Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation
A model developed in this paper describes the transport of nonequilibrium (produced by irradiation) point defects across a coherent interface in a heterophase medium. In the framework of this model we derive a kinetic equation for the distribution function of spherical nanoparticles of the second ph...
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irk-123456789-1952032023-12-03T17:24:00Z Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation Borisenko, Alexander Physics of radiation damages and effects in solids A model developed in this paper describes the transport of nonequilibrium (produced by irradiation) point defects across a coherent interface in a heterophase medium. In the framework of this model we derive a kinetic equation for the distribution function of spherical nanoparticles of the second phase in a solid solution, which accounts for the flow and diffusion of nanoparticles in the dimension space as well as their dissolution in atomic collision cascades. We obtain an analytical form and study the stationary solution of this equation. The result obtained fits well to experimental data [A. Certain et al. Journal of Nuclear Mate 2013) 434, 311] on distribution of Y-Ti-O nanoparticles in the oxide dispersion strengthened ferritic steel 14YWT, irradiated with nickel ions up to 100 dpa at different temperatures. We conclude that in this case irradiation affects the distribution of fine oxide nanoparticles by creating nonequilibrium point defects rather than by cascade mixing. Модель, що розроблена в цій статті, описує транспорт нерівноважних (створених опроміненням) точкових дефектів через когерентну границю в гетерофазному середовищі. У рамках цієї моделі отримано кінетичне рівняння для функції розподілу сферичних наночастинок другої фази в твердому розчині, що включає плин та дифузію наночастинок у просторі розмірів, а також їх розчинення в каскадах атомних зіткнень. Отримано аналітичну форму та досліджено стаціонарний розв’язок цього рівняння. Отриманий результат гарно узгоджується з експериментальними даними [A. Certain et al. Journal of Nuclear Materіals. (2013) 434, 311] щодо розподілу Y-Ti-O-наночастинок у дисперсійно-зміцненій оксидами феритній сталі 14YWT, опро-міненій іонами нікелю до 100 зна за різних температур. Зроблено висновок, що в цьому випадку опромінення впливає на функцію розподілу наночастинок завдяки виникненню нерівноважних точкових дефектів, а не каскадному перемішуванню. Модель, разработанная в этой статье, описывает транспорт неравновесных (созданных облучением) точечных дефектов через когерентную границу в гетерофазной среде. В рамках этой модели получено кинетическое уравнение для функции распределения сферических наночастиц второй фазы в твердом растворе, которое включает течение и диффузию наночастиц в пространстве размеров, а также их растворение в кас-кадах атомных соударений. Получена аналитическая форма и исследовано стационарное решение этого уравнения. Полученый результат хорошо согласуется с экспериментальными данными [A. Certain et al. Journal of Nuclear Materіals. (2013) 434, 311] относительно распределения Y-Ti-O-наночастиц в дисперсно-упрочненной оксидами ферритной стали 14YWT, облученной ионами никеля до 100 сна при разных темпе-ратурах. Сделан вывод, что в этом случае облучение влияет на функцию распределения наночастиц благо-даря возникновению неравновесных точечных дефектов, а не каскадному перемешиванию. 2019 Article Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation / Alexander Borisenko // Problems of atomic science and technology. — 2019. — № 5. — С. 3-10. — Бібліогр.: 16 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/195203 669.017:621.039.53 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Physics of radiation damages and effects in solids Physics of radiation damages and effects in solids Borisenko, Alexander Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation Вопросы атомной науки и техники |
| description |
A model developed in this paper describes the transport of nonequilibrium (produced by irradiation) point defects across a coherent interface in a heterophase medium. In the framework of this model we derive a kinetic equation for the distribution function of spherical nanoparticles of the second phase in a solid solution, which accounts for the flow and diffusion of nanoparticles in the dimension space as well as their dissolution in atomic collision cascades. We obtain an analytical form and study the stationary solution of this equation. The result obtained fits well to experimental data [A. Certain et al. Journal of Nuclear Mate 2013) 434, 311] on distribution of Y-Ti-O nanoparticles in the oxide dispersion strengthened ferritic steel 14YWT, irradiated with nickel ions up to 100 dpa at different temperatures. We conclude that in this case irradiation affects the distribution of fine oxide nanoparticles by creating nonequilibrium point defects rather than by cascade mixing. |
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Article |
| author |
Borisenko, Alexander |
| author_facet |
Borisenko, Alexander |
| author_sort |
Borisenko, Alexander |
| title |
Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation |
| title_short |
Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation |
| title_full |
Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation |
| title_fullStr |
Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation |
| title_full_unstemmed |
Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation |
| title_sort |
theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2019 |
| topic_facet |
Physics of radiation damages and effects in solids |
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http://dspace.nbuv.gov.ua/handle/123456789/195203 |
| citation_txt |
Theoretical evidence for stationary size distribution of oxide nanoparticles in dispersion strengthened steel under cascade producing irradiation / Alexander Borisenko // Problems of atomic science and technology. — 2019. — № 5. — С. 3-10. — Бібліогр.: 16 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-16T23:03:41Z |
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2025-07-16T23:03:41Z |
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1837846518684450816 |
| fulltext |
ISSN 1562- -10.
SECTION 1
PHYSICS OF RADIATION DAMAGES
AND EFFECTS IN SOLIDS
UDC 669.017:621.039.53
THEORETICAL EVIDENCE FOR STATIONARY SIZE DISTRIBUTION
OF OXIDE NANOPARTICLES IN DISPERSION STRENGTHENED
STEEL UNDER CASCADE PRODUCING IRRADIATION
Alexander Borisenko
National Science Center Kharkov Institute of Physics and Technology , Kharkiv, Ukraine
E-mail: borisenko@kipt.kharkov.ua
A model developed in this paper describes the transport of nonequilibrium (produced by irradiation) point de-
fects across a coherent interface in a heterophase medium. In the framework of this model we derive a kinetic equa-
tion for the distribution function of spherical nanoparticles of the second phase in a solid solution, which accounts
for the flow and diffusion of nanoparticles in the dimension space as well as their dissolution in atomic collision cas-
cades. We obtain an analytical form and study the stationary solution of this equation. The result obtained fits well to
experimental data [A. Certain et al. Journal of Nuclear Mate 2013) 434, 311] on distribution of Y-Ti-O nano-
particles in the oxide dispersion strengthened ferritic steel 14YWT, irradiated with nickel ions up to 100 dpa at dif-
ferent temperatures. We conclude that in this case irradiation affects the distribution of fine oxide nanoparticles by
creating nonequilibrium point defects rather than by cascade mixing.
INTRODUCTION
Phase transformations in solids under irradiation
have been subject to intensive studies for a long time
(see e.g. [1, 2] for reference). Besides, these studies are
actual for R&D of novel structural materials for nuclear
applications. During previous decades a lot of effort has
been applied to investigation of the so-called Oxide
Dispersion Strengthened (ODS) steels. These steels are
considered prospective because they preserve satisfacto-
ry mechanical characteristics at temperatures up to
700 and radiation damage doses up to 150 displace-
ments per atom (dpa) due to the presence of a dence
(concentration ) dispersion of nanosized par-
ticles based on Y-Ti-O composition in the matrix. Now-
adays active studies of irradiation impact on the stability
of this dispersion are underway.
If irradiation is absent, the second phase dispersion
in a supersaturated solid matrix is usually unstable: the
cube of the mean particle size grows linearly with time
due to the effect of Ostwald ripening [3]. The effect of
cascade-producing irradiation is as follows: atomic col-
lision cascades, produced by high-energy neutrons or
ions, give rise to mixing of solute and matrix atoms and
trap the temporal evolution of nanoparticles to the cycle:
nucleation diffusion growth cascade dissolution.
The model of homogeneous semicoherent intephase
boundary [4] has been previously proposed to explain
the effect of nonequilibrium point defects (PD) on the
kinetics of phase transformations in solids under cas-
cadeless (electron) irradiation. The recent papers [5 7]
study the contribution of heterophase fluctuations to
solubility and nucleation rate of second phases in solid
solutions at non-radiation conditions. In this paper we
develop the above models and apply them to study the
stability of the second phase dispersion with coherent
interface in an alloy under cascade-producing (ion or
neutron) irradiation.
1. A MODEL OF THE SOLUTE ATOM
TRANSPORT IN A HETEROPHASE
STRUCTURE WITH A COHERENT
INTERFACE
Consider an interphase boundary (Gibbs interface)
between a stoichiometric particle (p), consisting of at-
oms of several types, labeled , and a solution of
these atoms in a solid matrix (m). Let the interface be-
tween the particle and the matrix be coherent, i.e. the
atomic planes be continuous across it. Since the bulk
physical properties of such heterophase structure are
discontinuous across the interface, the PD number densi-
ty (concentration) profiles are expected to be discontin-
uous as well. The PD can penetrate across the interface
via a thermal activation mechanism. Therefore, the PD
transfer across the interface can be considered as a re-
versible surface chemical reaction.
1.1. TRANSFER OF POINT DEFECTS ACROSS
THE INTERFACE
An interstitial atom of the type j, located at one side
of the interface, can transfer to the other side of the in-
terface and vice versa. This process can be represented
in the form of a reversible chemical reaction:
, (1)
where denotes an interstitial of the type j in the parti-
cle or in the matrix .
In this way, the rate of transitions, represented by
Eq. (1), in each direction, is proportional to the concen-
tration of the interstitials in the corresponding
phase and the normal component of the flux of j-type at-
oms across the interface via the interstitial mechanism is
as follows (hereinafter the normal unit vector is sup-
posed to be directed from the particle into the matrix):
, (2)
where is a mean unit atomic volume. The kinetic co-
efficients in Eq. (2) are assumed to depend on tempera-
ture according to the Ar
, where is an activation en-
ergy of transfer across the interface of the j-type intersti-
tial in the corresponding phase;
constant; is temperature.
An atom of the type j, located at a regular lattice site
at one side of the interface, can transfer to a neighboring
vacant site at the other side of the interface and vice ver-
sa. This process can be represented in the form of a re-
versible chemical reaction:
, (3)
where is an atom of the type j at the matrix lattice
site; is a vacant lattice site in the j-th sublattice of
the particle; is an atom in the j-th sublattice of the
particle; is a vacant lattice site in the matrix.
Therefore, the rate of transitions, represented by
Eq. (3), in each direction, should be bilinear in the con-
centrations of the corresponding species and the normal
component of the flux of j-type atoms across the inter-
face via the vacancy mechanism is as follows:
. (4)
Here is a concentration of vacancies in the j-th sub-
lattice of the particle; is a concentration of the at-
oms j at lattice sites of the matrix; is a concentration
of vacancies in the matrix, and is a concentration of
the atoms j at lattice sites of the particle.
Similarly to Eq. (2), the kinetic coefficients in
Eq. (4) are assumed to depend on temperature according
to the Ar , but with a different activation
energy .
An interstitial atom located at one side of the inter-
face can recombine with a vacancy located at the other
side:
. (5)
These are irreversible reactions because an energy
threshold for production of the Frenkel pairs is usually
large. The normal component of the flux of j-type atoms
across the interface via the recombination mechanism
(5) is as follows:
, (6)
where is a phenomenological recombination kinet-
ic coefficient in the corresponding phase.
Therefore, the partial flux of j-type atoms across the
interface is a sum of the contributions, given by Eqs. (2),
(4), and (6):
. (7)
A total concentration of j-type atoms in the corre-
sponding phase is a sum of the concentrations of the at-
oms in both the interstitial and regular positions:
. One can consider the following relations
between the concentrations of solute atoms in different
lattice positions:
, (8)
where is a dimensionless constant, taking its value
from the range . The lower and upper limiting
values correspond to the cases when the solute atoms re-
side only in the regular and interstitial lattice positions,
respectively. The value of depends on both intera-
tomic potential and irradiation conditions.
Now, taking into account Eqs. (2), (4), (6), and (8),
one can represent Eq. (7) as follows:
.1
1
0
0
p
v
m
j
m
R
m
j
p
v
m
j
m
i
m
j
m
v
p
j
p
R
p
j
m
v
p
j
p
i
p
jj
jjjj
jjj
cxxxc
cxxxcJ
(9)
The total atomic flux across the interface is
. (10)
As a next approximation, we require that a chemical
composition (stoichiometry) of the particle is conserved:
. (11)
Then the partial flux of atoms j across the interface (9) is
related to the total atomic flux (10) as follows:
. (12)
Therefore, employing Eq. (9), one can represent the
total atomic flux across the interface as follows:
(13)
The state of kinetic equilibrium at the interface is de-
termined by the condition that the total atomic flux
across it turns to zero:
. (14)
Taking into account Eq. (13), one can find from
Eq. (14) a relation between the kinetically equilibrium
solute and PD concentrations at the interface:
.(15)
In the absence of external perturbations, which vio-
late the conservativity of the system (e.g. irradiation),
the conditions of kinetic and thermodynamic equilibrium
are equivalent and, therefore, the values entering
Eq. (15) can be considered as thermodynamically equi-
librium ones. In this paper we consider thermodynami-
cally nonequilibrium situation caused by cascade-
producing irradiation. Nevertheless, the state of kinetic
equilibrium (14) in this case is still possible. Therefore,
all the values entering Eq. (15) are generally considered
as kinetically rather than thermodynamically equilibrium
ones.
1.2. DIFFUSION OF SOLUTE IN THE MATRIX
The steady-state solute concentration profile in the
matrix is subject to the next diffusion equation:
, (16)
where is a solute diffusion coefficient in the matrix.
The normal component of the solute flux across the
interface is given by Eq. (9). In the first order in a devia-
tion of the solute concentration at the interface from its
kinetic equilibrium value (15), for a spherical particle of
radius , Eq. (9) becomes
, (17)
where
eqp
v
eqm
j
m
R
eqm
j
p
v
eqm
j
m
i
m
j
jjjj
cxxx
D
l
10
(18)
is a characteristic length in the model.
One can consider the second boundary condition as
follows:
, (19)
where is an average solute concentration in the ma-
trix.
The diffusion equation (16) with the boundary condi-
tions, given by Eqs. (17), (19), has the next solution:
. (20)
With the above results in mind, one can derive the
particle velocity in the dimension space as follows:
. (21)
From Eq. (20) one can see that
(22)
and this expression can be used instead of Eq. (17) as
the first boundary condition for the diffusion equation
(16). This means that, for big particles , the
model considered here asymptotically gives the same re-
sult as the diffusion problem with a given (kinetically
equilibrium) solute concentration at the interface, given
by the Gibbs-Thomson relation (see e.g. [8]):
, (23)
where is a kinetically equilibrium solubility limit
of solute in the matrix,
(24)
and is a specific nanoparticle-matrix interface energy.
2. SIZE DISTRIBUTION
OF NANOPARTICLES
In the absence of irradiation the temporal evolution
of the size distribution function of nanoparticles is sub-
ject to the Becker- [9].
High-energy heavy particles with energies of about
several megaelektronvolt in a solid create primary recoil
atoms with energies of up to tens kiloelektronvolt which,
in their turn, initiate atomic displacement cascades (see
e.g. [10]). After the cascade relaxation, some of the in-
volved atoms appear at different from the initial ones
spatial positions. This effect is known as cascade mixing
(see e.g. [2]). As a result, a minor fraction of atoms is
displaced on distances exceeding the period of crystal-
line lattice. The effect of atomic collision cascades on
heterophase nanoparticles was recently studied for the
case of copper clusters in iron by the means of molecu-
lar dynamics method [11]. The results obtained demon-
strate that, if the primary recoil atom is within or on the
surface of the nanoparticle, a certain average number of
atoms leave the nanoparticle and move to the matrix.
This process is called cascade dissolution.
In this way, the kinetic equation for the nanoparticle
distribution function with respect to the number of
atoms n, taking into account the effect of irradiation that
creates atomic displacement cascades with a volume rate
, is the Becker-
cascade term in its right-hand side:
, (25)
where and are the rates of emission and
adsorption of atoms at the nanoparticle interface respec-
tively, and is a probability that the atomic
collision cascade kicks k atoms out of the nanoparticle
consisting of initially atoms. This probability is
subject to the next normalization condition:
. (26)
Assuming that , in Eq. (25) one can
change from the discrete variable n to the continuous
one z and expand its right-hand side into the Taylor se-
ries up to the second order to obtain the second order
differential equation:
, (27)
where and .
Taking into account that , from Eqs. (12) and (17), one can derive (see also [5, 6]):
, (28)
. (29)
Variable z is related to the nanoparticle radius as follows:
, (30)
which allows to change in Eq. (27) to the size distribution function :
, (31)
where and is assumed for simplicity.
Using Eq. (23) one can express the velocity and the diffusion coefficient in the dimension space in Eq. (31) as
follows:
, (32)
, (33)
where the critical radius (which turns the rhs of Eq. (32) to zero) is
. (34)
Below it is convenient to change to dimensionless variables and .
For the new distribution function Eq. (31) takes the form
, (35)
where
, (36)
, (37)
, (38)
, , and the inverse number of atoms
in a critical cluster is
. (39)
We aim to find a solution of the stationary
variant of Eq. (35):
, (40)
where
,
18633
1
3
2
22
2 kCBkCBkC
p
(41)
(42)
and the prime sign means the derivative over .
Provided that , in the WKB approximation
(see e.g. [12]), the asymptotic form of the physically rel-
evant fundamental solution of Eq. (40) is
, (43)
where is a normalization coefficient.
For further applications it is convenient to change back to the absolute size distribution function. Then Eq. (43)
takes the form
, (44)
where is a normalization coefficient,
, (45)
, (46)
. (47)
When irradiation is absent and the solute
concentration is undersaturated , Eq. (44)
reduces to the equilibrium distribution function of sub-
critical nanoparticles (heterophase fluctuations) (see also
[5, 6]).
3. DISCUSSION AND FITTING OF THE
MODEL RESULTS TO SOME
EXPERIMENTAL DATA
In this Section we illustrate the model results by fit-
ting them to some experimental data of the atom-probe
tomography for the ODS steel 14YWT irradiated with
5 MeV Ni ions at different temperatures [13, 14].
Analysis of these experimental data demonstrates
that, under intensive cascade-producing irradiation with
5 MeV Ni ions, the oxide nanoparticles in the 14YWT
ODS steel are characterized by the stationary size distri-
bution. Therefore, in this case one can use the stationary
distribution function (44) derived in Section 3 for fitting.
For this purpose it is necessary to estimate numerically
the distribution parameter (47).
Consider the grows rate of the average nanoparticle
size at the coarsening stage, given by the Lifshits-
Slyozov formula [3]:
. (48)
Eq. (48) can be transformed as follows:
, (49)
where is an average number of atoms in the nanopar-
ticle. Eq. (49) may be considered as an average growth
rate due to diffusion process. From the other hand,
atomic collision cascades kick atoms from the nanopar-
ticle with an average rate
. (50)
Since, as it was mentioned before, the experimental-
ly obtained distribution function under irradiation
demonstrates stationary behavior, the average number of
atoms in the nanoparticle must conserve. Therefore, the
sum of rhs of Eqs. (49) and (50) is zero and from (47)
one finds
. (51)
In Figure, a f we plot the function (44) at finite and
zero values of the parameter together with the exper-
imental data [13], obtained by the atom-probe tomogra-
phy, on the size distribution of oxide nanoparticles in
14YWT ODS steel irradiated with 5 MeV Ni ions to
100 dpa at different temperatures.
The next values of the material parameters are used
in calculations. We take for the specific
nanoparticle-matrix interface energy. The mass density
of Y2Ti2O7 nanoparticles is ac-
cording to [15]. Therefore, the mean atomic volume in
this oxide is , almost equal to that of
bcc iron. This fact explains the high coherency of the
nanoparticle-matrix interface. The average number of
atoms, ejected from the nanoparticle by the atomic colli-
sion cascade, and its average square are taken and
respectively, according to results of molecular
dynamics simulations [11]. The values of parameters l
and at different temperatures are given in Table. To
find the limit of applicability of the discrete-continuous
transformation, one should consider that the radius of
the sphere of the unit atomic volume is
. A nanoparticle with con-
tains about 44 atoms. We take the lower limit radius
, corresponding to atoms
in the nanoparticle. The value of the critical radius
was considered infinitely large, corresponding to under-
saturated or saturated solute concentrations.
Distribution function parameters
600 450 300 100 75
850
(annealing)
0.035 0.045 0.05 0.1 3 0.045
0.354 0.587 1.054 1.619 40.836 0
From Figure, a c one can see that, at chosen values
of the model parameters, the calculated stationary distri-
bution functions with zero (squares) and non zero
(curves) are only slightly different in the 1...2 nm
range at 600, 450, and 300 C. In Figure, d the distribu-
tions at 100 C with zero and non zero are visually
indifferent. In Figure, f bars show the experimental dis-
tribution formed after annealing for 5 hours at 850 C in
the preliminary irradiated to 100 dpa at -75 C sample.
In the same Figure, f squares show the plot of stationary
distribution without irradiation, in a good agreement
with experimental data.
After irradiation to 100 dpa at -75 C, the atom
probe tomography does not detect any oxide nanoparti-
cles. The authors of [13] conclude that they are com-
pletely dissolved under irradiation. In Figure, e the dis-
tributions calculated with zero and non zero demon-
strate that at these conditions the oxide nanoparticles
become very small, with average size ,
corresponding to the average number atoms in
the nanoparticle. This value is very close to the minimal
number of solute atoms in a cluster , used as a
parameter in the cluster search algorithm employed [13].
In this way, the negative experimental result at these
conditions can be explained.
Normalized to its maximum size distribution function of oxide nanoparticles at temperatures:
a 600 ; b 450 ; c 300 ; d 100 ; e -75 ; f 850 . Bars correspond to atom probe
tomography data on 5 MeV Ni ion irradiated to 100 dpa ODS steel 14YWT [13]. Squares and curves are given
by Eq. (44) with zero and non zero values of the cascade parameter respectively.
The values of the model parameters are given in Table and in the main text
As it was mentioned before, at chosen values of the
model parameters, the cascade term has only a small
numerical effect on the calculated distribution function.
Nevertheless, it does not mean that irradiation does not
affect the distribution function at all. As follows from
Eq. (18), irradiation, which produces nonequilibrium
point defects and changes the solute diffusion coefficient
in the matrix, affects the value of the distribution param-
eter l. Really, from Table one can see that the value
is found both at 850 C without irradiation
and at 450 C under irradiation (see also Figure, b and
f). Therefore, it is tempting to assume that changes in
the nanoparticle distribution observed after irradiation
result from the effect of nonequilibrium point defects ra-
ther than from cascade mixing. This assumption is sup-
ported by the previous experimental [16] and theoretical
[4] findings. This assumption could be checked by per-
forming cascadeless (electron) irradiation experiments at
the same temperatures and dose rates.
CONCLUSIONS
Using the previously proposed model of point defect
transport in a heterophase medium [4 6], we obtain ex-
pressions for absorption and emission rates of solute at-
oms at the coherent interface as functions of steady state
concentrations of nonequilibrium point defects at the
opposite interface sides, providing the possibility to
study the kinetics of diffusion transformations in solids
under irradiation.
For the size distribution function of heterophase na-
noparticles under irradiation we obtain a kinetic equa-
tion with the cascade term and find its stationary solu-
tion in analytical form.
With the proper choice of the parameter values, the
model allows a good fit to experimental data [13] on the
size distribution function of oxide nanoparticles after
5 MeV Ni ion irradiation to 100 dpa at 600, 450, 300,
and 100 C and after post-irradiation annealing at
850 C.
a b c
d e f
We conclude that irradiation affects the distribution
of fine oxide nanoparticles by creating nonequilibrium
point defects rather than by cascade mixing. The exper-
imental check of this assumption is desirable.
ACKNOWLEDGEMENT
The author thanks to Prof. Oleksandr Bakai for dis-
cussion of the present result. This work was funded by
the National Academy of Science of Ukraine, Grant
# X-2-13-10/2019.
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[A. Certain 2013) 434, 311]
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