show
outside contrast when (g·b)sg>0[71]. The Burgers vectors
of loops that gives outside contrast are determined to be
a/3[¯
1¯
120], or a/3[¯
2110]. Since (B·b)<0(Bis the electron
beam direction that is nearly parallel to the [11¯
23] zone axis)
for both determined Burgers vectors, this implies that all the
loops showing outside contrast are vacancy type. Vacancy
loops near a grain boundary grow at a higher rate than those
further away. As indicated by green arrows, only a few small
interstitial loops can be determined in Fig. 3. Note that the
observation was conducted in situ with video recording; no
dislocation loop disappeared during annealing. In addition, a
quantitative analysis provided in the Supplemental Material
[64] suggests that almost all small loops present prior to
annealing are also vacancy type in nature. Therefore, the
overall numerical distribution of interstitial and vacancy loops
determined after annealing is believed to be representative
of the as-irradiated microstructure. These results are in good
agreement with both Holt’s study [72] and that of Griffiths
et al. [4], which showed a depletion of interstitial loops near
the GBs. That is, a dominance of vacancy loops as found near
GBs by TEM experiments is supported by the MD results
reported above which show that a-type vacancy loops are
more stable than a-type interstitial loops near the GBs.
D. Growth of vacancy loops under locally high
concentration of vacancies
As shown in the previous section, supersaturation of va-
cancies near GBs leads to the growth of vacancy loops. MD
was performed to simulate the loop growth in a supersaturated
vacancy environment. In the vicinity of either one a-type
or one c-component vacancy loop, 0.5% atoms (∼287 400
atoms in total) were randomly deleted to create a locally high
concentration of vacancies, which is much higher than the
equilibrium vacancy concentration [72], but roughly repre-
sentative of typical point-defect concentrations observed in
irradiated environments [73,74].
Three annealing temperatures (600, 800, and 1000 K) were
employed. The area of dislocation loops A(t) was measured
at each time step, and the difference of the area in regard to
its initial size (A0) is defined as A=A(t)−A0. Figure 4(a)
shows Aas a function of time for a-type and c-component
vacancy loops. The growth rate of vacancy loops increases
as temperature increases. The growth rate of a-type loops
is greater than that of c-component vacancy loops at low
temperatures. At 600 K, both vacancy loops show a very fast
growth rate for the first 1 ns, and a slower rate afterward.
Visual inspection of loop growth at 600 K indicates that the
loop can easily absorb nearby vacancies in the beginning of
the simulation, which leads to a faster growth rate. Once
the nearby vacancies are captured, the growth rate slows. At
1000 K, the growth rates for both loops are roughly constant
with time over the period investigated.
Vacancy clusters formed throughout the simulation boxes.
In addition, a-type vacancy loops can locally glide and tilt,
while c-component vacancy loops remain fixed in the basal
plane. This suggests that a-type vacancy loops may be more
readily absorbed by the GBs as loops tilt and glide.
In addition to growth rates, atomic diffusion was moni-
tored. Mean square displacements (MSD) in different direc-
tions are summed over all the atoms in the system, and illus-
trated in Fig. 5. The time-dependent MSD shows three main
features: (i) the MSD at 600 and 800 K exhibits a supralinear
trend; (ii) at 600 K, the MSD for a system having an a-type
(c-component) vacancy loop is highest in the y(z) direction;
and (iii) the MSD traces in all directions are similar at 1000 K,
but the slope of the MSD slightly decreases after 1.5 ns.
The supralinear MSD trend at lower temperatures might be
due to interaction between the loops and the vacancy clusters.
FIG. 4. Loop area differences as a function of annealing time at (a) 600, (b) 800, and (c) 1000 K.
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DAI, LONG, SAIDI, BÉLAND, YAO, AND DAYMOND PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
FIG. 5. The mean square displacement (MSD) in different directions during the growth of a-type and c-component vacancy loops at
different temperatures, as a function of time.
Visual inspection indicates that the vacancy absorption occurs
at the edge of the loop. Let us therefore further examine
the MSD results at a low temperature in Fig. 5. The normal
direction of the a-type vacancy loop’s plane at 600 K is
parallel to the ydirection, which is the same direction for
the highest MSD in Fig. 5(a). At the same temperature, the
habit plane of c-component vacancy loops is the basal plane,
and its normal direction is also the same direction for the
highest MSD in Fig. 5(d). This suggests that the loop affects
the neighboring vacancies along the normal direction of the
loop’s plane. However, at a higher temperature of 1000 K, i.e.,
Figs. 5(c) and 5(f), the MSD values for both systems are very
similar, and the MSD in the x-yplane is higher than that along
the zdirection. Further, as reported by [40,67,68], vacancies
FIG. 6. The number of monovacancies as a function of time during the growth of a vacancy loop at (a) 600 and (d) 1000 K; the number
of divacancies as a function of time at (b) 600 and (e) 1000 K; the number of total vacancy clusters as a function of time at (c) 600 and (f)
1000 K.
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PRIMARY DAMAGE PRODUCTION IN THE PRESENCE OF … PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
Normal-direction distance (A)
Radial-direction distance (A)
Normal-direction distance (A)
Radial-direction distance (A)
EF (eV)
Normal direction
Radial direction
(a)
Dislocation loop
R=40 A
0 50 100 150
0
20
40
60
80 (b)
0 50 100 150
0
20
40
60
80 (c)
FIG. 7. (a) The normal direction and the radial direction relative to the dislocation loop. The formation energy of a vacancy as a function
of the normal direction and the radial direction to the a-type vacancy loop (b) and c-component vacancy loop (c). In (b,c), the connected red
dots indicate steepest-gradient paths. In (b) these paths indicate a driving force toward the edge of the loop and a driving force away from the
loop, in its normal direction. In (c), the path indicates a driving force toward the edge of the loop.
in α-Zr show faster diffusion in the basal plane (x-yplane in
this study) than along the caxis (zdirection). Thus, at higher
temperatures the effect of the loop on the vacancies and the
surrounding clusters becomes much less important than the
“inherent” mobility of vacancies and their clusters, while at
lower temperatures the loop effect is more significant.
The vacancy loop growth mechanism in a vacancy-
supersaturated environment is further analyzed by determin-
ing the number of monovacancies NM, divacancies ND, and
clusters NCduring the loop growth, which is reported in
Fig. 6. Note that monovacancies are not considered as vacancy
clusters, but that divacancies are. Both NMand NDfor a-
type vacancy loops at 600 K decrease during the first 1 ns,
in tandem with loop growth as shown in Fig. 4(a).Atthe
same temperature for c-component vacancy loops, only ND
decreases while NMis relatively constant. This suggests that at
600 K, a-type vacancy loops absorb both monovacancies and
divacancies, while c-component vacancy loops mainly absorb
divacancies. As shown in the animations of the Supplemental
Material [64], at 600 K, a-type vacancy loops are more
mobile than c-component vacancy loops, which allows a-type
vacancy loops to glide to other positions and hence absorb
monovacancies. At 1000 K, as illustrated in Figs. 6(d)–6(f)
the values of NM,ND, and NCfor both loops continuously
decrease, which is consistent with the loop growth behavior in
Fig. 4(c). Thus, at such high temperature, the growth of a-type
and c-component vacancy loops occurs by absorbing all kinds
of vacancies and clusters. The ability of a-type vacancy loops
to tilt and glide in such a way, increasing its probability of
capturing defects, is typically not embedded in kMC and rate-
theory models. Our results suggest that the capture radius of
a-type loops in the prismatic direction should be larger than in
the basal direction or than the capture radius of c-component
loops of the same diameter.
Visual inspection of the animation presented in the Supple-
mental Material [64] shows that the growth of both a-type and
c-component dislocation loops is mainly due to the absorption
of vacancies and vacancy clusters at the edge of the loops. We
then calculated the formation energy of vacancies at different
positions relative to the loop (R=40 Å) as a function of the
radial and normal direction, which is shown in Fig. 7(b) for
an a-type vacancy loop, and in Fig. 7(c) for a c-component
vacancy loop. There are clear minima of the potential energy
of vacancies at the edge of the loops, which is consistent with
the visual inspection of the animations. This result is also
consistent with previous calculations of the strain and stress
fields around dislocation loops and lines [75–77]. Similarly,
vacancies preferentially form near grain boundaries in Zr [65]
and in cubic materials [78,79].
As observed by Harte et al. [80] and in other experimental
studies [5,81], a-type loops are randomly distributed at lower
irradiation doses, and the loops’ separation distances are rel-
atively large. Thus, the growth of a population of dislocation
loops at low dose will be similar to a single loop growth. The
density of a-type loops increases with dose. Moreover, these
studies [5,80,81] also found that the majority of a-type loops
at higher irradiation doses share a basal plane, which would
decrease their separation distance. By observing the vacancy
formation energy map in Fig. 7, it seems that vacancies that
follow an energy gradient will either converge toward the edge
of the loop or be driven away in a direction perpendicular
to the loop. Two such possible trajectories are illustrated in
Fig. 7(b). Also, notice that the gradient pushing vacancies
away from the loop in a direction normal to it tends to
(a) Time: 0 ps (b) Time: 0.1 ps (c) Time: 301.2 ps (d) Time: 501.2 ps (e) Time: 601.2 ps
GBs
a-type vacancy loop
FIG. 8. Snapshots illustrating a cascade-induced absorption of an a-type vacancy dislocation loop by the GB at 573 K. The clock starts as
the PKA is introduced. Only displaced atoms are shown based on the common neighbor analysis.
043602-7

DAI, LONG, SAIDI, BÉLAND, YAO, AND DAYMOND PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
<1120>
<1100>
<1120>
<1100>
10 nm
(a) (b)
FIG. 9. Snapshots (a) before and (b) after a 50-keV collision
cascade overlapping with an Fe-enriched a-type vacancy loop at
573 K. Fe atoms are colored in red, and the displaced atoms are
shown here based on common neighbor analysis.
push it in a position where it shares the loop’s basal plane.
These energy gradients may explain why multiple loops are
experimentally found lying on the same basal plane, roughly
equally spaced apart.
As a whole, our results show that properly modeling
the temperature and strain dependence of the diffusion
coefficients of vacancies and vacancy clusters is crucial in
capturing different growth regimes at different temperatures.
In the particular case of zirconium, this means that different
relative populations of a-type and c-component vacancy loops
will be observed near GBs for a given dose, depending on
irradiation temperature.
It should be noted that the MA07 potential (No. 3) selected
in the present study is notorious for underestimating the
binding energies of divacancies and trivacancies [40]. Since
vacancy loop growth is affected by the number of neigh-
boring monovacancies, divacancies, and other small clusters,
the vacancy loop growth simulations presented here are not
quantitatively accurate. For example, the real transition from
one growth regime to another will most likely not take place
at 800 K. Instead, it will take place at another temperature.
E. Stability of an a-type vacancy loop near the tilt GB during
collision cascades
The interaction of an a-type vacancy loop with a 50-keV
PKA near the GB was investigated in pure α-Zr. The collision
cascade induced a rotation and gradual glide toward the GB,
which led to eventual absorption by the GB as illustrated in
Fig. 8(e).
The importance of Fe alloying on the formation of c-
component loops was discussed by de Carlan et al. [82], and a
recent study by Topping et al. [83] suggested that an increase
in Fe content in a Zr alloy delays c-loop nucleation. Here,
the effect of Fe on the loop stability is explored. Collision
cascades were performed near a preexisting a-type vacancy
loop with a 50-keV PKA in the vicinity of a tilt GB. The
distance between the GB and the loop is about 10 nm, which
is the same size as the diameter of the loop [see Fig. 9(a)].
The hybrid MD/MC method was applied to perform Fe segre-
gation to the loop.
Fe segregation stabilized the loop, which did not undergo
cascade-induced absorption, as illustrated in Fig. 9. An anima-
tion of the cascade simulation is provided in the Supplemental
Material [64]. PKAs at multiple different locations and direc-
tions were simulated in this system, and all the results showed
that the loop was stabilized by the segregated Fe atoms during
the collision cascade.
Naturally, the strain field induced by the GBs gradually
decreases as the separation distance between the loop and
the GBs increases. One would expect the loop to have a
decreased propensity to be absorbed by the GBs. However,
this relation has not been systematically studied. In addition,
different types of GBs may produce different levels of strain
field that would then affect their sink strength as suggested
in [32,65]. In the future, studying these factors would be of
interest.
To understand how alloying Fe can stabilize the dislocation
loop, the effect of Fe segregation on the loop’s stress field
was explored. Fivenanometer-radius interstitial and vacancy
loops of a-type and c-component were equilibrated at 573 K.
Figure 10 illustrates the Fe concentration in the plane of the
dislocation loop after the relaxation, and Fe segregation was
greatest at interstitial loops. Fe atoms segregated to the inside
of the a-type interstitial loop, as illustrated in Fig. 10(a).The
concentration of Fe in the center of the loop is expected to de-
crease as loop size increases [76]. Figure 10(d) shows that the
c-component vacancy loop exhibits the least Fe segregation.
The distortion of the dislocation loop (elastic energy) and
segregation of Fe atoms (chemical energy) results in a change
of the system energy. Defect energy (EDef ) is defined as
the energy variation of an atom due to interaction with the
dislocation loop or chemical segregation: EDef =EPot −EPer,
where EPot is the potential energy of any atom, and EPer is
1
0.8
0.6
0.4
0.2
0
(b) (c) (d)
(a)
5 nm
[0002]
[2110]
[0002]
[2110]
[0110]
[2110]
[0110]
[2110]
FIG. 10. The Fe concentration map of dislocation loops in the Zr-Fe system: (a) a-type interstitial loop; (b) a-type vacancy loop; (c)
c-component interstitial loop; (d) c-component vacancy loop.
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PRIMARY DAMAGE PRODUCTION IN THE PRESENCE OF … PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
(d)
(b) (c)
(a)
5 nm
(eV/A)
3
o
0.016
0.012
0.008
0.004
0
(e) (f) (g) (h)
[0002]
[2110]
[0002]
[2110]
[0110]
[2110]
[0110]
[2110]
[0002]
[2110]
[0002]
[2110]
[0110]
[2110]
[0110]
[2110]
FIG. 11. The defect energy map of dislocation loops either without Fe segregation (a–d) or with Fe segregation (e,f): (a,e) are a-type
interstitial loop; (b,f) are a-type vacancy loop; (c,g) are c-component interstitial loop; (d,h) are c-component vacancy loop.
the averaged potential energy of an atom in a perfect Zr
lattice at equilibrium. The defect energy maps in the loop’s
plane without Fe segregation are shown in Figs. 11(a)–11(d),
and they are an a-type interstitial loop, a-type vacancy loop,
c-component interstitial loop, and c-component vacancy loop,
respectively. Figures 11(e)–11(h) are the defect energy maps
corresponding to Figs. 11(a)–11(d), which are in the condi-
tion of Fe segregation. By comparing Figs. 11(a)–11(d) and
Figs. 11(e)–11(h), we can see that the values of EDef are
greatly reduced by Fe segregation to the loops, which explains
why loops become more stable after Fe segregation.
The radial-direction component of atomic stresses (Srr)
of dislocation loops with and without Fe segregation were
calculated in Fig. 12.TheSrr of interstitial loops is greatly
affected by Fe segregation. Without Fe segregation, as il-
lustrated in Figs. 12(a) and 12(c),Srr is compressive inside
the loop. However, Srr became tensile after Fe segregation
to the loops; see Figs. 12(e) and 12(g). Moreover, the mag-
nitude of the contrast of Srr near the edge of the loop is
dramatically decreased after Fe segregation for all types of
the loop.
Recently, Harte et al. [80] hypothesized that a row of a-type
loops that are aligned parallel to the trace of the basal plane
might collapse into a coarse c-component loop. Fe segregation
can stabilize a-type loops as evidenced by the results above,
which may impede the transition from the aligned a-type
loops to a coarse c-component loop. This may explain why
c-component loop nucleation is delayed in Zr-Fe alloys, in
comparison to pure Zr, as reported by Topping et al. [83].
In addition, the enhanced stabilization of a-type loops by the
presence of Fe could prevent the absorption of a-type loops
by the GBs. This may affect the number of free vacancies and
their clusters near the GBs due to the growth of a-type loops.
IV. CONCLUSION
Primary cascade production of various defects has been
statistically studied. The defect production results in the cur-
rent study provide valuable additions to databases of pro-
duction rates which are necessary for kMC and rate-theory
models in the context of predicting radiation damage buildup.
The following conclusions were drawn:
FIG. 12. The radial-direction component of stress (Srr ) for dislocation loops either without Fe segregation (a–d) or with Fe segregation
(e,f): (a,e) are a-type interstitial loop, (b,f) are a-type vacancy loop; (c,g) are c-component interstitial loop; (d,h) are c-component vacancy
loop. Red is tensile and blue is compressive.
043602-9

DAI, LONG, SAIDI, BÉLAND, YAO, AND DAYMOND PHYSICAL REVIEW MATERIALS 3, 043602 (2019)
(1) Fewer residual defects remain after collision cascades
in the presence of extended defects, compared to a perfect
crystal. In particular, this will contribute to vacancy super-
saturation near the GBs, which could promote vacancy loop
growth.
(2) a-type vacancy loops are more stable than a-type
interstitial loops near the GBs. Also, simple diffusion sink
arguments predict a surplus of vacancy-type defects near GBs.
The hypothesis of vacancy supersaturation near GBs is further
supported by the results of in situ TEM experiments in this
study, which found a high density of vacancy loops near the
GBs.
(3) Vacancy loops have been simulated in a vacancy-
supersaturated environment. Growth of a-type vacancy loops
is faster than that of c-component loops at low tempera-
tures (e.g., 600 K), but both types of loops have similar
growth rates at high temperatures (e.g., 1000 K) since the
diffusion of vacancies and their clusters is dominant at high
temperatures. Thus, the ratio of a-to-cloops observed near
GBs is expected to be dependent on temperature. Also, the
simulations suggests that the capture radius of a-type vacancy
loops in the prismatic direction should be larger than that
in the basal direction and that of c-component loops. Fur-
thermore, analyzing the energy landscape around an a-type
vacancy loop also gave insights into why they are typically
observed lying on common basal planes, with somewhat
regular interloop spacing.
(4) a-type vacancy loops can be absorbed by GBs during
cascade-induced events.
(5) Fe segregation to the loops enhances loop stability by
reducing their defect energies and the magnitude of the stress
field at the edge of the loops. This effect impedes cascade-
induced absorption of loops by the GBs.
ACKNOWLEDGMENTS
The authors thank Compute Canada for generous allo-
cation of computer resources. The research was supported
by the NSERC (RGPIN-2018-04463) and NSERC/UNENE
Industrial Research Chair in Nuclear Materials (NIRC 345857
- 11) at Queen’s.
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