On the kinetics of phase transformation of small particles in Kolmogorov's model
The classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory is generalized to the case of a finite-size system. The problem of calculating the new-phase volume fraction in a spherical domain is solved within the framework of geometrical-probabilistic approach. The solutions are obtained for both h...
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| Цитувати: | On the kinetics of phase transformation of small particles in Kolmogorov's model / N.V. Alekseechkin // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 597-613. — Бібліогр.: 20 назв. — англ. |
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irk-123456789-1195732017-06-08T03:07:00Z On the kinetics of phase transformation of small particles in Kolmogorov's model Alekseechkin, N.V. The classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory is generalized to the case of a finite-size system. The problem of calculating the new-phase volume fraction in a spherical domain is solved within the framework of geometrical-probabilistic approach. The solutions are obtained for both homogeneous and heterogeneous nucleations. It is shown that the finiteness property results in a qualitative distinction of the volume-fraction time dependence from that in infinite space: the Avrami exponent in the process of homogeneous nucleation decreases with time from 4 to 1, i.e. a slowing down of the transformation process takes place. The obtained results can be used, in particular, for controlling the crystallization kinetics in amorphous powders. Класична теорiя Колмогорова-Джонсона-Мейла-Аврами узагальнюється на випадок обмеженої системи. У рамках геометрико-ймовiрнiсного пiдходу вирiшується задача обчислення об’ємної частки нової фази в сферичнiй областi. Одержано розв’язки для випадкiв гомогенної i гетерогенної нуклеацiї. Показано, що властивiсть обмеженостi системи приводить до якiсної вiдмiнностi часової залежностi об’ємної частки вiд такої в необмеженому просторi: показник Аврами в процесi гомогенної нуклеацiї зменшується з часом вiд 4 до 1. 2008 Article On the kinetics of phase transformation of small particles in Kolmogorov's model / N.V. Alekseechkin // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 597-613. — Бібліогр.: 20 назв. — англ. 1607-324X PACS: 05.70.Fh, 68.55.Ac, 81.15.Aa DOI:10.5488/CMP.11.4.597 http://dspace.nbuv.gov.ua/handle/123456789/119573 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory is generalized to the case of a
finite-size system. The problem of calculating the new-phase volume fraction in a spherical domain
is solved within the framework of geometrical-probabilistic approach. The solutions are obtained
for both homogeneous and heterogeneous nucleations. It is shown that the finiteness property
results in a qualitative distinction of the volume-fraction time dependence from that in infinite
space: the Avrami exponent in the process of homogeneous nucleation decreases with time from
4 to 1, i.e. a slowing down of the transformation process takes place. The obtained results can be
used, in particular, for controlling the crystallization kinetics in amorphous powders. |
| format |
Article |
| author |
Alekseechkin, N.V. |
| spellingShingle |
Alekseechkin, N.V. On the kinetics of phase transformation of small particles in Kolmogorov's model Condensed Matter Physics |
| author_facet |
Alekseechkin, N.V. |
| author_sort |
Alekseechkin, N.V. |
| title |
On the kinetics of phase transformation of small particles in Kolmogorov's model |
| title_short |
On the kinetics of phase transformation of small particles in Kolmogorov's model |
| title_full |
On the kinetics of phase transformation of small particles in Kolmogorov's model |
| title_fullStr |
On the kinetics of phase transformation of small particles in Kolmogorov's model |
| title_full_unstemmed |
On the kinetics of phase transformation of small particles in Kolmogorov's model |
| title_sort |
on the kinetics of phase transformation of small particles in kolmogorov's model |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119573 |
| citation_txt |
On the kinetics of phase transformation of small particles in Kolmogorov's model / N.V. Alekseechkin // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 597-613. — Бібліогр.: 20 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT alekseechkinnv onthekineticsofphasetransformationofsmallparticlesinkolmogorovsmodel |
| first_indexed |
2025-07-08T16:11:51Z |
| last_indexed |
2025-07-08T16:11:51Z |
| _version_ |
1837095836646899712 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 4(56), pp. 597–613
On the kinetics of phase transformation of small
particles in Kolmogorov’s model
N.V.Alekseechkin
Akhiezer Institute for Theoretical Physics, National Science Centre
“Kharkov Institute of Physics and Technology”, Akademicheskaya Str. 1,
Kharkov 61108, Ukraine
Received May 16, 2007, in final form March 15, 2008
The classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory is generalized to the case of a
finite-size system. The problem of calculating the new-phase volume fraction in a spherical domain
is solved within the framework of geometrical-probabilistic approach. The solutions are obtained
for both homogeneous and heterogeneous nucleations. It is shown that the finiteness property
results in a qualitative distinction of the volume-fraction time dependence from that in infinite
space: the Avrami exponent in the process of homogeneous nucleation decreases with time from
4 to 1, i.e. a slowing down of the transformation process takes place. The obtained results can be
used, in particular, for controlling the crystallization kinetics in amorphous powders.
Key words: KJMA theory, volume fraction, nucleation, Avrami exponent
PACS: 05.70.Fh, 68.55.Ac, 81.15.Aa
1. Introduction
The kinetics of a phase transformation process in infinite space is described by the
well-known expression of Kolmogorov [1] for the volume fraction XK(t) of the material
transformed:
XK(t) = 1 − exp
[
−
∫ t
0
I(t′)V (t′, t)dt′
]
, (1)
where V (t′, t) is the volume at time t of the nucleus appearing at time t′; I(t) is the nucle-
ation rate. For the spherical shape of nuclei, V (t′, t) = (4π/3)R3(t′, t), R(t′, t) =
∫ t
t′ u(τ)dτ ,
where R(t′, t) is the radius of a nucleus, u(t) is its growth velocity. This expression was
also derived by Johnson, Mehl and Avrami [2,3] with the use of a different approach at
constant values of I and u.
Under the restrictions of Kolmogorov’s model [1], or the K-model [4] (these restric-
tions are considered in detail in this monograph), the KJMA expression is exact in the
case of an infinite system. In practice, the fulfillment of the inequality L � R0 is re-
quired for its validity, where R0 is the size of the system considered, L is the mean
grain size. This inequality is satisfied in many cases, and the KJMA formula is widely
used for the analysis of experimental data. However, in the case of a system of suffici-
ently small size and at certain values of nucleation and growth rates, the deviation of the
c© N.V.Alekseechkin 597
N.V.Alekseechkin
volume fraction X(t) from XK(t) is possible. In particular, such a situation can occur
in the process of crystallization of a liquid drop or a small amorphous ball, especially
at the temperature at which the nucleation rate is small while the growth velocity is
large.
Therefore, consideration of the problem of calculating the new-phase volume fraction in
a finite system and derivation of the criteria for the applicability of the KJMA expression
are of interest. Until recently, the inclusion of finite-size effects into the KJMA theory was
performed mainly for thin films [5–8]. In reference [5], the time cone method is used for
this purpose. In reference [7], the anisotropy of nuclei (ellipsoids) is also included in this
problem. The detailed calculation of transformation kinetics in thin films for a spherical
shape of nuclei is given in reference [8].
In the present report, a rigorous solution of this problem is presented for a spherical
domain. The cases of both homogeneous and heterogeneous nucleations are considered.
The time dependence of the volume fraction X(t) is shown to differ qualitatively from
that in infinite space; therefore, it cannot be derived from the latter, equation (1), by the
use of correction factors. The obtained dependencies X(t) are in qualitative agreement
with the corresponding results of reference [8]. The solution is obtained with the help
of the critical region method which was earlier applied by the author to solving other
problems of calculating the volume fractions [9,10].
2. Calculating the volume fraction in a spherical domain.
Homogeneous nucleation
Consider the process of phase transformation of the spherical domain of volume V0 =
(4π/3)R3
0 at homogeneous nucleation inside the new-phase centres with the nucleation
rate I(t) and growth velocity u(t). Let the nuclei be of spherical shape.
Figure 1. The domain and the critical region for the point O′. The critical region part
of volume Ω(r ; t′, t) = v(r ; t′, t) is marked out.
Take at random the point O′ in the domain. Let it be at a distance r from the centre
of the domain which is the point O (figure 1). We find the probability Q(r, t) that the
point O′ will be non-transformed at time t. Let us specify the critical region for the point
O′ – the sphere of radius R(t′, t). At time t′, the boundary of this region moves at the
598
On the kinetics of phase transformation of small particles in Kolmogorov’s model
velocity u(t′), so that in the time interval 0 6 t′ 6 t its radius decreases from the greatest
value R(0, t) ≡ Rm(t) up to R(t, t) ≡ 0. In order for the point O ′ to be non-transformed,
it is necessary and sufficient that no centre of a new phase should be formed within the
critical region in the time interval 0 6 t′ 6 t. The probability of this event is [1]
Q(r, t) = exp [−Y (r, t)] . (2)
In the case of infinite space, the function Y does not depend on r and has the following
form [1]:
Y (t) =
∫ t
0
I(t′)V (t′, t)dt′, (3)
where V (t′, t) = (4π/3)R3(t′, t) is the critical region volume at time t′.
In the considered case, the new-phase centres can appear only within the domain.
At the same time, in general, only a part of the critical region for the point O ′ lies
within this domain. Let us denote the volume of this part by Ω(r ; t′, t) (figure 1). Hence,
calculating the probability Q(r, t), we must take Ω(r ; t′, t) instead of V (t′, t). Accordingly,
the expression for Y (r, t) has the following form:
Y (r, t) =
∫ t
0
I(t′)Ω(r ; t′, t)dt′. (4)
The volume fraction Q(t) of the material non-transformed at time t is the probability
for the point O′ to fall in the non-transformed part of the domain:
Q(t) =
1
V0
∫ R0
0
Q(r, t)(4πr2)dr, (5)
accordingly, the volume fraction of the material transformed is X(t) = 1 −Q(t).
Furthermore, the problem is how to find an explicit form of the function Ω(r ; t ′, t)
depending on t, t′ and r. For this purpose the following expression will be used. For two
overlapping spheres of radii r1, r2 and the spacing between centres h, the volume of the
second sphere lying within the first sphere is equal to
v(r1, r2;h) = π
{
2
3
(
r31 + r32
)
+
1
12
h3 − 1
2
h
(
r21 + r22
)
− 1
4
(
r21 − r22
)2
h
}
. (6)
Determine the times t1 and t2 by the equations
Rm(t1) = R0 , Rm(t2) = 2R0 . (7)
The following three cases with respect to time t arise.
1) Rm(t) < R0: t < t1
Let us determine the distance r0 by the equality
r0 = R0 −Rm . (8)
At 0 6 r 6 r0, the critical region lies entirely within the domain in the whole time interval
0 6 t′ 6 t ; accordingly, Ω(r ; t′, t) = V (t′, t). Let us determine the time tm(r, t) by the
equation
R(tm, t) = R0 − r. (9)
599
N.V.Alekseechkin
At r0 < r 6 R0 the critical region lies partially within the domain in the interval 0 6
t′ < tm(r, t); accordingly, Ω(r ; t′, t) = v(R0, R(t′, t); r) ≡ v(r ; t′, t). Further, in the interval
tm 6 t′ 6 t the critical region is entirely within the domain; hence, Ω(r ; t′, t) = V (t′, t).
Thus,
Y1(r, t) =
∫ t
0 I(t
′)V (t′, t)dt′ , 0 6 r 6 r0 ,
∫ tm(r,t)
0 I(t′)v(r ; t′, t)dt′ +
∫ t
tm(r,t) I(t
′)V (t′, t)dt′ , r0 < r 6 R0 ;
(10)
2) R0 6 Rm(t) 6 2R0: t1 6 t 6 t2
Figure 2. Positions of the critical region boundary at different times t′: 1) 0 6 t′ <
t′m(r, t); 2) t′ = t′m(r, t); 3) t′m(r, t) < t′ < tm(r, t) 4) t′ = tm(r, t).
Let us determine the distance r′0 by the equality
r′0 = Rm −R0 , (11)
and the time t′m(r, t) by the equation
R(t′m, t) = R0 + r. (12)
Consider the case 0 < r < r′0. In figure 2, the positions of the critical region boundary
are shown for this case at different times t′. In the time interval 0 6 t′ 6 t′m(r, t) the
domain lies entirely within the critical region; accordingly, Ω(r ; t′, t) = V0. In the interval
t′m(r, t) < t′ 6 tm(r, t), Ω(r ; t′, t) = v(r ; t′, t). And in the remaining interval tm(r, t) <
t′ 6 t, Ω(r ; t′, t) = V (t′, t). At r′0 6 r 6 R0 we have:
Ω(r ; t′, t) =
{
v(r ; t′, t) , 0 6 t′ < tm(r, t),
V (t′, t) , tm(r, t) 6 t′ 6 t.
(13)
Thus,
Y2(r, t) =
V0
∫ t′m(r,t)
0 I(t′)dt′ +
∫ tm(r,t)
t′m(r,t) I(t
′)v(r ; t′, t)dt′ +
∫ t
tm(r,t) I(t
′)V (t′, t)dt′ ,
0 6 r < r′0 ,
∫ tm(r,t)
0 I(t′)v(r ; t′, t)dt′ +
∫ t
tm(r,t) I(t
′)V (t′, t)dt′ r′0 6 r 6 R0 ;
(14)
600
On the kinetics of phase transformation of small particles in Kolmogorov’s model
3) Rm(t) > 2R0: t > t2
As it follows from the foregoing, in this case
Ω(r ; t′, t) =
V0 , 0 6 t′ 6 t′m(r, t),
v(r ; t′, t) , t′m(r, t) < t′ < tm(r, t),
V (t′, t) , tm(r, t) 6 t′ 6 t
(15)
for arbitrary r-value. Accordingly,
Y3(r, t) = V0
∫ t′m(r,t)
0
I(t′)dt′ +
∫ tm(r,t)
t′m(r,t)
I(t′)v(r; t′, t)dt′
+
∫ t
tm(r,t)
I(t′)V (t′, t)dt′, 0 6 r 6 R0 . (16)
The volume fraction of the material transformed in every case is
Xi(t) = 1 − 1
V0
∫ R0
0
e−Yi(r,t)(4πr2)dr , i = 1, 2, 3. (17)
The volume fraction at any time t is given by the following expression:
X(t) = η(t1 − t)X1(t) + η(t2 − t)η(t− t1)X2(t) + η(t− t2)X3(t) , (18)
where η(x) is the symmetric unit function [13].
The case of arbitrary shape of the domain as well as arbitrary nucleus shape permit-
ted by Kolmogorov’s model can be considered in a similar way, following the procedure
described above. The distinction will be only in determining Ω(r; t′, t).
3. The case of constant nucleation and growth rates
In order to analyse the effect of the finiteness of the system on the rate of a phase
transformation process, we consider the case of time-independent nucleation and growth
rates. First, we study the time dependence of the volume fraction at fixed a value of R0.
Let us introduce the following dimensionless variables: time τ = ut/R0 = t/t∗, t∗ = R0/u
, distance x = r/R0 and the parameter α = (π/3)(I/u)R4
0 . The calculation of integrals in
the expressions for Yi yields the following expressions for the volume fraction of the initial
phase for each case described above:
1) τ < 1
Q1(t) = (1 − τ)3 e−ατ
4
+ 3
∫ 1
1−τ
e−αφ1(x,τ)x2dx, (19)
where φ1(x, τ) =
∑4
k=−1 Pk(τ)x
k and the coefficients Pk(τ) are as follows: P−1(τ) =
−(3/20)τ 5 + (1/2)τ 3 − (3/4)τ + 2/5 ; P0(τ) = (1/2)τ 4 + 2τ − 3/2 ; P1(τ) = −(1/2)τ 3 −
(3/2)τ + 2 ; P2 = −1 ; P3(τ) = (1/4)τ ; P4 = 1/10.
2) 1 6 τ 6 2
Q2(τ) = 3
{
∫ τ−1
0
e−αφ2(x,τ)x2dx+
∫ 1
τ−1
e−αφ1(x,τ)x2dx
}
, (20)
where φ2(x, τ) =
∑4
k=0 Pk(τ)x
k and P0(τ) = 4τ−3 ; P1 = 0 ; P2 = −2 ; P3 = 0 ; P4 = 1/5.
601
N.V.Alekseechkin
3) τ > 2
Q3(τ) = 3
∫ 1
0
e−αφ2(x,τ)x2dx. (21)
The volume fraction of the material transformed is given by expression (18) with the
replacements t → τ , t1 → τ1 = 1 , t2 → τ2 = 2. The KJMA expression in this notation
has the following form:
XK(τ) = 1 − e−ατ
4
. (22)
Figure 3. The volume fractions X(τ) (full lines), XK(τ) (dashed lines) and ∆X(τ) =
XK(τ)− X(τ) (dotted lines) in the process of homogeneous nucleation. The groups of
curves (1, 1′, 1′′) and (2, 2′, 2′′) are for α = 0.1 and α = 10, respectively.
In figure 3, the dependence X(τ) at different values of α is shown in comparison with
that given by expression (22) for infinite space. Also, the function ∆X(τ) = XK(τ)−X(τ)
that gives the error caused by the use of (22) is presented.
Furthermore, consider the dependence of the volume fraction on radius R0 of the
domain at fixed time. To this end, let us introduce the following dimensionless quantities:
radius ρ = R0/ut, distance y = r/ut and the parameter β = (π/3)Iu3t4. The expressions
for the volume fraction Qi(ρ) of the initial phase in three cases described above are as
follows:
1) ρ > 1
Q1(ρ) =
(
ρ− 1
ρ
)3
e−β +
3
ρ3
∫ ρ
ρ−1
e−βψ1(y,ρ)y2dy , (23)
where ψ1(y, ρ) =
∑4
=−1 Pk(ρ)y
k: P−1(ρ) = (2/5)ρ5 − (3/4)ρ4 + (1/2)ρ2 − 3/20; P0(ρ) =
−(3/2)ρ4 + 2ρ3 + 1/2; P1(ρ) = 2ρ3 − (3/2)ρ2 − 1/2; P2(ρ) = −ρ2 ; P3 = 1/4; P4 = 1/10.
2) 1/2 < ρ < 1
Q2(ρ) =
3
ρ3
{
∫ 1−ρ
0
e−βψ2(y,ρ)y2dy +
∫ ρ
1−ρ
e−βψ1(y,ρ)y2dy
}
, (24)
where ψ2(y, ρ) =
∑4
=0 Pk(ρ)y
k and P0(ρ) = −3ρ4 + 4ρ3 ; P1 = 0 ; P2(ρ) = −2ρ3 ; P3 = 0
; P4 = 1/5.
3) 0 < ρ 6 1/2
Q3(ρ) =
3
ρ3
∫ ρ
0
e−βψ2(y,ρ)y2dy. (25)
602
On the kinetics of phase transformation of small particles in Kolmogorov’s model
The volume fraction of the transformed material is
X(ρ) = η
(
1
2
− ρ
)
X3(ρ) + η
(
ρ− 1
2
)
η (1 − ρ)X2(ρ) + η (ρ− 1)X1(ρ) , (26)
where Xi(ρ) = 1 −Qi(ρ).
Figure 4. The volume fractions X(ρ) (full lines) and XK(ρ) (dashed lines) at fixed time.
The pairs of curves (1, 1′) and (2, 2′) are for β = 1 and β = 2, respectively.
The KJMA formula in this notation is as follows
XK(ρ) = exp(−β). (27)
In figure 4, the dependence X(ρ) is shown for different values of β.
4. Heterogeneous nucleation
Let us derive the expression for the volume fraction in the case of nucleation at fixed
points randomly distributed over the domain (for example, on foreign particles) under the
condition that all the new-phase centres appear at t′ = 0. Two variants of the problem are
possible. In the first one, the points are distributed with the mean density n; their number
in the domain is a random quantity. For this case, the dependence of volume fraction
on time is derived from the general solution of Section 2 with the use of a δ-shaped
representation of the nucleation rate:
I(t′) = nδ+(t′). (28)
We use the dimensionless variables τ and x as well as the parameter γ = nV0 =
(4π/3)nR3
0 which is the mean number of particles in the domain. Substituting r1 = R0 ,
r2 = Rm(t) and h = r into expression (6) and going to dimensionless variables, we obtain:
v(r1, r2;h) = V0f(x, τ), where
f(x, τ) =
1
2
(
1 + τ3
)
+
1
16
x3 − 3
8
x
(
1 + τ2
)
− 3
16
(
1 − τ2
)2
x
. (29)
603
N.V.Alekseechkin
The expressions for Qi(τ) are as follows:
1) τ < 1
Q
(h)
1 (τ) = (1 − τ)3 e−γτ
3
+ 3
∫ 1
1−τ
e−γf(x,τ)x2dx, (30)
2) 1 6 τ 6 2
Q
(h)
2 (τ) = (τ − 1)3 e−γ + 3
∫ 1
τ−1
e−γf(x,τ)x2dx, (31)
3) τ > 2
Q
(h)
3 (τ) = e−γ . (32)
The volume fraction X (h)(τ) of the material transformed is given by expression (18)
as before, while in the case of infinite space it is given by the following one:
X
(h)
K (τ) = 1 − e−γτ
3
. (33)
In figure 5, the dependencies X (h)(τ) and X
(h)
K (τ) together with ∆X (h)(τ) = X
(h)
K (τ)−
X(h)(τ) are shown for different γ-values.
Figure 5. The volume fractions X (h)(τ) (full lines), X
(h)
K (τ) (dashed lines) and
∆X(h)(τ) = X
(h)
K (τ)− X(h)(τ) (dotted lines) in the first type of heterogeneous nucleation
at γ = 1 (1, 1′, 1′′) and γ = 10 (2, 2′, 2′′).
In the second variant of heterogeneous nucleation, the number N of particles in the
domain is fixed. We assume a uniform distribution of particles over the volume: the prob-
ability for any particle to be in the volume ∆v is equal to ∆v/V0 and does not depend on
either the shape or position of this volume. In this case, the binomial distribution applies:
the probability of m particles being located within the volume ∆v is
PN (m) = Cm
N
(
∆v
V0
)m [
1 − ∆v
V0
]N−m
. (34)
In particular, the probability that there are no particles within the volume ∆v is equal to
PN (0) =
[
1 − ∆v
V0
]N
. (35)
604
On the kinetics of phase transformation of small particles in Kolmogorov’s model
Setting ∆v = Ω(t, r), Ω(t, r) is equal to either Vm(t), or v(R0, Rm(t); r), or V0; we
see that expression (35) replaces the expression (2) in the case given. For the three cases
described above it is not difficult to get the following:
1) τ < 1
Q
(N)
1 (τ) = (1 − τ)3
[
1 − τ3
]N
+ 3
∫ 1
1−τ
[1 − f(x, τ)]N x2dx , (36)
were f(x, τ) is given by expression (29).
2) 1 6 τ 6 2
Q
(N)
2 (τ) = 3
∫ 1
τ−1
[1 − f(x, τ)]N x2dx, (37)
3) τ > 2
Q
(N)
3 (τ) = 0. (38)
The volume fraction of the transformed material is
X(N)(τ) = η(1 − τ)X1(τ) + η(τ − 1)η(2 − τ)X2(τ) + η(τ − 2). (39)
In the case N = 1, where the nucleation occurs at the centre of the domain,
X(1)
c (τ) = η(1 − τ)τ 3 + η(τ − 1). (40)
In figure 6, the dependence X (N)(τ) for different N -values is shown.
Figure 6. The volume fractions X (N)(τ) in the second type of heterogeneous nucleation.
The curves 1, 2, 3 correspond to N = 1, 10 and 100. The dashed line represents the
dependence X
(1)
c (τ), equation (40).
605
N.V.Alekseechkin
5. Crystallization of a liquid drop
As illustrated in figure 3, the departure of X(τ) from XK(τ) increases with a decrease
in α, which, in particular, corresponds to a decrease in R0. The parameter α has the
following meaning: 4α = IV0t
∗ is the mean number of centres formed in the domain
during time t∗. For increasing α, the curves X(τ) and XK(τ) come closer together in the
sense of numerical values, but the qualitative distinction between them remains. However,
this distinction can no longer be detected experimentally at sufficiently large α, since the
phase transformation is finished (X(τ) → 1) at small τ . So the process is described with
great accuracy by the KJMA expression over the whole time interval. The condition of the
process being completed may be formulated as ατ 4
f ' 1, where τf is the transformation
time. It is not difficult to derive that the mean linear size L̄ of a grain in the system in
the final state is of the order of (I/u)−1/4. Hence, we get another representation for α:
α ' (R0/L̄)4. For τf , we have τf ' L̄/R0. Consequently, the condition for applicability of
the KJMA expression with respect to the α-values, α� 1, becomes R0/L̄� 1. It is easy
to show that the condition γ � 1 for the applicability of the KJMA expression to the first
type of heterogeneous nucleation is reduced to the same inequality.
Let us consider the conditions under which the deviation from the KJMA law takes
place, by looking at the example of a metastable (supercooled) liquid crystallization. In
reference’s [14,15], the kinetics of crystallization and amorphization of Pd82Si18 alloy was
described. The values of the parameters included in the expressions given below are taken
from these works. The expression for the nucleation rate of the crystalline nuclei and that
for their growth velocity (for the approximation of planar interface) [12] may be presented
in the following form [15]:
I(T ) = 2Nν
(σ
T
)1/2
exp
[
−∆g + ∆Gc(T )
T
]
, (41)
u(T ) = aν exp
(
−∆g
T
)[
1 − exp
(
−∆µ(T )
T
)]
. (42)
Hereafter, all the quantities with energy dimensions are reduced to kTm, k is the
Boltzmann constant, Tm is the melting temperature. Accordingly, the temperature T is
dimensionless (in units of Tm); Tm = 1. The meaning of the quantities in (41) and (42) is
as follows.
∆Gc(T ) is the work done in achieving the critical size (Rc(T )) at nucleus formation:
Rc(T ) =
2σ
∆µ(T )
a, (43)
∆Gc(T ) =
16π
3
σ3
∆µ2(T )
, (44)
where σ ≡ σ′a2/kTm (σ′ is the surface tension on the crystal-liquid interface); ∆µ(T )
is the difference in chemical potential of atoms in the two phases. There are different
approximations for this significant function, e.g. the expression of Thomson and Spaepan
[16]:
∆µ(T ) =
2hmT (1 − T )
1 + T
, (45)
606
On the kinetics of phase transformation of small particles in Kolmogorov’s model
where hm = 0.92 is the heat of melting per atom. The expansion of ∆µ(T ) in terms of T
in the vicinity of Tm [15] may be also used instead of (45).
The remainder of the parameters are as follows: a = 2.26 · 10−8 cm is the interatomic
distance; N = 8.7 · 1022 cm−3 is the number of atoms in unit volume; ν = 1013 s−1 is the
frequency of atom vibrations; ∆g = 10 (this corresponds to about 1 eV) is the energy of
activation of atom jumps; σ ' 0.43hm [17].
Figure 7. The temperature dependencies of the nucleation rate (I), the growth velocity
(u) and the critical radius (Rc) – left Y -axis. Dashed lines represent the dependence (46)
for α = 1 (1) and α = 104 (2) – right Y -axis. The region above the curve 2 is that of the
MD regime. The marked deviations from the KJMA law occur in the region near and
below the curve 1.
In figure 7, the temperature dependencies of the nucleation rate, growth velocity and
critical radius are shown on the same scales (Imax ' 1019 cm−3s−1 and umax ' 0.3 cm s−1
are the maximal values of I(T ) and u(T ); R′ = Rc(0.995) ' 172a). The characteristic
feature of this figure is that the maxima of I(T ) and u(T ) are separated: the former
is located at a deeper supercooling than the latter. The critical radius decreases rather
rapidly with a decrease in temperature. The nucleation rate becomes appreciable when
the critical radius becomes small (this corresponds to the fact that the probability of
large fluctuations is vanishingly small). At the same time, the nucleation rate remains
small enough up to some temperature. Thus, there is a temperature interval in which
the values of the parameter α are sufficiently small for macroscopic values of the system
size R0. Notice that the condition α ' (R0/L̄)4 � 1 considered above corresponds to
the multidroplets (MD) regime [18–20]. Accordingly, the deviation from the KJMA law
considered here occurs beyond this regime, when this condition is not satisfied.
Consider the following dependence:
R0(T ) = α1/4L̄(T ) =
(
αu(T )
I(T )
)1/4
. (46)
For definiteness, let α = 1 (in this case, the deviation from the KJMA law is distinct
– see figure 3 – max(∆X(τ)) ' 0.25). Letting, as an example, T ' 0.84, we get, from
607
N.V.Alekseechkin
(41) and (42), I(0.84) ' 16 cm−3s−1, u(0.84) ' 0.23 s−1cm, and it follows from (46) that
R0 ' 0.34 cm , which is a macroscopic size. At the same time, Rc(0.84) ' 6a� L̄.
In figure 7, the dependence (46) for α = 1 is shown by dashed line 1.The deviation
from the KJMA law at some temperature may be observed for sizes R0 near this curve
and below it. If we assume α = 104 (R0/L̄ = 10, max(∆X(τ)) ' 0.02) as the lowest
limit of α for the MD regime, then this regime is operative for all temperatures in the
region above curve 2 in figure 7 (this is the dependence (46) for α = 104 ). It should
be noted that the condition Rc � L̄ is satisfied here down to the deepest supercooling
(L̄(0.6)/Rc(0.6) ' 70), though, as a rule, such supercooling is not achievable in practice.
6. Discussion
The expressions obtained above yield the volume-fraction value, X(t), averaged over an
ensemble of identical systems. The volume-fraction value, Xi(t), in the individual system
is obviously a random quantity, since it is a result of the realization of the random process.
If the probability for the volume fraction at time t to be in the interval [Xi, Xi + dXi] is
denoted by p(Xi, t), then the mean value is
X(t) =
∫ 1
0
Xip(Xi, t)dXi . (47)
In terms of an ensemble of Na systems, expression (47) is replaced by the following:
X(t) = lim
Na→∞
X(Na, t) , X(Na, t) =
1
Na
Na
∑
i=1
Xi(t). (48)
Let ∆Xi(t) ≡ Xi(t) − X(t) be the deviation from the mean value and let σ(t) ≡
M{[Xi(t)−X(t)]2} be the variance. Then, for the deviation δ(Na, t) of the volume fraction
X(Na, t) from its mean we have
δ(Na, t) ≡ X(Na, t) −X(t) =
σ(t)√
Na
χ(Na, t) ,
χ(Na, t) ≡ 1√
Naσ(t)
Na
∑
i=1
∆Xi(t). (49)
According to Lyapunov’s theorem [11], χ(Na, t) has the normal distribution for Na →
∞, so the following estimate for the probability of the inequality χ(Na, t) < ε is applicable
for sufficiently large Na:
p {χ(Na, t) < ε} ' 1√
2π
∫ ε
−∞
e−(z2/2)dz. (50)
Hence, we find
p {δ(Na, t) < ε} ' 1√
2π
∫
√
Naε/σ(t)
−∞
e−(z2/2)dz. (51)
The right-hand side of this equality is practically equal to unity at
√
Naε/σ(t) ' 3. Hence,
for the given accuracy ε, the corresponding value of Na is about (3σ(t)/ε)2.
608
On the kinetics of phase transformation of small particles in Kolmogorov’s model
The two variants of heterogeneous nucleation considered above may correspond to the
following experimental conditions in the case of crystallization of liquid drops. In the first
variant the drop is extracted at random from the bulk of a liquid in which the foreign
particles are distributed with the mean density n. In this case, the number of particles in
the drop is random including zero. The latter case leads to the final value of X (h)(τ) being
less than unity (figure 5).
In the second variant of heterogeneous nucleation, it is known that there are N particles
in the drop. Evidently, the expressions for X (h)(t) may be obtained by averaging the
expressions for X (N)(t) over all N :
X(h)(t) =
∞
∑
N=0
P (N)X(N)(t) , P (N) =
γN
N !
e−γ . (52)
In particular, at t > 2t∗ ≡ tf X(N)(t) = 1 at N > 1 and X (0) = 0. Consequently,
X(h)(t > tf ) =
∞
∑
N=1
P (N) = 1 − e−γ .
In turn, the expressions for X (N)(t) may be obtained by averaging the volume fraction
X(N)(~r1, ~r2, · · · , ~rN ; t), at the location of the particles at the fixed points ~rk, over all ~rk. It is
seen in figure 6 for the case N = 1 in what way this averaging changes the time dependence
of the volume fraction (curves 1 and dashed). In particular, the transformation time tf is
doubled. This is due to the particle positions near the domain boundary being taken into
account.
Consider some limiting cases of the expressions derived. At τ → 0 the addend in (19)
has the following expansion in terms of τ : 3τ − 3τ 2 + τ3 +O(τ5). Accordingly, the volume
fraction is
Q(τ) = Q1(τ) = e−ατ
4 [
1 +O(ατ 5)
]
. (53)
That is, we get the KJMA expression, as we should. Small τ -values correspond to small
times and large R0. The condition τ � 1 (Rm � R0) has the following meaning. It is
proposed in deriving the expression for the volume fraction in reference [1] that the point
O′ should lie at a distance greater than Rm(t) from the domain boundary. In order to
allow the volume ∆V = (4π/3)[R3
0 − (R0 −Rm)3] of the boundary layer to be neglected,
the condition ∆V/V0 � 1 or Rm/R0 � 1 should be obeyed.
At τ > 2 we have
Q(τ) = Q3(τ) = J(α)e−4ατ = J(α)e−IV0t, (54)
where
J(α) ≡ 3
∫ 1
0
eα(3+2x2−x4/5)x2dx.
At α → 0, J(α) → 1. Thus, we get the following result: the time dependence of the
volume fraction qualitatively differs from that in infinite space (it cannot be derived from
the latter, equation (22), by the use of correction factors). That is, the Avrami exponent
nA decreases with time from 4 to 1. The Avrami exponent [3,12] is the slope of the tangent
to the plot of ln(− ln(1−X(t))) against ln t (figures 8, 9). In other words, the finiteness of
the domain leads to a slowing down of the transformation process in comparison with the
609
N.V.Alekseechkin
Figure 8. The function ln(− ln(1 − X(τ))) versus ln τ for the case of homogeneous nu-
cleation with α = 0.1. The dashed line is for XK(τ).
Figure 9. The dependence of the Avrami exponent nA on ln τ corresponding to figure 8.
nucleation in infinite space. It is not difficult to derive from (54) the following expression
for nA(ξ), where ξ ≡ ln τ :
nA(ξ) =
1
1 − z(α) exp(−ξ) . (55)
Here, ξ > ln 2 and z(α) ≡ [ln J(α)]/4α ' 1.
Similarly, it is seen from (30)–(32) that the Avrami exponent decreases with time from
3 to 0 in the first variant of heterogeneous nucleation.
610
On the kinetics of phase transformation of small particles in Kolmogorov’s model
At ρ→ 0 we get from (25):
Q(ρ) = Q3(ρ) = e−4βρ3
[
1 +O(ρ4)
]
. (56)
This is expression (54) again, with J(α) = 1 (α → 0 at ρ → 0). The function Q(t) =
exp(−IV0t) is the probability that no centre of a new phase will appear in the domain by
time t. This is a volume fraction at small R0 and at large t, since averaging over r is not
essential in this case.
In the case ρ→ ∞ we obtain from (23)
Q(ρ) = Q1(ρ) = e−β . (57)
That is, we have the KJMA expression XK(t) again.
As is evident from the foregoing, the transformation time tf is infinite in the case
of homogeneous nucleation in a finite domain. Moreover, the finiteness of the domain
leads to the slowing down of the transformation process in comparison with nucleation in
infinite space. The transformation time is finite in the process of heterogeneous nucleation
considered above, tf = 2t∗, since all the centres appear at t′ = 0. The meaning of this time
in the context of the ensemble of Na systems is as follows: at t > tf the volume fraction in
each system is equal to either 1 or 0 with the probability equal to unity, for the number
of particles N > 1 and N = 0, respectively. In a more general process of heterogeneous
nucleation, where the centres appear at arbitrary times t′, the transformation time is
infinite.
The pattern shown in figure 7 is typical of the phenomena of crystallization and amor-
phization of a supercooled liquid. Amorphization is the result of nucleation and growth
of non-crystalline nuclei – clusters [14,15]. Correspondingly, for this process, the depen-
dencies in figure7 are shifted to the left and start at the melting temperature T̃m for the
infinite cluster [14,15]. The pattern of figure 7 is apparently rather general which also ari-
ses in other cases of phase transformations with homogeneous nucleation rate. In reference
[20], the KJMA model was applied to the description of the Ising lattice-gas kinetics at
mesoscopic scales of length and time. An excellent agreement of these two models was
demonstrated in two dimensions and for moderately strong fields. A metastable phase
decay picture of reference[20] is similar to the one just considered. The magnetic field
H plays the role of temperature in figure 7 (the temperature itself is a fixed parameter
therein). A test of the KJMA picture was carried out in the MD regime. Equation (46)
for α = 1 limiting this regime is that for the dynamic spinodal (DSP) [20]. This equation
determines the R
(DSP)
0 (H,T ) dependence, so the noticeable deviations from the KJMA law
should take place at the system sizes R0 ∼ R
(DSP)
0 and to a greater extent at R0 < R
(DSP)
0 .
At the same time, these sizes may be still macroscopic, 1 � Rc � L̄ ∼ R0, since R
(DSP)
0 is
large in the region of weak fields (which corresponds to slight supercooling in figure 7). In
this case, expressions (19)–(21) for the volume fraction should be used instead of (22).
611
N.V.Alekseechkin
References
1. Kolmogorov A.N., Izv. AN SSSR, Ser. matem., 1937, No. 3, 355.
2. Johnson W.A., Mehl R.F., Trans. AIME, 1939, No. 135, 416.
3. Avrami M., J. Chem. Phys., 1939, 7, 1103; 1939, 8, 212; 1941, 9, 177.
4. Belen’kiy V.Z. Geometrical Probability Models of Crystallization. Moscow, Nauka (in Russian),
1980.
5. Cahn J.W. – In: Materials Research Society Symposium Proceedings, 1996, 398, 425.
6. Weinberg M.C., Birnie III D.P., Shneidman V.A., J. Non- Cryst. Solids, 1997, 219, 89.
7. Weinberg M.C., Birnie III D.P., J. Non- Cryst. Solids, 1996, 196, 334.
8. Trofimov V.I., Trofimov I.V., Kim J.-I., Nucl. Instr. Met. in Phys. Res. B, 2004, 216, 334.
9. Alekseechkin N.V., Fizika Tverdogo Tela, 2000, 42, No. 7, 1316 (Engl. Transl. Rus. J. Physics
of the Solid State, 2000, 42, No. 7, 1354).
10. Alekseechkin N.V. Kristallografia, 2003, 48, No. 4, 760 (Engl. Transl. Rus. J. Crystallography
Reports, 2003, 48, No. 4, 707).
11. Gnedenko B.V. Course in the Theory of Probability. Moscow, Fizmatgiz (in Russian), 1961.
12. Cristian J.W. The Theory of Transformations in Metals and Alloys. Oxford, Pergamon, 1965.
13. Corn G.A., Corn T.M., Mathematical Handbook. New York, McGraw-Hill, 1968; Moscow,
Nauka (in Russian), 1984.
14. Alekseechkin N.V., Bakai A.S., Lazarev N.P., Fizika Nizk. Temp., 1995, 21, 565 (Engl. Transl.
Ukr. J. Low Temperature Physics, 1995, 21, 440).
15. Alekseechkin N.V., Bakai A.S., Abromeit C., Metallofizika i Noveishie Technologii, 1998, 20,
15 (Engl. Transl. Ukr. J. Met. Phys. Adv. Tech., 1999, 18, 619).
16. Thomson C.W., Spaepen F. Acta Met., 1979, 27, 1855.
17. Spaepen F., Acta Met., 1975, 23, 729.
18. Tomita H., Miyashita S., Phys.Rev. B, 1992, 46, 8886.
19. Rikvold P.A., Tomita H., Miyashita S., Sides S.W., Phys. Rev E, 1994, 49, 5080.
20. Ramos R.A., Rikvold P.A., Novotny N.A., Phys.Rev. B, 1999, 59, 9053.
612
On the kinetics of phase transformation of small particles in Kolmogorov’s model
Про кiнетику фазового перетворення малих часток в моделi
Колмогорова
М.В.Алєксєєчкiн
Iнститут теоретичної фiзики iм. А.I. Ахiєзера Нацiонального Наукового Центру
“Харкiвський фiзико-технiчний iнститут” , вул. Академiчна, 1, Харкiв 61108, Україна
Отримано 16 травня 2007 р., в остаточному виглядi – 15 березня 2008 р.
Класична теорiя Колмогорова-Джонсона-Мейла-Аврами узагальнюється на випадок
обмеженої системи. У рамках геометрико-ймовiрнiсного пiдходу вирiшується задача об-
числення об’ємної частки нової фази в сферичнiй областi. Одержано розв’язки для ви-
падкiв гомогенної i гетерогенної нуклеацiї. Показано, що властивiсть обмеженостi си-
стеми приводить до якiсної вiдмiнностi часової залежностi об’ємної частки вiд такої в
необмеженому просторi: показник Аврами в процесi гомогенної нуклеацiї зменшується
з часом вiд 4 до 1.
Ключовi слова: KJMA-теорiя, об’ємна частка, науклеацiя, показник Аврами
PACS: 05.70.Fh, 68.55.Ac, 81.15.Aa
613
614
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