On the kinetics of binary nucleation
Two problems of the theory of binary nucleation are solved: normalization of the equilibrium distribution function of nuclei and correct transition to the one-dimensional theory. Classification of multivariable nucleation processes is carried out and it is shown how to convert binary nucleation in...
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nasplib_isofts_kiev_ua-123456789-963502025-02-10T01:34:25Z On the kinetics of binary nucleation О кинетике бинарной нуклеации Про кінетику бінарної нуклеації Alekseechkin, N.V. Физика радиационных повреждений и явлений в твердых телах Two problems of the theory of binary nucleation are solved: normalization of the equilibrium distribution function of nuclei and correct transition to the one-dimensional theory. Classification of multivariable nucleation processes is carried out and it is shown how to convert binary nucleation into a process with linked fluxes by means of the corresponding transformation of the variables describing a nucleus. Just the use of the variables (total number of monomers, composition) makes it possible to solve the given problems. Two transitions to the one-dimensional nucleation are described. One of them corresponds to the formation of nuclei with stoichiometric composition. The other transition is that to unary (single-component) nucleation. Вирішено дві задачі теорії бінарної нуклеації: нормировка рівноважної функції розподілу зародків і коректний перехід в одномірну теорію. Проведено класифікацію багатомірних процесів зародження і показано, як перетворити бінарну нуклеацію в процес зі зв’язаними потоками за допомогою відповідного перетворення перемінных опису зародка. Саме використання перемінних (повне число мономерів, склад) дозволяе вирішити дані задачі. Описано два переходи в одномірну нуклеацію. Один з них відповідає утворенню зародків стехіометричного складу. Другий є перехід в однокомпонентну нуклеацію. Решены две задачи теории бинарной нуклеации: нормировка равновесной функции распределения зародышей и корректный переход в одномерную теорию. Проведена классификация многомерных процессов зарождения и показано, как представить бинарную нуклеацию в виде процесса со связанными потоками посредством соответствующего преобразования переменных описания зародыша. Именно использование переменных (полное число мономеров, состав) позволяет решить данные задачи. Описаны два перехода в одномерную нуклеацию. Один из них соответствует образованию зародышей стехиометрического состава. Другой есть переход в однокомпонентную нуклеацию. 2009 Article On the kinetics of binary nucleation / N.V. Alekseechkin // Вопросы атомной науки и техники. — 2009. — № 4. — С.170 -174. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 64.60.-I, 64.60.Qb, 05.20.Dd, 05.10Gg https://nasplib.isofts.kiev.ua/handle/123456789/96350 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Физика радиационных повреждений и явлений в твердых телах Физика радиационных повреждений и явлений в твердых телах |
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Физика радиационных повреждений и явлений в твердых телах Физика радиационных повреждений и явлений в твердых телах Alekseechkin, N.V. On the kinetics of binary nucleation Вопросы атомной науки и техники |
| description |
Two problems of the theory of binary nucleation are solved: normalization of the equilibrium distribution
function of nuclei and correct transition to the one-dimensional theory. Classification of multivariable nucleation
processes is carried out and it is shown how to convert binary nucleation into a process with linked fluxes by means
of the corresponding transformation of the variables describing a nucleus. Just the use of the variables (total number
of monomers, composition) makes it possible to solve the given problems. Two transitions to the one-dimensional
nucleation are described. One of them corresponds to the formation of nuclei with stoichiometric composition. The
other transition is that to unary (single-component) nucleation. |
| format |
Article |
| author |
Alekseechkin, N.V. |
| author_facet |
Alekseechkin, N.V. |
| author_sort |
Alekseechkin, N.V. |
| title |
On the kinetics of binary nucleation |
| title_short |
On the kinetics of binary nucleation |
| title_full |
On the kinetics of binary nucleation |
| title_fullStr |
On the kinetics of binary nucleation |
| title_full_unstemmed |
On the kinetics of binary nucleation |
| title_sort |
on the kinetics of binary nucleation |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2009 |
| topic_facet |
Физика радиационных повреждений и явлений в твердых телах |
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https://nasplib.isofts.kiev.ua/handle/123456789/96350 |
| citation_txt |
On the kinetics of binary nucleation / N.V. Alekseechkin // Вопросы атомной науки и техники. — 2009. — № 4. — С.170 -174. — Бібліогр.: 10 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT alekseechkinnv onthekineticsofbinarynucleation AT alekseechkinnv okinetikebinarnoinukleacii AT alekseechkinnv prokínetikubínarnoínukleacíí |
| first_indexed |
2025-12-02T12:47:30Z |
| last_indexed |
2025-12-02T12:47:30Z |
| _version_ |
1850400728753897472 |
| fulltext |
ON THE KINETICS OF BINARY NUCLEATION
N.V. Alekseechkin
Akhiezer Institute for Theoretical Physics,
National Science Center "Kharkov Institute of Physics and Technology", Kharkov, Ukraine
E-mail: n.alex@kipt.kharkov.ua
Two problems of the theory of binary nucleation are solved: normalization of the equilibrium distribution
function of nuclei and correct transition to the one-dimensional theory. Classification of multivariable nucleation
processes is carried out and it is shown how to convert binary nucleation into a process with linked fluxes by means
of the corresponding transformation of the variables describing a nucleus. Just the use of the variables (total number
of monomers, composition) makes it possible to solve the given problems. Two transitions to the one-dimensional
nucleation are described. One of them corresponds to the formation of nuclei with stoichiometric composition. The
other transition is that to unary (single-component) nucleation.
PACS: 64.60.-I, 64.60.Qb, 05.20.Dd, 05.10Gg
INTRODUCTION
The phenomena of binary and multicomponent
nucleation are studied rather intensively both
theoretically and experimentally. The examples of such
processes include the formation of gas bubbles in a two-
component solution of vacancies and gas atoms in
solids under irradiation [1,2], the nucleation of two- and
multicomponent precipitates in alloys and solid
solutions, the formation of aerosols in atmosphere
(nucleation in a gas mixture). The theory of these
phenomena relates to more general multivariable theory
of nucleation [3] which is an extension of Zel'dovich-
Frenkel’ one-dimensional theory [4,5] to the
multivariable case. The phenomenological approach
used in this theory is based on the expression for the
work ),...,,( 21 pξξξΔΦ of the new-phase nucleus
formation and the Fokker-Planck kinetic equation for
the distribution function (DF) );,...,,( 21 tf pξξξ in the
space of the variables }{ iξ that describe a nucleus.
Among the unsolved problems of binary nucleation, the
following two can be singled out: normalization of the
equilibrium DF and correct passage to the one-
dimensional limit. Notice that the steady state
nucleation rate and the DF can not be calculated
correctly without exact value of the normalization
constant. In the present report, both these problems are
solved with the use of the results of general
multivariable nucleation theory developed in Ref. [3].
MODEL AND MAIN RESULTS
OF THE MULTIVARIABLE THEORY
OF NUCLEATION
Near the saddle point , where a nucleus is
assumed to be a macroscopic subsystem, the work
∗ξ
ΔΦ
can be represented as a quadratic form,
),...,,(
2
1)( 21 pH ξξξ+ΔΦ=ΔΦ ∗ξ ,
kiikp hH ξξξξξ =),...,,( 21 , (1)
where kiikh ξξ ∂∂ΔΦ∂= 2 at ∗= ξξ , , )( ∗∗ ΔΦ=ΔΦ ξ
and all variables are measured from their critical
values; so, we have 0=∗ξ .
Upon being reduced to the sum of squares, this form
has one negative term. This is a characteristic feature of
the processes of multivariable nucleation; the
corresponding variable is called “unstable”. The nuclei
which have passed over the energetic barrier in the
vicinity of the saddle point as a result of Brownian
motion in the space }{ iξ are the viable fragments of a
new phase, so that the main problem of the theory is to
calculate their flux over this barrier (the nucleation
rate).
The work )(ξΔΦ determines the equilibrium DF of
nuclei as heterophase fluctuations [5]
⎥⎦
⎤
⎢⎣
⎡ ΔΦ
−=
kT
constf )(exp)(0
ξξ . (2)
As mentioned above, the evolution of the DF is
described by the Fokker-Planck equation
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
∂
∂
∂
∂
=
∂
∂ ),(),(),( tftfd
t
tf
i
j
ij
i
ξξξ ξ
ξξ
& . (3)
The condition of equality of the flux to zero in the
equilibrium state makes it possible to obtain an equation
for . Substituting into eq. (3), we get iξ& )(0 ξf
k
ik
kjk
ij
j
ij
i kT
z
h
kT
d
kT
d
ξξ
ξ
ξ −≡−=
∂
ΔΦ∂
−=& , , (4)
Hence
DHZ =
1−= ZHD . (5)
This significant relationship shows that macroscopic
equations of movement of a nucleus in its phase space
}{ iξ ( i.e. the matrix ) allow us to determine the
matrix of diffusivities in the Fokker-Planck equation.
Z
In Ref. [3], the following equation for the steady
state nucleation rate has been obtained:
( ) kThkTNI
∗ΔΦ
−−−= e2 1
1
11
21 λπ , (6)
where is an element of the matrix ; is the
number of monomers in unit volume of the initial
phase;
1
11
−h 1−H N
1λ is the negative eigenvalue of the matrix Z .
170 ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2009. №4-2.
Серия: Физика радиационных повреждений и радиационное материаловедение (94), с. 170-174.
An equation for the steady state DF is as follows [3]:
≡= ∫
∞+
−
)(
2
)(
0
2
2
)(
)()(
e
eHξ
e
e
e
ξξ
κ
κ
π
κ
dy
kT
ff
y
kT
s
,
)(2
)(
2
1
0 ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−≡
e
eHξξ
κkT
erfcf (7)
where )(1)( ξξ erferfc −≡ ; is the eigenvector of the
matrix corresponding to
e
Z 1λ (this is the flux
direction); )(eκ is the curvature of the normal section of
the saddle surface along the direction . ΔΦ e
Equations for the flux direction in processes of two-
variable nucleation have been derived in Ref. [3] for
different relationships between the nucleation
parameters. In the case of binary nucleation, the
following equation holds:
γ
γγ
θ +⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
−
−
=
2
12
1122
12
12
1122
2
)(
2 h
hhsignh
h
hhtg , (8)
where 1122 / dd=γ .
CLASSIFICATION OF MULTIVARIABLE
NUCLEATION PROCESSES
All multivariable processes of nucleation can be
divided into two classes: (i) processes with independent
changes of variables and (ii) processes with linked
fluxes [6,7]. In the first case, a variation in the variable
1ξ in an elementary event does not affect the value of
the variable 2ξ and vise versa; both the variables are
physically equivalent. The processes of binary
nucleation are of this type: a nucleus is characterized by
the numbers of monomers of each kind, 11 n≡ξ
and 22 n≡ξ ; the elementary event is the attachment or
detachment of a monomer. More general case is the
multicomponent nucleation; correspondingly, a nucleus
is characterized by the numbers 11 n≡ξ , 22 n≡ξ , ...,
pp n≡ξ of the monomers of each species.
In the second case, a change in the variable 1ξ in an
elementary event leads to a change in the variable 2ξ .
The variation in 2ξ can be represented as the sum of a
regular part and a fluctuating part . In
addition, the fluctuations of
)(
2
rδξ )(
2
fδξ
2ξ independent of 1ξ are
possible. An example is non-isothermal nucleation in a
mixture of a vapor and an inert gas [6]. The variables
are the number of vapor molecules in a cluster ( n≡1ξ )
and the cluster energy ( εξ ≡2 ). When a molecule is
attached to a nucleus, the average energy of the latter
likewise increases.
Of course, the combined processes are also possible,
e.g. when non-isothermal effects are taken into account
for the condensation of a vapor mixture; in this case, the
variables are , , and 1n 2n ε .
As is evident, the roles of variables are different in
the processes with linked fluxes. Consequently, there is
no need to bring the quadratic form H to the canonical
form in order to single out the unstable variable: the
number n≡1ξ of monomers in a nucleus or its size
(radius, volume) is a natural unstable variable, as in the
one-dimensional theory. Definitive signs of
correspond to this fact. So, e.g. in the two-dimensional
case, the following signs have to be:
ikh
011 <h and . So, the normalization of the
equilibrium DF for these processes with respect to
stable variables [3] is carried out with the use of the
fluctuation theory [8]. The normalization in respect to
the unstable variable is carried out in the same way,
as in the one-dimensional theory [5].
022 >h
n
In the processes of binary, as well as -component,
nucleation, the situation is quite different. The variables
and are physically equivalent. Accordingly,
and have to be of the same sign. It is not difficult to
establish from the analysis of directions of the flux of
nuclei on the ( , )-plane that the only possible
physical situation is and at
p
1n 2n 11h
22h
1n 2n
011 >h 022 >h 012 <h .
These signs also can be obtained in direct calculations
of for concrete mixtures using the experimental
dates for the corresponding thermodynamic quantities.
So, we can not apply the algorithm described above for
the normalization of the function . The
extension of the one-dimensional algorithm of
normalization [5] to this case can not be performed also,
so the normalization constant equal to the
total number of monomers of both kinds, which is
widely used for binary-nucleation processes in
literature, is incorrect. Consequently, in order to
normalize the function we have to convert
the given process to a process with linked fluxes, i.e. to
change the roles of variables by their corresponding
transformation. Apparently, the total number of
monomers in a nucleus,
ikh
),( 210 nnf
21 NNN +=
),( 210 nnf
qnnnn +++= ...21 , and
compositions, nnc /22 = , ..., , must be
taken as new variables. In the variables ( n , , ..., ),
the - component nucleation will be a process with
linked fluxes with the corresponding signs of (the
variable is unstable, as before; the variables are
stable).
nnc pp /=
2c pc
p
ikh
n ic
NORMALIZATION OF THE EQUILIBRIUM
DISTRIBUTION FUNCTION
As an example of a binary nucleation process, we
consider the condensation of the mixture of vapors of
two substances into ideal solution. The work of nucleus
formation has the following form [9]:
,)(ln
ln),(
3/2
2211
21
2
2
21
1
1221121
nvnv
nn
n
kTn
nn
n
kTnnnnn
++
+
+
+
+
++=ΔΦ
α
χχ
)/ln()( 00
iiiii PPkTPPv +−=χ , (9)
171
where is the molecular volume of the th species in a
nucleus,
iv i
α is the surface tension, is the
sum of partial pressures of vapors, and is the vapor
pressure of pure i th component at the temperature .
21 PPP +=
0
iP
T
We pass to the new variables and
. Eq. (9) takes the form
21 nnn +=
)/( 212 nnnc +=
3/2)()(),( ncscncn αμ +Δ−=ΔΦ (10)
similar to the one-dimensional one. Correspondingly,
and now. 0),(
11 <cnh 0),(
22 >cnh
In the vicinity of the saddle point ( , ), the
quadratic form in eq. (1) can be identically transformed
to the following form:
∗n ∗c
2)(),(
22
2
),(
22
),(
))((det),( ncchn
h
cnH ecn
cn
cn
−+=
H , (11)
where is
determined from the condition of equilibrium
with respect to composition c .
ntgnhhnc e
c
ncnce )(),(
22
),(
12
)( )/()( θ≡−=
0/),( =∂ΔΦ∂ ccn
So, the equilibrium DF splits into two parts, ),(0 cnf
)()(),( 00 cncnf flψρ= :
kT
nh cncn
NNn
2),(
22
),( )2/(det
210 e)()(
H+ΔΦ
− ∗
+=ρ ,
kT
ncchcn
fl
ecn
kT
h
c 2
))((),(
22
2)(),(
22
e
2
)(
−
−
=
π
ψ . (12)
The “fluctuating” part )(cflψ of the equilibrium DF
is normalized according to the theory of fluctuations
[8], whereas the function )(0 nρ is normalized at
as in the one-dimensional theory [5],
acquiring the factor
)(ncc e=
21 NNN += . As it is known,
),(
22
2)( /))(( cne hkTncc =− . This equality will be used
below for describing the limiting cases.
If the variables ( , ) are used nevertheless, then
the DF is got by the reverse transition
:
1n 2n
),( 210 nnf
),(),( 21 nncn →
kT
nncn
kT
h
nn
NNnnf
),(),(
22
21
21
210
21
e
2
),(
ΔΦ
−
+
+
=
π
, (13)
where the multiplier is the Jacobian of this
transition. This equation solves the problem of
normalization of the function .
)/(1 21 nn +
),( 210 nnf
The matrix has the simplest form in the variables
( , ): there are only diagonal elements and
. The matrix in the variables ( , c ) can be
obtained by means of the transformation of initial
movement equations (4) to these variables. Doing so,
we find:
D
1n 2n 11d
22d ),( cnD n
⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
−+−+−
−+−
+
=
∗
∗∗
∗
∗∗
∗
∗∗
2
22
2
11
2
2211
2211
2211
),(
)1()1(
)1(
n
dcdc
n
dcdc
n
dcdcdd
cnD
(14)
so that . Also, the following
relationships can be derived:
, ,
. Hence, we also can calculate the steady
state nucleation rate in the variables ( , ) using the
normalizing constant from (13); the nucleation rate
value (6) is invariant with respect to the transformations
2),(),( /detdet 21
∗= nnncn DD
),(2),( 21detdet nncn n HH ∗= ),(),( 21detdet nncn ZZ =
),(
1
),(
1
21 nncn λλ =
1n 2n
),(),( 21 cnnn ↔ , as it must from the physical point of
view.
LIMITING CASES
The equations 0/),( 21 =∂ΔΦ∂ innn , 1, 2, define
the lines and of the equilibrium of a nucleus
with respect to the variables and . They have the
following directions in the vicinity of the saddle point:
=i
)1(
eL )2(
eL
1n 2n
12
11)1(
h
htg e −=θ ,
22
12)2(
h
htg e −=θ . (15)
There are kinetic and thermodynamic limits. Let us
consider the former, 0/ 1122 →= ddγ and ∞→γ . The
fact of the same signs of and reflects the
certain symmetry of a system in respect to both the
species. It leads to the symmetry of expression (9) with
respect to and as well as to the following
symmetry. In the limit
11h 22h
1n 2n
∞→γ , we have
22111 /det hd H→λ , and (see
eq. (8)). The kinetics of the second species
predominates in this case, so that the equilibrium in
respect to the variable has a chance to be
established. Therefore, the flux vector tends to the line
. In the opposite case
)2(
2212 / etghhtg θθ ≡−→
2n
)2(
eL 0→γ , we have
11221 /det hd H→λ , and , i.e.
the replacement of indexes
)1(
1211 / etghhtg θθ ≡−→
21↔ in the expression for
1λ takes place. So, the flux vector is enclosed between
the equilibrium lines and , as it must from the
physical point of view.
)1(
eL )2(
eL
Now let us pass to the variables ( n , ) and consider
the thermodynamic limits. In binary nucleation, two
transitions to the one-dimensional theory are possible.
The first of them, (the variance of c
tends to zero), is general for the processes with linked
fluxes [3]. In this limit, the variable has the same,
equilibrium, value for all the nuclei of size .
It is seen from (12), that can be represented in this
case as ; also,
, and , i.e.
c
∞→kTh cn /),(
22
c
)()( nc e n
0f
))(()(),( )(c e
00 ncncnf −= δρ
0)()( →nc e ),(
11
),(
22
),( )/(det cncncn hh →H
)(0 nρ converts to the DF of the one-dimensional
theory. So, the variable has the same, critical, value
for all the nuclei in this limit, i.e. it converts to the
constant and falls out from consideration. This
transition corresponds to the formation of the two-
c
∗c
172
component embryos with the fixed (stoichiometric)
composition . ∗= cc
The second transition is peculiar to this process.
This is the passage to unary nucleation, when one of the
components vanishes. Let us assume . Since
, and , we have:
02 →N
22 ~ NP 222 ~ Pd 0→γ , and
consequently 0→θtg in this limit. Using (10), it is not
difficult to obtain the following equation for : ∗c
1
2
1 P
P
c
c ωω
ϑ=
− ∗
∗ , 21 vv≡ω ,
( ) kT
PPv
PP
)(
0
2
0
1
0
1
0
21
e
−
−
≡
ω
ϑ . (16)
Taking an interest only in qualitative picture let us
put 1=ω . Then one follows from (16) :
( )
( )ϑ
ϑ
′+
′
=∗
12
12
1 PP
PPc . (17)
So, at , or, what is the same, at , we
have , i.e. the saddle point “drives down” to the
-axis.
02 →N 02 →P
0→∗c
n
Further, in the same approximation, vvv ≡= 21 , one
obtains from eq. (10):
3/43/2),(
11 9
2 −
∗−= nvh cn α ,
)1(
),(
22
∗∗
∗
−
=
cc
kTnh cn ,
∗
∗
−
+Δ=
c
ckTh cn
1
ln),(
12 χ , (18)
where 12 χχχ −≡Δ .
From these expressions one follows: at
, and , i.e.
; has the same form as
in the one-dimensional theory. Thus, this transition
formally proceeds in accordance with the general way,
, however, at , i.e. this is the
transition to unary nucleation in the framework of the
saddle-point theory.
∞→),(
22
cnh
0→∗c 0/)( ),(
22
2),(
12 →cncn hh
),(
11
),(
22
),( /det cncncn hh →H ),(
11
cnh
∞→kTh cn /),(
22 0→c
In closing, let us find the contours of constant ratio
[10] using from eq. (7).
Applying the expansion
rff s =0/ ),( 21 nnfs
xxerf )/2()( π≈ , we find
that in the vicinity of the saddle point the following
equality holds:
( ) rnana
f
f s =⎥
⎦
⎤
⎢
⎣
⎡
+−= 2211
0
21
2
1
π
,
( )
( )θ
θκ
tghh
tgkT
a 121121
)(12
1
+
+
−= ,
( ) ( θ
θκ
tghh
tgkT
a 221222
)(12
1
+
+
−= ) . (19)
From this equation one obtains:
( r
a
n
a
an 21
2 2
1
2
1
2 −+−=
π ) . (20)
So, we have the set of parallel straight lines (for
different r ) with slope tangent
θ
θ
τ
tghh
tghh
a
atg
2212
1211
2
1
+
+
−=−= . (21)
As it was stated above, in the limit
0/ 1122 →= ddγ , 1211 / hhtg −→θ . It follows from
(21) that 0→τtg in this case. In other words, in the
case of slow kinetics of the second species the contours
of rff s =0/ are almost parallel to the -axis. So, the
theory confirms the results obtained by the authors of
Ref. [10] via numerical solution of binary-nucleation
equations.
1n
CONCLUSIONS
At first glance it would seem that processes of
binary (multicomponent) nucleation and those with
linked fluxes are physically different. However, the
passage to the variables (total number of monomers,
composition) converts binary nucleation into a process
with linked fluxes. So, we conclude that the latter is the
general case of nucleation processes.
Representation of binary nucleation as a process
with linked fluxes makes it possible to normalize the
equilibrium DF and thereby to calculate correctly the
stationary nucleation rate and size distribution of nuclei.
Also, it allows us to investigate the transitions to the
one-dimensional nucleation.
The normalization factor for the equilibrium DF has
been obtained in the framework of this approach. The
nucleation rate value does not depend on the variables
used, as it must from the physical point of view.
The use of the variables (total number of monomers,
composition) makes it possible to reveal and correctly
describe all the limits, both the kinetic and
thermodynamic ones. Two one-dimensional limits for
binary nucleation have been described here. One of
them corresponds to the formation of nuclei with
stoichiometric composition. The second limit is the
transition to unary (single-component) nucleation.
The results of the theory which concern the steady
state DF confirm the results of numerical solution of the
binary-nucleation equations given in literature.
REFERENCES
1. N.V. Alekseechkin and P.N. Ostapchuk.
Homogeneous nucleation of gaseous pores in a two-
component solution of vacancies and gas atoms // Fizika
Tverdogo Tela. 1993, v. 35, p. 929-940 (Physics of the
Solid State. 1993, V. 35, p. 479-484).
2. A.E.Volkov, A.I. Ryazanov. Theory of gas
bubble nucleation in supersaturated solution of
vacancies, interstitials and gas atoms // J. Nucl. Mater.
1999, v. 273, p. 155-163.
3. N.V. Alekseechkin. Multidimensional kinetic
theory of first-order phase transitions // Fizika Tverdogo
Tela. 2006, v. 48, p. 1676-1685 (Physics of the Solid
State. 2006, v. 48, p. 1775-1785.).
4. Ya.B. Zel’dovich. To the theory of new-phase
formation. Cavitation // JETP. 1942, № 12, p. 525-538.
5. Ya.I. Frenkel’. Kinetic Theory of Liquids.
Lеningrad: «Nauka», 1975, 592 p.
6. J. Feder, K.C. Russell, J. Lothe, and G.M.
Pound. Homogeneous nucleation and growth of
173
droplets in vapours // Adv. Phys. 1966, v. 15, p. 111-
178.
9. H. Reiss. The kinetics of phase transitions in
binary systems // J. Chem. Phys. 1950, v. 18, p. 840-
848. 7. K.C.Russell. Linked flux analysis of nucleation
in condensed phases // Acta Met. 1968, v. 16, p. 761-
769.
10. B.E. Wyslouzil and G. Wilemski. Binary
nucleation kinetics. III. Transient behavior and time
lags // J. Chem. Phys. 1996, v. 105, p. 1090-1100. 8. L.D. Landau and E.M. Lifshits. Statistical
physics. Moscow: «Nauka», 1976, Part 1, 584 p.
Статья поступила в редакцию 19.05.2009 г.
О КИНЕТИКЕ БИНАРНОЙ НУКЛЕАЦИИ
Н.В. Алексеечкин
Решены две задачи теории бинарной нуклеации: нормировка равновесной функции распределения
зародышей и корректный переход в одномерную теорию. Проведена классификация многомерных
процессов зарождения и показано, как представить бинарную нуклеацию в виде процесса со связанными
потоками посредством соответствующего преобразования переменных описания зародыша. Именно
использование переменных (полное число мономеров, состав) позволяет решить данные задачи. Описаны
два перехода в одномерную нуклеацию. Один из них соответствует образованию зародышей
стехиометрического состава. Другой есть переход в однокомпонентную нуклеацию.
ПРО КІНЕТИКУ БІНАРНОЇ НУКЛЕАЦІЇ
М.В. Алєксєєчкін
Вирішено дві задачі теорії бінарної нуклеації: нормировка рівноважної функції розподілу зародків і
коректний перехід в одномірну теорію. Проведено класифікацію багатомірних процесів зародження і
показано, як перетворити бінарну нуклеацію в процес зі зв’язаними потоками за допомогою відповідного
перетворення перемінных опису зародка. Саме використання перемінних (повне число мономерів, склад)
дозволяе вирішити дані задачі. Описано два переходи в одномірну нуклеацію. Один з них відповідає
утворенню зародків стехіометричного складу. Другий є перехід в однокомпонентну нуклеацію.
174
MODEL AND MAIN RESULTS
OF THE MULTIVARIABLE THEORY
OF NUCLEATION
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