Atomic Structure in the Vicinity of Nanovoids and Features of These Defects
Many properties of metals are determined by the defects, such as point defects, their complexes and nanovoids, whereas properties of these defects are generally related to the changes in atomic structure in the vicinity of these defects. In this work, recently developed approach is applied to simula...
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Germanov, A.B. Ershova, I.V. Kislitskaya, E.V. Nazarov, A.V. Zaluzhnyi, A.G. 2016-07-04T19:42:00Z 2016-07-04T19:42:00Z 2013 Atomic Structure in the Vicinity of Nanovoids and Features of These Defects / A.B. Germanov, I.V. Ershova, E.V. Kislitskaya, A.V. Nazarov, A.G. Zaluzhnyi // Металлофизика и новейшие технологии. — 2013. — Т. 35, № 10. — С. 1319-1331. — Бібліогр.: 14 назв. — рос. 1024-1809 PACS numbers: 61.72.Bb, 61.72.jd, 65.40.De https://nasplib.isofts.kiev.ua/handle/123456789/104223 Many properties of metals are determined by the defects, such as point defects, their complexes and nanovoids, whereas properties of these defects are generally related to the changes in atomic structure in the vicinity of these defects. In this work, recently developed approach is applied to simulate vacancy complexes and nanovoids. A developed model on the basis of Molecular Statics is used to investigate the atomic structure peculiarities in the vicinity of vacancy complexes and nanovoids, and the atomic displacements in the elastic medium surrounding the computational cell are determined in a self-consistent manner. The second part of the work is concerned with the study of atomic structure changes under temperature increasing within the new model based on Molecular Dynamics. Within the scope of this model, coordinates of the atoms in the area nearby of vacancy complex or nanovoid surface are averaged, during a simulation. Obtained mean positions of atoms are used for calculation of averaged interatomic distances; that allows determining lattice-parameter temperature dependence and then temperature-determined changes of atomic structure in the defects’ vicinity. Simulation is performed for various f.c.c. and b.c.c. metals. For these metals, thermal expansion data are obtained, and the change of atomic structure in the defects’ vicinity is determined from temperature increase. Багато властивостей металів визначаються дефектами, їхніми комплексами і нанопорами, тоді як властивості цих дефектів, взагалі кажучи, пов’язані зі змінами атомної структури поблизу цих дефектів. У даній роботі нещодавно запропонований підхід застосовується для симуляції вакансійних комплексів та нанопор. Нещодавно запропонована модель на основі методи молекулярної статики використовується для дослідження особливостей атомної структури в околі вакансійних комплексів і нанопор, а також уможливлює самоузгодженим чином знайти атомні зміщення у пружньому середовищі, яке оточує розрахункову комірку. Другу частину роботи присвячено вивченню змін атомної структури зі зростанням температури, шляхом застосування нової моделі, яка ґрунтується вже на молекулярній динаміці. В рамках даної моделі координати атомів поблизу вакансійних комплексів або поверхні нанопор усереднюються в процесі симуляції. Одержані середні положення атомів використовуються для розрахунку усереднених міжатомних відстаней, що уможливлює спочатку одержати температурну залежність параметра ґратниці, а потім обумовлені температурою зміни атомної структури в околі дефектів. Моделювання проводилося для різних ГЦК- та ОЦК-металів. Для цих металів одержано дані для теплового розширення, а також визначено зміну атомної структури в околі дефектів при зростанні температури. Многие свойства металлов определяются дефектами, их комплексами и нанопорами, тогда как свойства этих дефектов, вообще говоря, связаны с изменениями атомной структуры вблизи этих дефектов. В данной работе недавно предложенный подход применяется для симуляции вакансионных комплексов и нанопор. Недавно предложенная модель на основе метода молекулярной статики используется для исследования особенностей атомной структуры в окрестности вакансионных комплексов и нанопор, а также позволяет самосогласованным образом найти атомные смещения в упругой среде, окружающей расчётную ячейку. Вторая часть работы посвящена изучению изменений атомной структуры с ростом температуры путём применения новой модели, основанной уже на молекулярной динамике. В рамках данной модели координаты атомов вблизи вакансионных комплексов или поверхности нанопор усредняются в процессе симуляции. Полученные средние положения атомов используются для расчёта усреднённых межатомных расстояний, что позволяет сначала получить температурную зависимость параметра решётки, а затем обусловленные температурой изменения атомной структуры в окрестности дефектов. Моделирование проводилось для различных ГЦК- и ОЦК-металлов. Для этих металлов получены данные для теплового расширения, а также определено измене en Інститут металофізики ім. Г.В. Курдюмова НАН України Металлофизика и новейшие технологии Дефекты кристаллической решётки Atomic Structure in the Vicinity of Nanovoids and Features of These Defects Атомная структура вблизи нанопор и особенности этих дефектов Article published earlier |
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| title |
Atomic Structure in the Vicinity of Nanovoids and Features of These Defects |
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Atomic Structure in the Vicinity of Nanovoids and Features of These Defects Germanov, A.B. Ershova, I.V. Kislitskaya, E.V. Nazarov, A.V. Zaluzhnyi, A.G. Дефекты кристаллической решётки |
| title_short |
Atomic Structure in the Vicinity of Nanovoids and Features of These Defects |
| title_full |
Atomic Structure in the Vicinity of Nanovoids and Features of These Defects |
| title_fullStr |
Atomic Structure in the Vicinity of Nanovoids and Features of These Defects |
| title_full_unstemmed |
Atomic Structure in the Vicinity of Nanovoids and Features of These Defects |
| title_sort |
atomic structure in the vicinity of nanovoids and features of these defects |
| author |
Germanov, A.B. Ershova, I.V. Kislitskaya, E.V. Nazarov, A.V. Zaluzhnyi, A.G. |
| author_facet |
Germanov, A.B. Ershova, I.V. Kislitskaya, E.V. Nazarov, A.V. Zaluzhnyi, A.G. |
| topic |
Дефекты кристаллической решётки |
| topic_facet |
Дефекты кристаллической решётки |
| publishDate |
2013 |
| language |
English |
| container_title |
Металлофизика и новейшие технологии |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| format |
Article |
| title_alt |
Атомная структура вблизи нанопор и особенности этих дефектов |
| description |
Many properties of metals are determined by the defects, such as point defects, their complexes and nanovoids, whereas properties of these defects are generally related to the changes in atomic structure in the vicinity of these defects. In this work, recently developed approach is applied to simulate vacancy complexes and nanovoids. A developed model on the basis of Molecular Statics is used to investigate the atomic structure peculiarities in the vicinity of vacancy complexes and nanovoids, and the atomic displacements in the elastic medium surrounding the computational cell are determined in a self-consistent manner. The second part of the work is concerned with the study of atomic structure changes under temperature increasing within the new model based on Molecular Dynamics. Within the scope of this model, coordinates of the atoms in the area nearby of vacancy complex or nanovoid surface are averaged, during a simulation. Obtained mean positions of atoms are used for calculation of averaged interatomic distances; that allows determining lattice-parameter temperature dependence and then temperature-determined changes of atomic structure in the defects’ vicinity. Simulation is performed for various f.c.c. and b.c.c. metals. For these metals, thermal expansion data are obtained, and the change of atomic structure in the defects’ vicinity is determined from temperature increase.
Багато властивостей металів визначаються дефектами, їхніми комплексами і нанопорами, тоді як властивості цих дефектів, взагалі кажучи, пов’язані зі змінами атомної структури поблизу цих дефектів. У даній роботі нещодавно запропонований підхід застосовується для симуляції вакансійних комплексів та нанопор. Нещодавно запропонована модель на основі методи молекулярної статики використовується для дослідження особливостей атомної структури в околі вакансійних комплексів і нанопор, а також уможливлює самоузгодженим чином знайти атомні зміщення у пружньому середовищі, яке оточує розрахункову комірку. Другу частину роботи присвячено вивченню змін атомної структури зі зростанням температури, шляхом застосування нової моделі, яка ґрунтується вже на молекулярній динаміці. В рамках даної моделі координати атомів поблизу вакансійних комплексів або поверхні нанопор усереднюються в процесі симуляції. Одержані середні положення атомів використовуються для розрахунку усереднених міжатомних відстаней, що уможливлює спочатку одержати температурну залежність параметра ґратниці, а потім обумовлені температурою зміни атомної структури в околі дефектів. Моделювання проводилося для різних ГЦК- та ОЦК-металів. Для цих металів одержано дані для теплового розширення, а також визначено зміну атомної структури в околі дефектів при зростанні температури.
Многие свойства металлов определяются дефектами, их комплексами и нанопорами, тогда как свойства этих дефектов, вообще говоря, связаны с изменениями атомной структуры вблизи этих дефектов. В данной работе недавно предложенный подход применяется для симуляции вакансионных комплексов и нанопор. Недавно предложенная модель на основе метода молекулярной статики используется для исследования особенностей атомной структуры в окрестности вакансионных комплексов и нанопор, а также позволяет самосогласованным образом найти атомные смещения в упругой среде, окружающей расчётную ячейку. Вторая часть работы посвящена изучению изменений атомной структуры с ростом температуры путём применения новой модели, основанной уже на молекулярной динамике. В рамках данной модели координаты атомов вблизи вакансионных комплексов или поверхности нанопор усредняются в процессе симуляции. Полученные средние положения атомов используются для расчёта усреднённых межатомных расстояний, что позволяет сначала получить температурную зависимость параметра решётки, а затем обусловленные температурой изменения атомной структуры в окрестности дефектов. Моделирование проводилось для различных ГЦК- и ОЦК-металлов. Для этих металлов получены данные для теплового расширения, а также определено измене
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1024-1809 |
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https://nasplib.isofts.kiev.ua/handle/123456789/104223 |
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Atomic Structure in the Vicinity of Nanovoids and Features of These Defects / A.B. Germanov, I.V. Ershova, E.V. Kislitskaya, A.V. Nazarov, A.G. Zaluzhnyi // Металлофизика и новейшие технологии. — 2013. — Т. 35, № 10. — С. 1319-1331. — Бібліогр.: 14 назв. — рос. |
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1319
ДЕФЕКТЫ КРИСТАЛЛИЧЕСКОЙ РЕШЁТКИ
PACS numbers: 61.72.Bb, 61.72.jd, 65.40.De
Atomic Structure in the Vicinity of Nanovoids
and Features of These Defects
A. B. Germanov*,**, I. V. Ershova**, E. V. Kislitskaya**, A. V. Nazarov*,**,
and A. G. Zaluzhnyi*,**
*SSC RF ‘Institute for Theoretical and Experimental Physics’,
25 Bolshaya Cheremushkinskaya Str.,
117218 Moscow, Russia
**National Research Nuclear University,
31 Kashirskoe shosse,
115409 Moscow, Russia
Many properties of metals are determined by the defects, such as point de-
fects, their complexes and nanovoids, whereas properties of these defects are
generally related to the changes in atomic structure in the vicinity of these
defects. In this work, recently developed approach is applied to simulate va-
cancy complexes and nanovoids. A developed model on the basis of Molecular
Statics is used to investigate the atomic structure peculiarities in the vicinity of
vacancy complexes and nanovoids, and the atomic displacements in the elastic
medium surrounding the computational cell are determined in a self-consistent
manner. The second part of the work is concerned with the study of atomic struc-
ture changes under temperature increasing within the new model based on Mo-
lecular Dynamics. Within the scope of this model, coordinates of the atoms in the
area nearby of vacancy complex or nanovoid surface are averaged, during a sim-
ulation. Obtained mean positions of atoms are used for calculation of averaged
interatomic distances; that allows determining lattice-parameter temperature
dependence and then temperature-determined changes of atomic structure in the
defects’ vicinity. Simulation is performed for various f.c.c. and b.c.c. metals.
For these metals, thermal expansion data are obtained, and the change of atomic
structure in the defects’ vicinity is determined from temperature increase.
Багато властивостей металів визначаються дефектами, їхніми комплек-
сами і нанопорами, тоді як властивості цих дефектів, взагалі кажучи,
пов’язані зі змінами атомної структури поблизу цих дефектів. У даній
роботі нещодавно запропонований підхід застосовується для симуляції
вакансійних комплексів та нанопор. Нещодавно запропонована модель на
основі методи молекулярної статики використовується для дослідження
особливостей атомної структури в околі вакансійних комплексів і нано-
Металлофиз. новейшие технол. / Metallofiz. Noveishie Tekhnol.
2013, т. 35, № 10, сс. 1319—1331
Оттиски доступны непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2013 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
1320 A. B. GERMANOV, I. V. ERSHOVA, E. V. KISLITSKAYA et al.
пор, а також уможливлює самоузгодженим чином знайти атомні зміщен-
ня у пружньому середовищі, яке оточує розрахункову комірку. Другу ча-
стину роботи присвячено вивченню змін атомної структури зі зростанням
температури, шляхом застосування нової моделі, яка ґрунтується вже на
молекулярній динаміці. В рамках даної моделі координати атомів поблизу
вакансійних комплексів або поверхні нанопор усереднюються в процесі си-
муляції. Одержані середні положення атомів використовуються для розра-
хунку усереднених міжатомних відстаней, що уможливлює спочатку одер-
жати температурну залежність параметра ґратниці, а потім обумовлені те-
мпературою зміни атомної структури в околі дефектів. Моделювання про-
водилося для різних ГЦК- та ОЦК-металів. Для цих металів одержано дані
для теплового розширення, а також визначено зміну атомної структури в
околі дефектів при зростанні температури.
Многие свойства металлов определяются дефектами, их комплексами и
нанопорами, тогда как свойства этих дефектов, вообще говоря, связаны с
изменениями атомной структуры вблизи этих дефектов. В данной работе
недавно предложенный подход применяется для симуляции вакансион-
ных комплексов и нанопор. Недавно предложенная модель на основе ме-
тода молекулярной статики используется для исследования особенностей
атомной структуры в окрестности вакансионных комплексов и нанопор, а
также позволяет самосогласованным образом найти атомные смещения в
упругой среде, окружающей расчётную ячейку. Вторая часть работы по-
священа изучению изменений атомной структуры с ростом температуры
путём применения новой модели, основанной уже на молекулярной ди-
намике. В рамках данной модели координаты атомов вблизи вакансион-
ных комплексов или поверхности нанопор усредняются в процессе симу-
ляции. Полученные средние положения атомов используются для расчёта
усреднённых межатомных расстояний, что позволяет сначала получить
температурную зависимость параметра решётки, а затем обусловленные
температурой изменения атомной структуры в окрестности дефектов.
Моделирование проводилось для различных ГЦК- и ОЦК-металлов. Для
этих металлов получены данные для теплового расширения, а также
определено изменение атомной структуры в окрестности дефектов при
росте температуры.
Key words: point defects, vacancy complexes, nanovoids, b.c.c. metals, f.c.c.
metals, Molecular Statics, Molecular Dynamics, simulation.
(Received June 12, 2013)
1. INTRODUCTION
A considerable amount of micro- and nanovoids is generated in systems
under extreme conditions such as irradiation. The defects play a sig-
nificant role in the processes of material structure forming, diffusion
phase transformations, swelling, etc. Therefore, it is necessary to de-
velop the methods of determining the defects characteristics. It is ob-
ATOMIC STRUCTURE IN THE VICINITY OF NANOVOIDS 1321
vious that defect characteristics are determined by the atomic struc-
ture. Atoms surrounding defect shift from the sites of ideal lattice, e.g.
defect atomic structure changes with respect to an ideal one; that, in
turn, leads to changes in interaction energy of neighbour atoms and
results in modification of defect energy characteristics and other fea-
tures. In general, displacement fields in the vicinity of voids and nano-
voids were determined by the solution of equations from the classical
theory of elasticity. However, the validity of the theory of elasticity in
the case of nanovoids appears to be doubtful as for this kind of defects,
as well as for point defects, the usual usability condition does not meet
the conditions of the above-mentioned theory R a, where R is the
nanovoid radius and a is the lattice parameter. In other words, a de-
scription of displacement fields near the nanovoids in the framework
of the theory of elasticity appears to be problematic, as it does not take
into account the discrete atomic structure of materials.
For the better understanding of the arising problem, let us look at
the results of the works that dealt with the determination of the dis-
placement fields near point defects. In the work of Dederichs and
Pollmann [1], the general equations for the displacement field of point
defects in cubic crystals are given and some exact results are presented
on the basis of anisotropic elasticity theory. One from these exact re-
sults, which concerns the displacement field of point defects in cubic
crystals in the direction (100), coincides with one for the cubic axes:
,
)2)((
))((
2
1
4
)100(
44121112114411
44121211
441211
21
ccccccc
cccc
c
d
ccr
P
S (1)
where cij are the elastic constants, d c11 c12 2c44, P is the spur of the
dipole forces tensor Pij.
In another work [2], the atomic structure in the vicinity of monovacancy
is determined, using the model [3, 4], which takes into account the atomic
displacements in the elastic continuum surrounding the computational cell.
In Figure 1, the simulation results for displacements of atoms in the
vicinity of vacancy in the directions (100), (110), (111) are presented.
In this figure, the values for displacements in the direction (100) are
shown as a dashed line, another curve shows the displacements being
calculated by Eq. (1), given in article [1] by Dederichs.
From Figure 1, it can be seen, that the magnitudes of atomic displace-
ments, obtained by Molecular Statics (MS), differ significantly from the
predictions of the anisotropic theory of elasticity. It should be emphasized
that the displacement fields in the vicinity of vacancy obtained from the
computer simulation take into account anisotropy of media as well as its
discreteness, and these results lead us to the conclusion that the discrete-
ness factor for this problem is more vital than the influence of anisotropy.
The main aim of displacement field investigation in the vicinity of
1322 A. B. GERMANOV, I. V. ERSHOVA, E. V. KISLITSKAYA et al.
nanovoids is related to the evaluation of displacement field influence
on the void growth rate (the kinetic equations for the growth rate of
the voids in cubic metals are obtained, taking into consideration the
elastic strains arising from voids in [5], based on the general equations
for the vacancy fluxes under strain obtained in [6]), and, therefore,
there is a vital necessity to know displacements at the surface and
within the thin surface layer. That is why the information about dis-
placement fields, which can be obtained with the help of anisotropic
elasticity theory, does not allow to solve the specified problem.
In our recent works [2—4], a new approach was developed. In particular, in
this approach an iterative procedure is used, in which the atomic structure in
the vicinity of point defect and C1 constant, determining the displacement of
atoms embedded into an elastic continuum, are obtained in a self-consistent
manner. The vacancy features (including formation volumes and migration
volumes) obtained for a number of cubic metals agree well with experimental
values [2]. In this work, we use our approach for direct investigation of the
atomic structure in the vicinity of vacancy complexes and nanovoids. Molec-
ular Dynamics is used in the second part to investigate the changes of atomic
structure with the temperature increase. Thus, the obtained atoms positions
give us the possibility to determine these defects characteristics and to per-
form more advanced level simulation of nanovoid growth in materials, su-
persaturated with vacancies, in particular under irradiation.
2. SIMULATION OF MOLECULAR STATICS OF ATOMIC STRUCTURE
The equilibrium positions of atoms in the computational cell are simu-
lated using a variational procedure, which is usually employed in the
Fig. 1. The displacement field for the vacancy in case of the simulation data
for the directions (markers) and calculated from the Eq. (1).
ATOMIC STRUCTURE IN THE VICINITY OF NANOVOIDS 1323
molecular statics method [2, 3].
The computational cell (zone I, Fig. 2) is a sphere containing more
than 60000 atoms. It is surrounded by atoms embedded into an elastic
medium (zone II, Fig. 2). The displacements u of these atoms, connect-
ed with the disturbances caused by a pore, are found on the basis of the
first term of the static isotropic elastic equation solution [2—4]:
u C1r/r
3. (2)
In this model, a self-consistent iterative procedure is introduced to
calculate the C1 constant as well as the atomic structure simulation in
the crystal with a defect. The constant C1 is calculated according to Eq.
(2) using the results of atomic displacements simulation in the compu-
tational cell for the atoms in the coordination shells that are located in
a spherical layer III. The constant C1 calculated at the previous step of
the iterative procedure is used to determine atomic displacements of
the elastic medium II. Then, the relaxation of the atoms of zone I is
carried out anew, and the constant C1 is calculated again.
Thus, obtained atomic structure in the vicinity of pores is compared
with the displacement field from the solution of equations from the the-
ory of elasticity. According to classical theory of elasticity, displace-
ments fields in a spherical layer G in region near void of radius R [7]:
0 1 0 13 3
3 33
0 13 3 3 3
, ,
( 2 / )1 2 ( 2 / ) 1
, ,
2
x y
G
G G
x y
u C x C u C y C
r r
p R R Rp R R
C C
E ER R R R
(3)
where is the Poisson ratio, E is the Young modulus, is the specific
surface energy, p is the internal pressure, RG is the radius of the sphere
Fig. 2. Scheme of computational cell: I–directly computational cell, II–
atoms embedded in an elastic continuum, III–atoms for C1 calculation.
1324 A. B. GERMANOV, I. V. ERSHOVA, E. V. KISLITSKAYA et al.
which volume equals to the average volume per void. In our calcula-
tions, only one void is taken into account and C0 equals to zero.
3. RESULTS OF MS SIMULATION
The simulation is done for vacancy complex of 15 vacancies (we may
consider it as nanovoid of r 5.72 Å) and for nanovoid of size r 8.08 Å
that has atoms of up to 2 and 10 coordination shells deleted. The pair-
wise potential for b.c.c. iron is used [8].
The results for displacement fields ux at z 0 obtained using MS
simulation and equations from the theory of elasticity (Eq. (3)) for two
pores are presented in Fig. 3 and Fig. 4. From the symmetry considera-
tions, it is clear that the displacement fields obtained for uy are similar
to ux and differ just by a 90 degrees rotation over z-axis. Predicted by
the MS simulation, displacement fields as a function of coordinates in
the vicinity of nanovoids have functionally complex non-monotonic
character, unlike the results from the theory of elasticity. Their shapes
are more complicated, include several differently localized maximums
and minimums, absolute value of which is larger in comparison with
the results, given by equations from the classical theory of elasticity.
These peculiarities of displacement fields should be taken into consid-
eration, when simulating void growth rate takes into account elastic
fields, generated by nanovoids.
From elastic theory, it is known that volume change connected with lattice
relaxation during defect formation linearly depends on C1 constant [2, 3]:
V 4C1. (4)
It should be emphasized that the results of self-consistent calculation
Fig. 3. The displacement field for the vacancy complex of 15 vacancies (nano-
void with radius r 2a 5.72 Å) as obtained, using MS simulation (left) and
using equation from the theory of elasticity (right).
ATOMIC STRUCTURE IN THE VICINITY OF NANOVOIDS 1325
for atomic structure in the vicinity of vacancy complexes and nanovoids
point out a non-monotonic character of size dependence of the relaxation
volume, unlike the results from the theory of elasticity (Eq. (3)).
3.1. Molecular Dynamics Simulation of Thermal Expansion and
Atomic Structure in the Vicinity of Vacancies, Vacancy Complexes,
and Nanovoids
Some time ago, we suggested a new model to simulate atomic structure
changes caused by temperature [12, 13]. At first, we describe main fea-
tures of the model in a short paragraph, and then simulation results
concerned with a thermal expansion of some metals and an atomic
structure in the vicinity of vacancies at finite temperatures are pre-
sented to illustrate abilities of our model. Last, results of simulation
for vacancy complexes and nanovoids are presented.
3.2. Model for Atomic Structure Simulation at Finite Temperature
Atomic structure of defectless crystal is studied by Molecular Dynamics
(MD) using velocity Verlet algorithm. The computational cell has free-
boundary conditions and a spherical shape and contains up to 50000 atoms.
Coordinates of the central atom and its neighbours inside the nearest shells
are averaged during a simulation run that lasts for large number of vibra-
tions of the atom on the lattice site (up to 1000 vibration periods) (Fig. 5).
This procedure of getting interatomic distances from averaged posi-
Fig. 4. The displacement field for the nanovoid with radius 3/2
(3 /2)r a
7.43 Å as obtained, using MS simulation (left) and using equation from the
theory of elasticity (right).
1326 A. B. GERMANOV, I. V. ERSHOVA, E. V. KISLITSKAYA et al.
tions of atom in the centre of MD cell is applied to lattice of defectless crys-
tal. Using interatomic distances and geometry of f.c.c. and b.c.c. struc-
tures, we directly get lattice parameter. Thus, lattice parameter is deter-
mined in the nearly the same way as it is done in X-Ray measurements: in
experiment, lattice parameter is obtained directly from such structure
characteristics as interplanar spacing, whereas in our simulation lattice
parameter is directly obtained from interatomic positions. Repeating this
simulation at different increasing temperatures, we get the temperature
dependence of the lattice parameter. These thermal expansion curves for
f.c.c. and b.c.c. metals are shown on Fig. 6 (for iron in the range from
1185 K to 1667 K the data are shown for metastable b.c.c. lattice).
The simulation data show good agreement with the results of experi-
ments (X-ray lattice parameter determination) [14]. As our model gives
reliable results concerning changes of interatomic distances and lattice
parameters with temperature in defectless lattice, i.e. describes tempera-
ture changes in distances between atoms, therefore, this model is ex-
pected to describe the changes of atomic structure in the presence of de-
fects.
3.2.1. Atomic Structure in the Vicinity of Vacancies at Finite Temperatures
Further, the method of getting lattice parameter via obtained intera-
tomic distances from mean positions of atoms is applied for vacancies.
In this case, the atom in centre of computational cell is deleted and dis-
tances between atoms in the nearest neighbour shells to the defect are
calculated (Fig. 5).
Fig. 5. B.c.c. structure with atoms of two nearest-neighbour shells: hollow–
atom in the centre of computational cell (site of the deleted atom in case of vacan-
cy), filled–nearest-neighbour atoms, striped–next-nearest neighbour atoms.
ATOMIC STRUCTURE IN THE VICINITY OF NANOVOIDS 1327
Using the results for perfect crystal and for crystal with vacancy,
we get the temperature dependence of the ratio of distances for some
nearest shells in the system with vacancy to lattice parameter obtained
in perfect lattice (Fig. 7).
Fig. 7. Temperature dependence of ratio of distances between the centre of
empty site (vacancy) and atoms in its nearest shells to lattice parameter for
f.c.c. Ni (potential [11]).
Fig. 6. Thermal expansion of b.c.c and f.c.c. metals: Fe (potential [9]), V (po-
tential [10]), Al, Ni (potentials [11, 12]).
1328 A. B. GERMANOV, I. V. ERSHOVA, E. V. KISLITSKAYA et al.
We obtain a rather not obvious result that, for the whole tempera-
ture range beginning with fifth nearest shells, the ratio under study
remains constant with temperature (see Fig. 7).
Our simulation is performed for the shells nearest to the vacancy.
Because quantity of atomic displacements decreases significantly with
the distance from vacant site increasing in comparison with the near-
est shells, it should be expected that for shells having larger radius the
ratio of these shells radius to the lattice parameter conserves as well.
Then, we may conclude that geometrical similarity of atomic structure
in the vicinity of vacancies beginning with fifth nearest shells pre-
serves as the temperature increases. Whence an interesting and vital
effect follows: ground state atomic structure from the Molecular Stat-
ics and thermal expansion data both give the possibility to calculate
the atomic structure in the vicinity of vacancy for any temperature
and, consequently, to calculate the temperature dependence of vacancy
formation characteristics such as relaxation volumes and formation
volumes [13].
3.2.2. Results of Simulation for Vacancy Complexes and Nanovoids
The same procedure is applied for vacancy complexes and nanovoids.
Similarly to the case of vacancy, the atoms in the centre of the compu-
tational cell are deleted to create a vacancy complex or a nanovoid.
From average atomic positions (Fig. 8), we calculate interatomic dis-
Fig. 8. Void representation: hollow–sites of deleted atoms, filled–nearest-
neighbour atoms, and striped–next-nearest neighbour atoms.
ATOMIC STRUCTURE IN THE VICINITY OF NANOVOIDS 1329
tances in the nearest shell to the void border.
The simulation is done for vacancy complexes and nanovoids of the
same size as in MS simulation. The results of simulation for nanovoids
are shown in Fig. 9.
The ratio of distances between nearest neighbour atoms and centre
of defect to lattice parameter beginning with fifth nearest shells re-
mains constant. As a result, we can state that in the case of vacancy
complexes as well as in the case of nanovoids the same regularities that
were held true for vacancies are satisfied. This fact and its very signif-
icant conclusions should be further investigated concerning the stud-
ies of void growth in the presence of elastic fields.
Then, it is shown that the ratio of relaxation volume to the atomic
volume (or in the units of atomic volume ) remains constant as the
a
b
Fig. 9. Temperature dependence of ratio of distances between defect centre
and atoms in its nearest shells to lattice parameter for b.c.c. Fe [8]: vacancy
complex of 15 vacancies (rvoid 2a) (a); nanovoid of radius 7.43 Å (
void
r
3/2
(3 /2)a ) (b).
1330 A. B. GERMANOV, I. V. ERSHOVA, E. V. KISLITSKAYA et al.
temperature changes. Therefore, the C1 value is temperature dependent:
3222
1 )1)(0()0()]1)(0()][1)(0([)()()( TruTrTuTrTuTC (5)
and
.)1)(0()( 3
11 TCTC
Therefore, all the quantities entering in the definition of C1 may also
depend on temperature.
4. CONCLUSION
A model based on Molecular Dynamics and Molecular Statics is intro-
duced which allows to determine atomic structure in the vicinity of ag-
gregate defects, such as vacancy complexes and nanovoids. More thor-
ough investigation of atomic structure using Molecular Statics permits
to find qualitatively new peculiarities in displacement fields from nano-
voids that cannot be correctly obtained with the use of equations from
the theory of elasticity, because they cannot be applied to the atomic
scale. The fields of displacements in the vicinity of nanovoids are signif-
icantly more complicated and with much bigger magnitudes of dis-
placements than in the vicinity of vacancies. The displacement fields
from our calculations should be taken into account in void growth simu-
lations with the effects of elastic fields, generated by nanovoids.
The results concerning geometrical similarity preservation of the
atomic structure beginning with the fifth nearest shells in the vicinity
of vacancy during the temperature changes in metals are shown to be
correct in the case of nanovoids as well. As the surface curvature for
voids of larger radius will decrease, we may suggest that the preserva-
tion of geometrical similarity in atomic structure will hold true.
REFERENCES
1. P. H. Dederichs and J. Pollmann, Z. Phys., 255, No. 4: 315 (1972).
2. I. V. Valikova and A. V. Nazarov, Phys. Met. Metallogr., 109, No. 3: 220 (2010).
3. I. Valikova and A. Nazarov, Defect Diffus. Forum, 277: 125 (2008).
4. I. V. Valikova and A. V. Nazarov, Phys. Met. Metallogr., 105: 544 (2008).
5. A. Nazarov, A. Mikheev, I. Valikova, and A. Zaluzhnyi, Solid State Phenom.,
172—174: 1156 (2011).
6. A. V. Nazarov and A. A. Mikheev, J. Phys.: Condens. Matter, 20: 485203 (2008).
7. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Oxford: Elsevier: 1986).
8. R. A. Johnson, Phys. Rev., 134, No. 5A: 1329 (1964).
9. M. I. Mendelev, S. Han, D. J. Srolovitz et al., Philos. Mag., 83: 3977 (2003).
10. P. M. Derlet, D. Nguyen-Manh, and S. L. Dudarev, Phys. Rev. B, 76: 054107 (2007).
ATOMIC STRUCTURE IN THE VICINITY OF NANOVOIDS 1331
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59: 3393 (1999).
12. E. Reshetnikova, A. Germanov, I. Valikova, and A. Nazarov, Proc. of Interna-
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13. Presented for publication.
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/TUR <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>
/UKR <FEFF04120438043a043e0440043804410442043e043204430439044204350020044604560020043f043004400430043c043504420440043800200434043b044f0020044104420432043e04400435043d043d044f00200434043e043a0443043c0435043d044204560432002000410064006f006200650020005000440046002c0020044f043a04560020043d04300439043a04400430044904350020043f045604340445043e0434044f0442044c00200434043b044f0020043204380441043e043a043e044f043a04560441043d043e0433043e0020043f0435044004350434043404400443043a043e0432043e0433043e0020043404400443043a0443002e00200020042104420432043e04400435043d045600200434043e043a0443043c0435043d0442043800200050004400460020043c043e0436043d04300020043204560434043a0440043804420438002004430020004100630072006f006200610074002004420430002000410064006f00620065002000520065006100640065007200200035002e0030002004300431043e0020043f04560437043d04560448043e04570020043204350440044104560457002e>
/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|