Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere
The problem of the diffusion evolution of a pore filled with molecular hydrogen in a spherical granule in a hydrogen medium is solved. The initial position of the pore is displaced relative to the center of the granule. A nonlinear system of equations is obtained, which describes the behavior of the...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere / M.I. Kopp, P.N. Ostapchuk, V.V. Yanovsky // Problems of Atomic Science and Technology. — 2022. — № 2. — С. 18-24. — Бібліогр.: 26 назв. — англ. |
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irk-123456789-1959392023-12-08T16:10:11Z Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere Kopp, M.I. Ostapchuk, P.N. Yanovsky, V.V. Physics of radiation damages and effects in solids The problem of the diffusion evolution of a pore filled with molecular hydrogen in a spherical granule in a hydrogen medium is solved. The initial position of the pore is displaced relative to the center of the granule. A nonlinear system of equations is obtained, which describes the behavior of the size of the gas-filled pore, the amount of gas in it and its position relative to the center of the bounded particle with time. Numerical calculations have shown the existence of two stages of evolution. The first (fast) stage is associated with the equalization of pressure in the pore with the external. The second is the slow diffusion “healing” of the pore, when the amount of gas adjusts to its size and the gas pressure is approximately equal to the external. Вирішено задачу про дифузійну еволюцію пори, наповненої молекулярним воднем в сферичній гранулі, що знаходиться у водневому середовищі. Початкове положення пори зміщене відносно центру гранули. Отримано нелінійну систему рівнянь, яка описує поведінку розмірів заповненої газом пори, кількості газу в ній та її положення відносно центру обмеженої частинки з часом. Чисельні розрахунки показали існування двох стадій еволюції. Перша (швидка) стадія пов’язана з вирівнюванням тиску в порі із зовнішнім. Друга – повільне дифузійне “загоювання” пори, коли кількість газу підлаштовується під її розмір, а тиск газу приблизно дорівнює зовнішньому. Решена задача о диффузионной эволюции поры, наполненной молекулярным водородом в сферической грануле, находящейся в водородной среде. При этом пора изначально смещена относительно центра гранулы. Получена нелинейная система уравнений, описывающая поведение со временем размера газонаполненной поры, количества газа в ней и ее положения относительно центра ограниченной частицы. Численные расчеты показали наличие двух стадий эволюции. Первая (быстрая) стадия связана с выравниванием давления в поре с внешним. Вторая – медленное диффузионное «залечивание» поры, когда количество газа подстраивается под ее размер, а давление газа приблизительно равно внешнему. 2022 Article Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere / M.I. Kopp, P.N. Ostapchuk, V.V. Yanovsky // Problems of Atomic Science and Technology. — 2022. — № 2. — С. 18-24. — Бібліогр.: 26 назв. — англ. 1562-6016 PACS: 61.46+w, 61.72-y DOI: https://doi.org/10.46813/2022-138-018 http://dspace.nbuv.gov.ua/handle/123456789/195939 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Physics of radiation damages and effects in solids Physics of radiation damages and effects in solids |
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Physics of radiation damages and effects in solids Physics of radiation damages and effects in solids Kopp, M.I. Ostapchuk, P.N. Yanovsky, V.V. Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere Вопросы атомной науки и техники |
| description |
The problem of the diffusion evolution of a pore filled with molecular hydrogen in a spherical granule in a hydrogen medium is solved. The initial position of the pore is displaced relative to the center of the granule. A nonlinear system of equations is obtained, which describes the behavior of the size of the gas-filled pore, the amount of gas in it and its position relative to the center of the bounded particle with time. Numerical calculations have shown the existence of two stages of evolution. The first (fast) stage is associated with the equalization of pressure in the pore with the external. The second is the slow diffusion “healing” of the pore, when the amount of gas adjusts to its size and the gas pressure is approximately equal to the external. |
| format |
Article |
| author |
Kopp, M.I. Ostapchuk, P.N. Yanovsky, V.V. |
| author_facet |
Kopp, M.I. Ostapchuk, P.N. Yanovsky, V.V. |
| author_sort |
Kopp, M.I. |
| title |
Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere |
| title_short |
Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere |
| title_full |
Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere |
| title_fullStr |
Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere |
| title_full_unstemmed |
Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere |
| title_sort |
diffusion evolution of a pore in bounded particle in a hydrogen atmosphere |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2022 |
| topic_facet |
Physics of radiation damages and effects in solids |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/195939 |
| citation_txt |
Diffusion evolution of a pore in bounded particle in a hydrogen atmosphere / M.I. Kopp, P.N. Ostapchuk, V.V. Yanovsky // Problems of Atomic Science and Technology. — 2022. — № 2. — С. 18-24. — Бібліогр.: 26 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT koppmi diffusionevolutionofaporeinboundedparticleinahydrogenatmosphere AT ostapchukpn diffusionevolutionofaporeinboundedparticleinahydrogenatmosphere AT yanovskyvv diffusionevolutionofaporeinboundedparticleinahydrogenatmosphere |
| first_indexed |
2025-07-17T00:15:57Z |
| last_indexed |
2025-07-17T00:15:57Z |
| _version_ |
1837851118740176896 |
| fulltext |
18 ISSN 1562-6016. ВАНТ. 2022. №2(138)
https://doi.org/10.46813/2022-138-018
DIFFUSION EVOLUTION OF A PORE IN BOUNDED PARTICLE
IN A HYDROGEN ATMOSPHERE
M.I. Kopp1, P.N. Ostapchuk2, V.V. Yanovsky1,3
1Institute for Single Crystals, National Academy of Science Ukraine, Kharkiv, Ukraine;
2Institute of Electrophysics and Radiation Technologies,
National Academy of Science Ukraine, Kharkiv, Ukraine;
3V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
E-mail: ostapchuk@kipt.kharkov.ua
The problem of the diffusion evolution of a pore filled with molecular hydrogen in a spherical granule in a
hydrogen medium is solved. The initial position of the pore is displaced relative to the center of the granule. A
nonlinear system of equations is obtained, which describes the behavior of the size of the gas-filled pore, the amount
of gas in it and its position relative to the center of the bounded particle with time. Numerical calculations have
shown the existence of two stages of evolution. The first (fast) stage is associated with the equalization of pressure
in the pore with the external. The second is the slow diffusion “healing” of the pore, when the amount of gas adjusts
to its size and the gas pressure is approximately equal to the external.
PACS: 61.46+w, 61.72-y
INTRODUCTION
The creation of new materials used in nuclear
technology, metallurgy, microelectronics, power
engineering, instrumentation, space and aviation
technology, electrochemical production, solar energy
and many other areas is to some extent associated with
the problem of porosity, which has a significant impact
on the service characteristics of materials. In the general
case, pores have an arbitrary form and size and can be
localized both within the elements of the structure of a
solid body (for example, inside crystallites, fragments,
blocks, cells, or granules) and along their boundaries
depending on the prehistory of the substance, its energy
balance, and structure [1].
Historically, theoretical calculations were initially
associated with the so-called diffusion porosity in an
unbounded homogeneous medium. According to
classical concepts, the diffusion porosity arises in a
solid phase supersaturated with point defects due to the
migration of excess vacancies and solute gas atoms and
includes the stage of pore nucleation and the stage of
their growth. As a rule, a phenomenological approach
is used for studying the kinetics of nucleation in
multicomponent systems, which is based on the
expression for the work ( )ix of formation of a
new phase nucleus and the Fokker-Planck kinetic
equation for the distribution function in the space of
variables ix . This approach is an extension of the
one-dimensional theory of Zeldovich and Frenkel to the
multidimensional case [2, 3]. As a result, the stationary
nucleation rate in the space of two or more variables is
calculated [4–9]. The general idea of all works is the
reduction of a multidimensional problem to a one-
dimensional. In this case, the methods of one-
dimensionalization are different, therefore, the pre-
exponential factors in the expression for the nucleation
rate also are different. As for the growth stage, here the
most complete theory has been developed for the so-
called coalescence stage. The main contribution to this
theory was made by V.V. Slezov and his students [10–
17].
However, the development of nanotechnologies
requires new theoretical researches of defect structures
in bounded particles of nano- and meso-scales. The
most widespread threedimensional defects in such
meso- and nanoparticles are vacancy pores, gas-filled
pores as well as new phase inclusions. The regularities
of diffusion growth, healing and motion of such defects
in nanoparticles is an important problem. Such defect
structure plays an important role for the possibility of
further compactification of nanoparticles and creating
new materials [18]. Establishing regularities of defect
structure evolution will enable one to control it as well
as to change properties of corresponding meso- and
nanoparticles. The creation of the theory of the diffusive
evolution of pores in bounded medias, for example, in
spherical nanoparticles, is a rather complicated task.
This problem is close to that of diffusion interaction of
pores in unbounded matrix [19]. Indeed, the role of
second object the pore interacts with in bounded
particles is played by matrix particle boundary.
Interaction with boundaries leads to principally different
pore behaviour as compared to that in unbounded
materials. The formation of pores in spherical
nanopaparticles was discovered experimentally in the
work [20]. In the review [21] the results are presented of
theoretical and numerical investigations related to
formation and disappearing of pores in spherical and
cylindrical nanoparticles. In [21], great attention is paid
to the problem of hole nanoshell stability, i. e. to the
case when in the nanoparticle center large vacancy
pores are situated. Analytical theory of diffusive
interaction of the nanoshell and the pore situated at
arbitrary distance from particle center was considered in
the works [22–24]. With the supposition of
quasiequilibrium of diffusive fluxes, the equations have
been obtained nonlinear equations for the change of the
radii of pore and spherical granule as well as of center-
to-center distance between the pore and the granule. It
was shown the absence of critical pore size unlike the
case of pores in an unbounded matrix [1]. In the case of
a general position, pore in such particles dissolves
mailto:ostapchuk@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2022. №2(138) 19
diffusively, while diminishing in size and shifting
towards granule center.
As is known, hydrogen almost always negatively
affects the service properties of metals and alloys.
Having high diffusion mobility, hydrogen penetrates
into metals to great thicknesses and a macroscale almost
evenly over the entire volume of the metal is distributed.
The harmful effect of hydrogen extends to the entire
volume of the metal. Although the destruction starts
from the most structurally dangerous places (stress
concentrators, the most stress parts of the product, etc.).
Hydrogen leads to many undesirable changes in the
mechanical properties of metals, which hydrogen
embrittlement of metals is called. If there is a
discontinuity in the material (for example, in the form of
a pore), hydrogen actively fills its volume creating high
pressure and facilitating the development of cracks.
Naturally, we can consider the problem of the
diffusion evolution of a pore filled with molecular
hydrogen in a spherical granule in a hydrogen medium.
STATEMENT OF THE PROBLEM
Let us consider the spherical granule of the radius
SR containing the gas-filled pore of the radius < SR R .
Granule and pore centers are separated from each other
by the distance l (Fig. 1). The surface of the granule is
surrounded by a one component molecular gas (for
example, hydrogen) with a constant pressure
0 =P const , which fills the pore. Let us assume that the
mechanism of pore filling with gas is as follows. The
hydrogen molecule 2H breaks up into atoms H on the
surface of granule, which penetrate into the matrix
forming a solid solution. There are exist various
mechanisms of gas atoms diffusion in the crystal lattice
of a bounded matrix. The examples of such mechanisms
are hopping of matrix atoms, as well as of complexes
formed by dopant atoms with vacancies or by other
ways. It is assumed that the solution is dilute enough to
ignore the interaction of dissolved atoms with each
other. Further, hydrogen atoms again collect into gas
molecules 2H on the surface of the pore, which fills the
pore. In this case, we assume that at the surface of pore
and outer boundary of granule in the matrix has been
maintained the local thermodynamic equilibrium
between the solid solution and molecular gas in the pore
and outside granule.
Fig. 1. On the left, the filling of a vacancy pore with molecular hydrogen is shown. There is a one atomic gas
(hydrogen) with partial pressure 0P at the granule boundary. On the right, a gas-filled pore in a bispherical
coordinate system is shown. The surface of pore and granule in this coordinate system are coordinate planes
const=
As can be seen from Fig. 1, the geometry of pore
and granule boundaries dictates the use of bispherical
coordinate system [25], as the most convenient one. In
bispherical coordinate system each point A of the
space is matched to three numbers ( , , ) , where
1
2
| |
= ln( )
| |
AO
AO
, 1 2= O AO , is polar angle.
Let us cite relations connecting bispherical coordinates
with Cartesian ones:
sin cos
= ,
cosh cos
a
x
−
sin sin
= ,
cosh cos
a
y
−
sinh
= ,
cosh cos
a
z
−
(1)
where a is the parameter, that at fixed values of pore
and granule radii as well as of their center-to-center
distance is determined by the relation
2 2 2 2[( ) ][( ) ]
=
2
s sl R R l R R
a
l
− − + −
.
Pore and granule surfaces in such coordinate system
are given by relations
1 =
a
arsinh
R
; 2 =
s
a
arsinh
R
. (2)
These relations determine values of 1 and 2 from
pore and granule radii, while includes additionally
center-to center distance l between the pore and the
granule.
Thus, the description of diffusion evolution of a gas-
filled pore in a bounded particle implies the existence of
equations for the rate of pore and granule volume
change, the distance between their centers, and the
number of gas molecules in the pore with time.
The equations describing the rate of pore and
granule volume change have the form [1, 26]:
=2 1
= |
4
vR n j dS
R
−
,
=2 2
= |
4
S v
S
R n j dS
R
−
. (3)
20 ISSN 1562-6016. ВАНТ. 2022. №2(138)
The rate of changing center-to-center distance between
the pore and the granule is determined by relation
=2 1
3
= ( ) |
4
vV n n j dS
R
−
. (4)
The integration is carried out over the surface of
pore and granule with the outer normal n to them;
is the volume per lattice node; =
1, 2
|vn j are the flux
densities of vacancies per pore and granule. The pore
filling rate with hydrogen has a form similar to (3):
=
1
1
2
= |HN n j dS− , (5)
where =
1
|Hn j is the flux density of atomic
hydrogen per pore. These fluxes comply Fick's first law
,
, ,=
v H
v H v H
D
j C−
and are determined from the solution of the diffusion
problem in the quasi-stationary approximation:
, 0v Hjdiv = (6)
,v HC – concentration of vacancies and atomic
hydrogen in the granule matrix; ,v HD are diffusion
coefficients.
The approximation (6) is valid when the
characteristic time for establishing the concentration
profile is much shorter than characteristic times for
changing the pore size and the volume of gas in it. We
note an important consequence (6): the total flow of
vacancies through any closed surface is conserved. It
means that
= =
1 2
| |v vn j dS n j dS =
and the rates of change in the pore and granule volumes
are related by
2 2
S SR R R R= or:
3 3
SR ( t ) V R( t )= + , (7)
where V is initial volume of granule material
(multiplier 4 3/ is omitted for convenience). The
existence of conservation law (7) enables us to reduce
the number of unknown quantities to three R , l , and
N .
The expression (6) takes the form in bispherical
coordinates
, ,
, ,
1 1 sin
= = 0.
cosh cos sin cosh cos
v H v H
v H
C C
C
+
− − .
(8)
Here we take into account that due to symmetry of the problem, vacancy concentration does not depend on variable
. The boundary conditions for vacancies are:
1
( , ) | = v
v RC C ;
2
( , ) | =
S
v
v RC C ,
v
RC ,
S
v
RC are equilibrium concentrations of vacancies near the spherical surface of pore and granule (see, for
example, [1, 26]):
0
2
= expv v
R
P
C C
kTR kT
−
,
0
0
2
= exp
S
v v
R
s
P
C C
kTR kT
− −
, (9)
where
0
vC is equilibrium vacancy concentration near
the plane surface, is surface energy, T is granule
temperature, ω is the volume per lattice site, P is gas
pressure inside the pore. For the simplicity we use state
equation of ideal gas:
34
=
3
P R N kT
. (10)
The boundary conditions for atomic hydrogen are
determined by the law of mass action:
1/2
H
R
P
C
kT
=
,
1/2
0
S
H
R
P
C
kT
=
, (11)
where is a constant characterizing the thermal
equilibrium at the surface of pore and granule of gas
molecules with respect to the chemical reaction of
dissociation into constituent atoms.
3/2 12
2/3
2
= exp expk H
k
h
m kT kT kT
−
− −
.
Here m is the mass of a gas molecule; h – Planck's
constant; k are the energy levels of the resting
molecule; the constant H characterizes the chemical
potential of hydrogen atoms for a dilute solid solution
= lnH H HkT C + .
Thus, the diffusion problem (8) is solved once for a
“faceless” concentration C with the same “faceless”
boundary conditions RC ,
SRC and then the
corresponding rates of change of the sought values are
written out taking into account specific expressions for
equilibrium concentrations (9)–(11).
ISSN 1562-6016. ВАНТ. 2022. №2(138) 21
SYSTEM OF EQUATIONS
General solution of the equation (8) with account of boundary conditions is determined as [25]
2
1
=0 1 2
sinh( 1/ 2)( )
( , ) = 2(cosh cos ) exp( ( 1/ 2) ) (cos )
sinh( 1/ 2)( )
R k
k
k
C C k P
k
+ −
− − + −
+ −
1
2
=0 1 2
sinh( 1/ 2)( )
exp( ( 1/ 2) ) (cos )
sinh( 1/ 2)( )SR k
k
k
C k P
k
+ −
− − +
+ −
, (12)
where ( )kP x are the Legendre functions. As a result, for the equations (3)–(5) we get:
02
2
( , ) ( , ) ,
S
v vv
R S R
S
D a a a
R C R P С R N
R R R
= −
; (13)
0
4
( ) ( , ) ,
S
H HH
R R
S
D a a a
N C P С R N
R R
= −
;
(2 1)
2
(2 1)( )
1 2
0 1
k
k
k
e
e
− +
+ −
=
=
−
; (14)
2 2
03 2
6
= ( , ) ( , ) 1
S
v vv
R S R
D a R
l C R P С R N
R a
− + −
;
(2 1)
2
(2 1)( )
1 2
=0
(2 1)
=
1
k
k
k
k e
e
− +
+ −
+
−
. (15)
Here it is taken into account that displacement
velocity rate along “z” coincides with dl / dt . The
details of the calculations are quite cumbersome and
included in the Appendix. The equations (9)–(11), (13)–
(15), and (7) with the appropriate initial conditions
completely determine the evolution of granule and gas-
filled pore with time. In the limiting case when the gas
is absent and there is no external pressure 0 0P = ,
equations (9), (13), (15) coincide with the results of
[22]. The numerical analysis shows that the function
and the expression in braces (15) are positive over the
entire physically reasonable ( SR R l + pore inside
the granule) range of SR , R , l .
Therefore, from (13) and (15) immediately follow that
the dissolving pore is displaced towards the center of
granule and vice versa. Next, we introduce
dimensionless variables for convenience:
0
R
r
R
= ;
0
S
S
R
r
R
= ;
0
l
L
R
= ;
0
a
R
= ;
0
N
n
N
= ;
0
2
A
kTR
= ;
0
3
0
3
4
N
B
R
=
;
0
0
P
p
kT
= ;
2
0
0
vv
v
R
t
D C
= ;
0
04
g
H
N
t
D R
=
;
g
t
t
= ;
v
g
t
q
t
= .
Here 00 |tR R == ; 00 |tN N == are the radius of pore and the number of gas molecules in at the initial moment
of time; vt and gt are the characteristic times of change in the size of pore and gas in it. The complete system of
equations (9)–(11), (13)–(15), (7) with initial conditions take the form in new variables:
02 3
2
exp exp ,
S S
dr A A n
p B
d q r r r r r r
= − − − −
; 0| 1r = = ; (16)
0 3
,
S
dn n
p B
d r r r
= −
; 0| 1n = = ;
2 2
03 3 2
6
= exp exp 1
S
dL A A n r
p B
d q r r r r
− − − − + −
; 0 0|L L= = ;
( )3
1
3
0
3
1S
/
Sr rr= + − ; 0 0|S Sr r= = ;
2
2 2 2 2 2
2 2 2 2
1 2
2
S
S S S S
r L r L r
L r r r r
= + − − +
.
Taking into account the equation of state of gas
3
=
n
p B
r
, we also have 0|= =p B . For further
numerical calculations, we will take: 1450T = K,
161 38 10k . −= erg/K is Boltzmann constant,
310 = erg/cm2 is surface energy density,
23 310 cm−= ,
4
0R 10 cm−= , which corresponds to
310A −= . We take the external gas pressure
22 ISSN 1562-6016. ВАНТ. 2022. №2(138)
10
0 10P = dyn/cm2 or in dimensionless
1
0 5 10p −= .
We set the dimensionless parameter 1q = . The values
0Sr , 0L , B will vary.
RESULTS
Let consider the case of a “small” pore ( 0 0Sr r )
with initial gas pressure 0 0|=p p , setting
0 100=Sr , 02.5=B p . We assume 0 10=L (pore
far from granule center) for definiteness. Fig. 2 shows
the dependence of pore radius r( ) on time (see
Fig. 2,a) and the change of hydrogen molecules number
n( ) in the pore on time (see Fig. 2,b) according to the
system of equations (16).
0 5 10 15 20
1
1.1
1.2
1.3
1.4
1.5
a)
r
a
0 5 10 15 20
0.5
0.6
0.7
0.8
0.9
1
b)
n
b
Fig. 2. Plots of the dependence of pore radius r (a) and the number of hydrogen molecules n (b) in the pore on
time at 02.5=B p , 0 100=Sr , 0 10=L
It can be seen that in a quite short period of time
( 8 ) the pore radius grows and the amount of gas
decreases in it reaching some quasi-stationary values
8| 1 432= =r . and 8| 0 588= =n . . In this case, the gas
pressure in the pore equal to outer pressure 8 0|= =p p .
And then there is a very slow dissolution of the pore but
with the condition ( ) 0 p p . It follows from system
(16) at 0=p p , n 0= . However, the pore radius
decreases a little r 0 causing an increase of pressure
in it and, as a result, a decrease the amount of gas.
There is a gas adjustment to the pore size for
maintaining the above condition.
In the other case 0 0|=p p (
22.5 10−= B ), the
process at the initial stage is reversed. The pore radius
rapidly decreases in front of increasing the amount of
gas and pressure in the pore. The dependences r( )
and n( ) on are shown in Fig. 3. The process
stabilizes at reaching pressure 0p , and we have a slow
dissolution of the pore. The amount of gas is controlled
size of pore.
It is clear that in the case 0 0|= =p p ( 0.5=B ) we
observe immediately regime of slow dissolution of the
pore. We note that this behavior of a “small” pore does
not depend on its location relative to the center of
granule. Similar results were obtained for 0 0 1=L .
(pore near the granule center) and 0 60=L (pore near
the granule boundary).
The same result is for the “large” pore
( 0 02= Sr r ). We must take into account the physical
condition S(L r) r+ (pore inside granule), so we set
0 0 15=L . .
0 5 10 15 20
0.2
0.4
0.6
0.8
1
a)
r
a
0 5 10 15 20
1
1.1
1.2
1.3
b)
n
b
Fig. 3. Plots of the dependence of pore radius r (a) and the number of hydrogen molecules n (b ) in the pore on
time at
2
02.5 10−= B p , 0 100=Sr , 0 10=L
ISSN 1562-6016. ВАНТ. 2022. №2(138) 23
0 5 10 15 20
1
1.1
1.2
1.3
1.4
1.5
a)
r
a
0 5 10 15 20
0.5
0.6
0.7
0.8
0.9
1
b)
n
b
0 5 10 15 20
0.15
0.16
0.17
0.18
0.19
0.2
c)
L
c
Fig. 4. Plots of the dependence of pore radius r (a) and the number of hydrogen molecules n (b) in the pore on
time and the position of pore L (c) relative to the center of granule at
2
02.5 10−= B p , 0 2Sr = , 0 0 15L .=
Fig. 4 shows the dependence of pore radius r( )
(see Fig. 4,a), the amount of gas n( ) in pore (see
Fig. 4,b) on time as well as the position L( ) of pore
relative to center of granule (see Fig. 4,c) at
02.5 .B p=
As expected, the pore quickly equalizes the pressure
( ( ) 0 p p ) and enters the stage of slow “healing”.
The exit time ( 3 ) is shorter than for the analogous
case of a “small” pore, although the situation is the
same. We note that the pore is displacement from
granule center (see Fig. 4,c) at the stage of intensive
growth of the pore. However, the above inequality is not
violated.
CONCLUSIONS
1. The problem of the diffusion evolution of a pore
filled with molecular hydrogen in a spherical granule in
a hydrogen medium is solved exactly. Initially, the pore
is displaced relative to the center of granule (see Fig. 1).
2. A nonlinear system of equations (16) is obtained,
wich describing the time behavior of the size of a gas-
filled pore, the amount of gas in pore, and the position
of pore relative to the center of a bounded particle.
3. The system (16) was solved numerically. The
calculations showed the presence of two stages of pore
evolution. The first (fast) stage is associated with the
equalization of pressure in the pore with the external
one. The second stage is the slow diffusion “healing” of
pore, when the amount of gas adjusts to its size, and the
gas pressure is approximately equal to the external one.
REFERENCES
1. P.G. Cheremskoy, V.V. Slyozov, V.I. Betehin.
Pores in Solid Matter. M.: “Energoatomizdat”, 1990,
360 p.
2. Ya.B. Zel’dovich // JETP. 1942, v. 12, p. 525.
3. Ya.I. Frenkel. Kinetic theory of Liquids. L.:
“Nauka”, 1975, 592 p.
4. H. Reiss // J. Chem. Phys. 1950, v. 18, p. 840.
5. J.S. Langer // Ann. Phys. 1969, v. 54, p. 258.
6. H. Trinkaus // Phys. Rev. B. 1983, v. 27, p. 7372.
7. F.M. Kuni, A.A. Melikhov // Theoret. and
Math. Phys.1989, v. 81, p. 247.
8. N.V. Alekseechkin, P.N. Ostapchuk // Solid State
Physics. 1993, v. 35, p. 929.
9. N.V. Alekseechkin // Solid State Physics. 2006,
v. 48, p. 1676.
10. Z.K. Saralidze, V.V. Slezov // Solid State
Physics. 1965, v. 7, p. 1605.
11. V.V. Slezov, V.B. Shikin // Euronuclear. 1965,
v. 2, N 3, p. 127.
12. V.V. Slezov // Metallofizika. 1981, v. 3, N 1,
p. 21 (in Russian).
13. V.V. Slezov, V.V. Sagalovich // Metallofizika.
1981, v. 3, N 2, p. 13 (in Russian).
14. V.V. Slezov, V.V. Sagalovich // The Physics of
Metals and Metallography. 1981, v. 52, issue 2, p. 263.
15. V.V. Slezov, E.F. Tichonov, M.А. Chusainov //
Metallofizika. 1982, v. 4, N 5, p. 3 (in Russian).
16. P.N. Ostapchuk, V.V. Slezov, V.V. Sagalovich
// Metallofizika. 1986, т.8, №2. p. 80 (in Russian).
17. P.N. Ostapchuk, V.V. Slezov, V.V. Sagalovich.
Gas porosity during diffusion decomposition: Rev. M.:
“TsNIIatominform”, 1988, 72 p.
18. A.V. Ragulya, V.V. Skhorohod. Consolidation
of nanostructural materials. Kiev: “Naukova dumka”,
2007, 374 p.
19. V.I. Dubinko, A.V. Tur, A.A.Turkin, and
V.V. Yanovsky // Phys. Met. Metallogr. 1989, v. 68,
p. 17.
20. Y. Yin, R.M. Rioux, C.K. Erdonmez, S. Hughes,
G.A. Somorjai, A.P. Alivisatos. Formation of hollow
nanocrystals through the nanoscale Kirkendall effect //
Science. 2004, v. 304 , p. 711.
21. T.V. Zaporozhets, A.M. Gusak, O.N. Podolyan //
Usp. Fiz. Met. 2012, v. 13, p. 1.
22. V.V. Yanovsky, M.I. Kopp, M.A. Ratner //
Functional Materials. 2012, v. 26, N 1, p. 131.
(arXiv:1809.06565v1 [cond-mat.mes-hall] 2018).
23. V.V. Yanovsky, M.I. Kopp, M.A. Ratner //
Functional Materials. 2020, v.27, N 3, p. 533.
24. M.A. Ratner, V.V. Yanovsky // Functional
Materials. 2021, v. 28, N 1, p. 151.
25. G. Arfken. Mathematical Methods in Physics.
M.: “Atomizdat”, 1970, 712 p.
26. Ja.E. Geguzin, M.A. Krivoglaz. Motion of
Macroscopic Inclusions in Solid Matter. M.:
“Metallurgy”, 1971.
24 ISSN 1562-6016. ВАНТ. 2022. №2(138)
APPENDIX
1. Auxiliary expressions:
( 1/2)
1
1
( ) 2
=
1/ 2cosh
k
kP t dt e
kt
− +
−
+−
. (17)
Via differentiating the relation (17) over parameter , one consequently finds
( 1/2)
1
3/21
( ) 2 2
=
sinh(cosh )
k
kP t dt e
t
− +
−
−
;
( 1/2)
1
5/2 31
( ) 4 2 (cosh ( 1/ 2)sinh )
=
3(cosh ) sinh
k
kP t dt e k
t
− +
−
+ +
−
. (18)
2. Dynamic growth equations:
1
= =
1 1
cosh cos
| = |
D c
n j
a
−
;
2
2
1
sin
=
(cosh cos )
a d d
dS
−
. (19)
=2 10
1
sin
= |
cosh cos2
va D C d
R
R
−
−
; =
10
1
sin
= |
cosh cos
2 HD d
N a
C
−
−
.
3. Substitute (19) in solution (12)
1
= 1 1
1
=01
1 2
1 2
=0 1 2
sinh
| = 2 (cos )exp( ( 1/ 2)) cosh cos
2 cosh cos
( 1/ 2) (cos ) ( 1/2) ( 1/2)
cosh( 1/ 2)( ) .
sinh( 1/ 2)( ) S
R
k
k
k
R R
k
CC
P k
k P k k
C k e C e
k
− −
− + + −
−
+ + + + − −
+ −
(20)
After replacing the signs of summation and integration, as well as using relations (17), (18), we obtain growth
rates (13), (14).
4. The rate of displacement of pore relative to the center of granule is determined by the relation
1
=2 210
1
3 cosh cos 1
= | sin
2 (cosh cos )
v
z
D a C
v e d
R
−
−
. (21)
After substituting relation (20) into (21) and corresponding calculations, we get (15).
Article received 17.02.2022
ДИФУЗІЙНА ЕВОЛЮЦІЯ ПОРИ В ОБМЕЖЕНІЙ ЧАСТИНЦІ
В АТМОСФЕРІ ВОДНЮ
М.І. Копп, П.М. Остапчук, В.В. Яновський
Вирішено задачу про дифузійну еволюцію пори, наповненої молекулярним воднем в сферичній гранулі,
що знаходиться у водневому середовищі. Початкове положення пори зміщене відносно центру гранули.
Отримано нелінійну систему рівнянь, яка описує поведінку розмірів заповненої газом пори, кількості газу в
ній та її положення відносно центру обмеженої частинки з часом. Чисельні розрахунки показали існування
двох стадій еволюції. Перша (швидка) стадія пов'язана з вирівнюванням тиску в порі із зовнішнім. Друга –
повільне дифузійне «загоювання» пори, коли кількість газу підлаштовується під її розмір, а тиск газу
приблизно дорівнює зовнішньому.
ДИФФУЗИОННАЯ ЭВОЛЮЦИЯ ПОРЫ В ОГРАНИЧЕННОЙ ЧАСТИЦЕ
В АТМОСФЕРЕ ВОДОРОДА
M.И. Koпп, П.Н. Oстапчук, В.В. Яновский
Решена задача о диффузионной эволюции поры, наполненной молекулярным водородом в сферической
грануле, находящейся в водородной среде. При этом пора изначально смещена относительно центра
гранулы. Получена нелинейная система уравнений, описывающая поведение со временем размера
газонаполненной поры, количества газа в ней и ее положения относительно центра ограниченной частицы.
Численные расчеты показали наличие двух стадий эволюции. Первая (быстрая) стадия связана с
выравниванием давления в поре с внешним. Вторая – медленное диффузионное «залечивание» поры, когда
количество газа подстраивается под ее размер, а давление газа приблизительно равно внешнему.
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