Modeling the precipitation kinetics in systems with strong heterophase fluctuations
The hierarchy of the Becker-Döring rate equations is used to consider the influence of heterophase fluctuations on the kinetics of precipitation in supersaturated solid solutions, including the nucleation stage and the crossover to the final Ostwald ripening regime. Numerical simulation shows that a...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Modeling the precipitation kinetics in systems with strong heterophase fluctuations / A.A. Turkin, A.S. Bakai // Вопросы атомной науки и техники. — 2007. — № 3. — С. 394-398. — Бібліогр.: 16 назв. — англ. |
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Turkin, A.A. Bakai, A.S. 2017-01-07T19:07:01Z 2017-01-07T19:07:01Z 2007 Modeling the precipitation kinetics in systems with strong heterophase fluctuations / A.A. Turkin, A.S. Bakai // Вопросы атомной науки и техники. — 2007. — № 3. — С. 394-398. — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 05.70.Fh, 64.60.Qb, 64.70.Kb, 64.75.+g, 68.35.Rh, 81.30.Mh https://nasplib.isofts.kiev.ua/handle/123456789/111047 The hierarchy of the Becker-Döring rate equations is used to consider the influence of heterophase fluctuations on the kinetics of precipitation in supersaturated solid solutions, including the nucleation stage and the crossover to the final Ostwald ripening regime. Numerical simulation shows that a quasi-stationary distribution of solute clusters forms in the region of small sizes. This population of small clusters (heterophase fluctuations) is separated by a gap from the distribution of coarsening precipitates. The scaling behaviour of the precipitate size distribution during the later stage of decomposition is consistent with the Lifshits-Slyozov theory of coarsening. Ієрархія рівнянь Бекерa-Дерингa використовується для розгляду впливу гетерофазних флуктуацій на кінетику розпаду пересичених твердих розчинів, включаючи стадію зародження і перехід до кінцевого режиму коалесценції. Числове моделювання показує, що в області малих розмірів формується квазістаціонарний розподіл дрібних кластерів (гетерофазні флуктуації), що відокремлений від розподілу зростаючих виділень. Скейлінгова поведінка розподілу виділень за розмірами на пізній стадії розпаду відповідає теорії коалесценції Ліфшица-Сльозова. Иерархия уравнений Бекерa-Дерингa используется для рассмотрения влияния гетерофазных флуктуаций на кинетику распада пересыщенных твердых растворов, включая стадию зарождения и переход к конечному режиму коалесценции. Численное моделирование показывает, что в области малых размеров формируется квазистационарное распределение мелких кластеров (гетерофазные флуктуации), которое отделено от распределения растущих выделений. Скейлинговое поведение распределения выделений по размерам на поздней стадии распада соответствует теории коалесценции Лифшица-Слезова. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Phase transformations in condensed matter Modeling the precipitation kinetics in systems with strong heterophase fluctuations Article published earlier |
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| title |
Modeling the precipitation kinetics in systems with strong heterophase fluctuations |
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Modeling the precipitation kinetics in systems with strong heterophase fluctuations Turkin, A.A. Bakai, A.S. Phase transformations in condensed matter |
| title_short |
Modeling the precipitation kinetics in systems with strong heterophase fluctuations |
| title_full |
Modeling the precipitation kinetics in systems with strong heterophase fluctuations |
| title_fullStr |
Modeling the precipitation kinetics in systems with strong heterophase fluctuations |
| title_full_unstemmed |
Modeling the precipitation kinetics in systems with strong heterophase fluctuations |
| title_sort |
modeling the precipitation kinetics in systems with strong heterophase fluctuations |
| author |
Turkin, A.A. Bakai, A.S. |
| author_facet |
Turkin, A.A. Bakai, A.S. |
| topic |
Phase transformations in condensed matter |
| topic_facet |
Phase transformations in condensed matter |
| publishDate |
2007 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| description |
The hierarchy of the Becker-Döring rate equations is used to consider the influence of heterophase fluctuations on the kinetics of precipitation in supersaturated solid solutions, including the nucleation stage and the crossover to the final Ostwald ripening regime. Numerical simulation shows that a quasi-stationary distribution of solute clusters forms in the region of small sizes. This population of small clusters (heterophase fluctuations) is separated by a gap from the distribution of coarsening precipitates. The scaling behaviour of the precipitate size distribution during the later stage of decomposition is consistent with the Lifshits-Slyozov theory of coarsening.
Ієрархія рівнянь Бекерa-Дерингa використовується для розгляду впливу гетерофазних флуктуацій на кінетику розпаду пересичених твердих розчинів, включаючи стадію зародження і перехід до кінцевого режиму коалесценції. Числове моделювання показує, що в області малих розмірів формується квазістаціонарний розподіл дрібних кластерів (гетерофазні флуктуації), що відокремлений від розподілу зростаючих виділень. Скейлінгова поведінка розподілу виділень за розмірами на пізній стадії розпаду відповідає теорії коалесценції Ліфшица-Сльозова.
Иерархия уравнений Бекерa-Дерингa используется для рассмотрения влияния гетерофазных флуктуаций на кинетику распада пересыщенных твердых растворов, включая стадию зарождения и переход к конечному режиму коалесценции. Численное моделирование показывает, что в области малых размеров формируется квазистационарное распределение мелких кластеров (гетерофазные флуктуации), которое отделено от распределения растущих выделений. Скейлинговое поведение распределения выделений по размерам на поздней стадии распада соответствует теории коалесценции Лифшица-Слезова.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111047 |
| citation_txt |
Modeling the precipitation kinetics in systems with strong heterophase fluctuations / A.A. Turkin, A.S. Bakai // Вопросы атомной науки и техники. — 2007. — № 3. — С. 394-398. — Бібліогр.: 16 назв. — англ. |
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2025-11-26T00:10:43Z |
| last_indexed |
2025-11-26T00:10:43Z |
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1850595359949062144 |
| fulltext |
MODELING THE PRECIPITATION KINETICS IN SYSTEMS
WITH STRONG HETEROPHASE FLUCTUATIONS
A.A. Turkin and A.S. Bakai
Akhiezer Institute for Theoretical Physics,
National Science Center “Kharkov Institute of Physics and Technology”,
Kharkov, Ukraine,
e-mail: a.turkin@kipt.kharkov.ua
The hierarchy of the Becker-Döring rate equations is used to consider the influence of heterophase fluctuations
on the kinetics of precipitation in supersaturated solid solutions, including the nucleation stage and the crossover to
the final Ostwald ripening regime. Numerical simulation shows that a quasi-stationary distribution of solute clusters
forms in the region of small sizes. This population of small clusters (heterophase fluctuations) is separated by a gap
from the distribution of coarsening precipitates. The scaling behaviour of the precipitate size distribution during the
later stage of decomposition is consistent with the Lifshits-Slyozov theory of coarsening.
PACS: 05.70.Fh, 64.60.Qb, 64.70.Kb, 64.75.+g, 68.35.Rh, 81.30.Mh
1. INTRODUCTION
The properties of many industrial alloys are
controlled by the precipitation of second phase
particles. Traditionally, the precipitation kinetics in
supersaturated metastable solid solutions is considered
in terms of three distinct stages: (i) nucleation of stable
clusters of the precipitating phase, (ii) independent
diffusion growth of these clusters, and (iii) coarsening
(Ostwald ripening) that involves the growth of large
precipitates at the expense of dissolving small ones,
driven by an overall reduction in the interface energy.
There exists an important class of alloys in which
precipitate-matrix interface energy is low. Usually these
precipitates are coherent with the matrix and resistant to
coarsening. Examples of such alloys include nickel
superalloys [1], copper bearing high strength low alloy
steels (HSLA) [2,3], aluminum alloyed with scandium
[4,5] etc. Due to precipitation of a high density of
coherent, nanometer sized particles these alloys exhibit
substantial strength. This effect can be beneficial or
detrimental depending on alloy application. For
example, model Fe-Cu alloys have received much
attention because of the influence of irradiation on the
precipitation of copper-rich particles in ferritic steels
and the related steel hardening and embrittlement of
reactor pressure vessels [6,7].
The processes of nucleation, growth and coarsening
in the system of coherent precipitates with low interface
energy can occur simultaneously, which makes
interpretation of the alloy behavior difficult. The
interface energy affects the nucleation rate, the
precipitate growth as well as drives the coarsening
process through the capillarity effect. Existing models
of the later stage of decomposition consider only large
coarsening precipitates and solute monomers, whereas
the population of subcritical clusters is ignored.
However, even in undersaturated solid solutions a
steady-state distribution of small-sized clusters forms
due to continuous nucleation and decay of unstable
clusters. In 1939 Frenkel has named this distribution the
heterophase fluctuations (HF) [8]. In this communi-
cation we consider the effect of HF on the kinetics of
precipitation in binary alloys. At a low interface energy
and/or high temperatures the HF may contribute
considerably to the overall balance of solute atoms and
influence both the nucleation and coarsening stages.
2. THE MODEL
Consider a binary alloy supersaturated with a
component that will be called the solute in the
following. The alloy may contain second phase
precipitates which are assumed to consist of the pure
solute component. All precipitates are approximated as
spherical in shape. Our goal is to describe the evolution
of the precipitate size distribution function (DF) which
is defined as the time-dependent concentration
of solute clusters of a given size varying over the
range from solute dimers to large precipitates. We will
use an approach known in the literature as the Bekker-
Döring model [9] or the Master equation approach.
According to this model the kinetic equations for cluster
concentrations are defined by the rates at which clusters
absorb a solute atom and lose a solute atom due to
thermal evaporation. Only solute monomers are
assumed to be mobile, i.e. direct collisions of clusters
are not considered within the Bekker-Döring model
),( tnf
n
....,3,2,),(
1,,1 =−=
∂
∂
+− nJJ
t
tnf
nnnn (1)
),()(),1()1( ,1 tnfnWtnfnWJ nn
−+
− −−−= , (2)
where is the concentration of mobile
monomers (all concentrations are defined in terms of
atomic fractions). In the diffusion-limited case the
absorption and emission rates W and W are
given by
1),1( Ctf ≡
)(n+ )(n−
1)1(42)1( DCRW
ω
π
=+ , (3)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 394-398. 394
1)(4)( DCnRnW
ω
π
=+ , , (4) 1>n
RDCnRnW 1)(4)(
ω
π
=− , , (5) 1>n
where is the atomic volume, is the radius of the
-atomic cluster, is the diffusion coefficient of
solute atoms. C is the thermal equilibrium
concentration of solute monomers at the precipitate
boundary given by the Gibbs-Thomson relation
ω )(nR
n D
R
1
=
TRk
CC
B
eR γω2exp 11 , (6)
where is the solute monomer concentration at
equilibrium with the bulk solute phase and is the
precipitate-matrix interface energy.
eC1
γ
The total number of solute atoms is kept constant
∑
∞
=
=+
2
1 ),(
n
QtnnfC , (7) 10
where is the atomic fraction of solute atoms. Q
The solid solution is supersaturated if . The
excess amount of solute atoms forms clusters of the new
phase. The well-known expression for the steady-state
nucleation rate of stable clusters can be obtained from
Eqs. (1) and (2)
eCC 11 >
( ) ,)n(f)n(WJJ
n
en,n
1
1
1
1
−∞
=
−+
−
== ∑ (8)
where
∏
=
−
+
+
+
=
n
i
e
iW
iW
nW
fWnf
2 )(
)(
)(
)1()1()( (9)
is the DF corresponding to the constrained equilibrium
. The function has the minimum at
the critical size n defined by the relation
0 ,1 =− nnJ )(nfe
crn=
)()( crcr nWnW −+ = . (10)
All clusters, smaller than the critical cluster, play an
important role providing a "path" for nucleation of
stable growing precipitates (Eq. (8) contains all cluster
sizes). Even saturated stable solution ( )
contains HF in the form of the equilibrium distribution
of clusters [8]. This means that the solute solubility is
not identical to the solubility of monomers. The
solubility should be defined as the total content of
solute atoms in the saturated solution including solute
atoms in subcritical clusters that form and decay due to
fluctuations, i.e. in the capillary approximation used
here the solute solubility is given by
∞=crn
,exp21
)( + =
2 2
3132
1
2
1
−+= ∑ ∑
∑
∞
= =
−
∞
=
n
n
i
e
n
e
ee
inC
nnfCC
β
(11)
where γωβ
TkR B1
2
= is the dimensionless interface
energy.
Fig. 1 shows the relative contributions of monomers
and HF (the second term in Eq. (11)) to the solute
solubility. It is seen that at small values of the parameter
the monomer concentration is lowβ 1. In alloy systems
with small values of the parameter the majority of
solute atoms belongs to immobile clusters and does not
participate directly in diffusion decomposition. This
means that the conventional models of nucleation and
coarsening are valid for systems with ; their
application to systems with small values of may
result in erroneous interpretation of experimental data.
β
1>>β
β
0.1 1 10
10-3
10-2
-1
100
HF
Monomers
Fr
ac
tio
n
Interface energy, β
Fig. 1. Relative contributions of monomers and HF
to the solute solubility versus the interface energy
To investigate the influence of HF on homogeneous
nucleation and coarsening of precipitates we track the
full particle size DF through the cause of
transformation. We use the numerical method described
in [11]. To reduce the number of equations solved
simultaneously we keep the original discrete Eqs. (1)
and (2) up to 200 atoms in clusters; for larger sizes
Eqs. (1) and (2) are transformed to a partial differential
equation of the Fokker-Planck type for a continuous
size variable, which allows coarse-graining of the
numerical mesh. The combined set - the discrete
equations and the continuous Fokker-Planck equation -
is solved numerically by the method of lines [12].
3. RESULTS AND DISCUSSION
In simulations we used dimensionless time
tDR 1
14 −= ωπτ . (12)
The solubility was fixed at the value = 0.2at%,
then the equilibrium concentration of solute monomers
was calculated as a function of the parameter
eC
eC1 β
1 As an example, the interfacial energy of coherent Ni3Al
particles precipitating in Ni–Al alloys is 14 mJ/m2 [10] and
the value of is estimated as β T700≈β .
395
using Eq. (11). Fig. 2 shows formation of the precipitate
size DF in the supersaturated solid solution during
aging. The DF is defined in terms of cluster radius
),(
3
43),( 32
31
τ
ω
π
ω
τ nfnRF
= , (13) = ug 4
where = 1.2x10ω
N
-29 m-3. In the initial state the matrix
contains only solute monomers with the concentration
= 2 at%. To analyze the simulation results the
population of precipitates is divided into two groups: (i)
small clusters (HF with R < = 1 nm); and (ii) large
precipitates. The density of HF , the precipitate
density and the mean precipitate radius are
given by relations
Q
*R
hfN
p pR
∫=
*
2
),(
R
R
hf dRRFN τ ,
∫
∞
=
*
),(
R
p dRRFN τ , , (14) ∫
∞
=
*
),(
R
p dRRFRR τ
where is the dimer radius. 2R
0.1 1 10
10-9
10-7
10-5
10-3
10-1
τ=106
τ=108
τ=1010
Fe
FLS
D
is
tri
bu
tio
n
fu
nc
tio
n,
n
m
-4
Radius, nm
Fig. 2. Evolution of DF at = 0.5 Boxes show the
quasi-stationary distribution of HF (see Eqs. (9),(13));
circles correspond to the Lifshitz-Slyozov distribution
β
It is seen that the quasi-stationary distribution of
solute clusters forms in the region of small sizes
(Fig. 2). In the end of the nucleation stage a gap in the
solute cluster distribution forms, which separates the
population of small clusters (HF) from the distribution
of large growing precipitates. The size width of the
region of HF depends on interface energy and
temperature (the parameter ); in particular, it increases
with decreasing the interface energy. The simulated DF
of small clusters is well described by Eq. (9) and (13).
The leading edge of the distribution advances with time
and large precipitates undergo a process of the Ostwald
ripening. At the later stage of aging the shape of the
precipitate DF is close to that predicted by the Lifshitz-
Slyozov (LS) theory of coarsening [13]
β
=
)()(
)(
),(
ττ
τ
τ
pp
p
LS R
Rg
R
N
RF , (16)
−
−
−
+ u
u
uu
u
5.1
exp
5.1
5.1
3
3
9
)(
31137
2 . (17)
Fig. 3 shows the time dependence of the mean
precipitate radius and the precipitate density calculated
for several values of the parameter that is
proportional to the interface energy. It is seen that
changing the parameter has a pronounced effect on
the onset of nucleation and crossover to the later
coarsening stage. What is important here is that the
incubation period for nucleation increases with
decreasing the interface energy; in other words, despite
low values of the critical size for nucleation the system
evolves slowly.
β
β
103 104 105 106 107 108 109 1010
1
10
100
β
β = 0.5
β = 1.0
β = 2.0
β = 4.0
β = 6.0
M
ea
n
ra
di
us
, n
m
Aging time
R~τ1/3
a
103 104 105 106 107 108 109 1010
1020
1021
1022
1023
1024
1025
b
N~τ-1
N
um
be
r d
en
si
ty
, m
-3
Aging time
β
Fig. 3. Time dependence of the mean precipitate
radius (a) and the precipitate density (b). The interface
energy increases in the direction of arrows
Intuitively, on the basis of the classical nucleation
theory [14], completely opposite tendency can be
expected. But this logic may lead to erroneous
396
conclusions. As an example, we would like to discuss
here some results of ref. [15], in which a numerical
precipitation model by Kampmann and Wagner [16]
(KWN model) has been used to investigate the effect of
interface energy, solute supersaturation, and diffusivity
on the kinetics of precipitation in binary alloys. In the
methodology of the KWN model the kinetics of small-
sized clusters (HF) is not considered. Instead, the
classical nucleation theory is used to derive the
nucleation rate as a function of time and solute
concentration. To simulate the precipitation kinetics the
continuous time evolution is divided up into a number
of small time steps. At each step new stable clusters are
allowed to nucleate; for the existing precipitates a
deterministic description is used. The radius of the
newly formed precipitates is set to be slightly larger
than the critical radius to enable these precipitates to
grow. The precipitate size DF is used to calculate the
precipitate volume fraction and the instantaneous mean
concentration of solute monomers. In the next time step
the updated concentration of monomers is used in
calculating the nucleation rate and the deterministic
growth rates of precipitates. It should be emphasized
that both the model of this communication and the
KWN model are based on the same physical
assumptions. However, in contrast to results depicted in
Fig. 3, in [15] in the framework of KWN model it has
been found that the incubation period for decomposition
decreases drastically when the parameter decreases
from 5 to 0.8. The reason of this discrepancy is that the
KWN model is not physically justified for systems in
which is less or of order unity. As it has been written
in [16] one of the main assumptions of the KWN model
is the following: those fluctuations formed in the
nucleation regime which do not grow beyond the
critical size do not influence the precipitation kinetics
significantly; i.e. the number of fluctuations with
subcritical size and, hence the fraction of solute atoms
contained in them is negligibly small.
β
β
According to Fig. 3; in the long-time limit the
asymptotic behavior is consistent with the LS
coarsening theory [13], i.e. 31τ∝pR and . 1−∝τpN
In the case of strong HF ( = 0.5) the atomic
fraction of HF remains constant up to = 10
β
τ 7 (Fig. 4b)
while in the system with small contribution of HF the
nucleation stage is already completed (Fig. 4,a). The
reason is that at low interface energy the driving force
for evolution is low. Note that in the final state at
= 10τ 10 in the system with strong HF the concentration
of mobile monomers is small (Fig. 4,b). This should be
taken into account when using formulas of the LS
theory for the experimental evaluation of material
parameters. The point is that the mean precipitate radius
obeys the asymptotic relation
tDC
Tk
R e
B
p 1
3
9
8 γω
= , (18)
which well agree with our simulation results. This
relation contains the equilibrium concentration of
monomers, but not the solute solubility. We have not
found in the literature discussion of this issue.
We have also performed simulations of precipitation
kinetics in the system with a size-dependent interface
energy. It is known that the interface energy usually
increases as the second phase particle grows from a
small cluster into a distinct precipitate. The interface
energy increases when precipitates lose coherency
during growth. This means that small-sized precipitates
may have low interface energy that controls the fraction
of HF and the nucleation rate, while large observable
precipitates have a higher value of the interface energy
that controls the coarsening behavior. We have found
that in this case (i) nucleation and coarsening processes
are slow and (ii) at the intermediate stage of coarsening
the precipitate DF is broader than the LS distribution
function. The detailed description of these results will
be presented elsewhere.
100 102 104 106 108 1010
0
1
2
C1
HF
P
a β = 6
Fr
ac
tio
n,
a
t%
Aging time
100 102 104 106 108 1010
0.0
0.5
1.0
1.5
2.0
C1
HF
P
b
β = 0.5
Fr
ac
tio
n,
a
t%
Aging time
Fig. 4. Time dependence of atomic fractions of
momomers ( C ), heterophase fluctuations (HF) and
precipitates (P) in the system without HF at =6 (a)
and in the system with strong HF at = 0.5 (b)
1
β
β
397
4. CONCLUSIONS
Simulation of the precipitation kinetics in systems
with strong HF ( 1) shows that: ~<β
1. HF inhibit the homogeneous nucleation even in
highly supersaturated solid solutions.
2. HF influence the shape of the precipitate size
distribution function.
3. In the long-time limit the precipitate size distribution
tends to a stable self-similar distribution close to the
LS distribution.
REFERENCES
1. M. Durand-Charre. The microstructure of super-
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МОДЕЛИРОВАНИЕ КИНЕТИКИ РАСПАДА В СИСТЕМАХ
С СИЛЬНЫМИ ГЕТЕРОФАЗНЫМИ ФЛУКТУАЦИЯМИ
А.А. Туркин, A.С. Бакай
Иерархия уравнений Бекерa-Дерингa используется для рассмотрения влияния гетерофазных флуктуаций
на кинетику распада пересыщенных твердых растворов, включая стадию зарождения и переход к конечному
режиму коалесценции. Численное моделирование показывает, что в области малых размеров формируется
квазистационарное распределение мелких кластеров (гетерофазные флуктуации), которое отделено от
распределения растущих выделений. Скейлинговое поведение распределения выделений по размерам на
поздней стадии распада соответствует теории коалесценции Лифшица-Слезова.
МОДЕЛЮВАННЯ КІНЕТИКИ РОЗПАДУ В СИСТЕМАХ
З СИЛЬНИМИ ГЕТЕРОФАЗНИМИ ФЛУКТУАЦІЯМИ
А.А. Туркін, О.С. Бакай
Ієрархія рівнянь Бекерa-Дерингa використовується для розгляду впливу гетерофазних флуктуацій на
кінетику розпаду пересичених твердих розчинів, включаючи стадію зародження і перехід до кінцевого
режиму коалесценції. Числове моделювання показує, що в області малих розмірів формується
квазістаціонарний розподіл дрібних кластерів (гетерофазні флуктуації), що відокремлений від розподілу
зростаючих виділень. Скейлінгова поведінка розподілу виділень за розмірами на пізній стадії розпаду
відповідає теорії коалесценції Ліфшица-Сльозова.
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