Continuum Mech. Thermodyn.
DOI 10.1007/s00161-012-0258-5
O R I G I NA L A RT I C L E
David Molnar · Christian Niedermeier · Alejandro Mora ·
Peter Binkele · Siegfried Schmauder
Activation energies for nucleation and growth and critical
cluster size dependence in JMAK analyses of kinetic
Monte-Carlo simulations of precipitation
Received: 30 October 2011 / Accepted: 25 June 2012
© Springer-Verlag 2012
Abstract Kinetic Monte-Carlo (KMC) methods are used as an approach to simulate precipitation in
Cu-alloyed bcc Fe. In order to characterize the process, transformed fractions, that is, the precipitated atoms,
are related to Johnson-Mehl-Avrami-Kolmogorov theory. However, simulated data often deviate from corresponding fit curves and so does the resulting growth exponent when compared to theoretical expectations.
Furthermore, some data may suggest the development of a metastable phase. In our study, we show that the
characteristics of the transformed fraction and, as a consequence, the derived growth exponents sensitively
depend on the number of atoms that are considered to form a particle and hence contribute to the transformed
fraction. With a temperature dependence of the critical cluster size and additionally accounting for severe
impingement of the particles, we obtain growth exponents which lie close to the expected range between
n = 1.5 and n = 2.5 for pre-existing nuclei or continuous nucleation, respectively. From these, we obtain
activation energies for nucleation and growth of precipitates. In this way, atomistic KMC simulations yield thermodynamical quantities, which can be valuable input parameters for larger length scale simulation methods,
for example, for Phase Field Method simulations.
Keywords Kinetic Monte-Carlo · Precipitation kinetics · Phase transformation · Nucleation and growth
1 Introduction
During annealing or service at elevated temperatures of above 300 ◦ C, the nanostructures of supersaturated
Cu-alloyed bcc-Fe changes due to the formation of small second phase particles that precipitate from the solid
solution. In case of Cu-alloyed α-Fe, Cu precipitates form within the Fe matrix yielding a strengthening of the
material at early stages.
The process of precipitation in the Fe-Cu system has been observed experimentally [1–3] and has also been
modelled computationally by the application of the kinetic Monte-Carlo (KMC) method [4,5]. The calculation
of site-exchange probabilities of a single vacancy with neighbouring lattice sites within the Fe-Cu matrix and
Communicated by Oliver Kastner.
D. Molnar (B) · S. Schmauder
Institute for Materials Testing, Materials Science and Strength of Materials
and SimTech Cluster of Excellence, University of Stuttgart,
70569 Stuttgart, Germany
E-mail: [email protected]
Tel.:+49-711-68563929
Fax:+49-711-68562635
C. Niedermeier · A. Mora · P. Binkele
Institute for Materials Testing, Materials Science and Strength of Materials,
University of Stuttgart, 70569 Stuttgart, Germany
D. Molnar et al.
subsequent vacancy jumps repeated many times yield a clustering of Cu atoms. This process can be followed
by considering the precipitated volume, which is also referred to as the transformed fraction. Typically, in the
literature, the transformed fractions obtained from experiments as well as from KMC simulations are compared
to the kinetics of Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory [6–8]. However, JMAK theory holds
only under strict requirements, which are often not satisfied [9]. Furthermore, KMC simulated data may show
remarkable deviations from JMAK-like behaviour [10], which even might suggest the existence of metastable phases. These metastable phases might be justifiable in multi-component systems [11,12] but are rather
unlikely to appear in binary alloys such as Cu-alloyed α-Fe. By carefully analysing KMC precipitation data
of 1 wt.% Cu and 2 wt.% Cu initially solved in α-Fe, these deviations can be addressed to the critical number
of atoms that are considered to form a precipitate and hence contribute to the transformed fraction.
Beneath the advantage of taking into account every single atom, KMC simulations allow for vast parameter
studies and the simulation of nucleation and growth of precipitates. However, KMC simulations are limited to
small simulation samples with length scales in the order of 10–100 nm. In order to reach larger length scales,
sequential multiscale materials modelling (MMM) is a promising approach where simulation methods applied
on different length or time scales are connected through transfer parameters. Regarding the further growth of
precipitates, a different method has to be applied as not only the size of the precipitates becomes too large
but also the precipitates start to change their structure from bcc to fcc via 9R and 3R intermediate structures
[13,14]. Most probably Phase Field Methods (PFM) will provide the framework for the next higher length
scale. Due to the fact that they are based on a thermodynamical description, two of the necessary transfer
parameters are the activation energies of nucleation and growth. They have been successfully obtained by
analysing the transformed fraction in experiments of, for example, Co precipitation in Cu [15]. Adapting
the procedure described in [9], we will obtain the activation energies for nucleation and growth from KMC
simulations.
In Sect. 2, the KMC method will be introduced. Section 3 will provide the simulation results. Analysing
the transformed fraction in dependence of the critical cluster size in Sect. 4 will reveal the sensitivity of the
transformed fraction curves on the preliminary assumptions. In Sect. 5, the activation energies for nucleation
and growth will be obtained by assuming JMAK-like behaviour. Accounting for a more severe impingement
of particles, a JMAK correction factor will be introduced in Sect. 5.2 in order to adjust the growth exponent
close to values between 1.5 and 2.5 and the corresponding activation energies for nucleation and growth will
be derived. Conclusions and an outlook will be given in Sect. 6.
2 The kinetic Monte-Carlo method
The process of Cu-precipitation in α-Fe is simulated by the application of a kinetic Monte-Carlo method
(KMC), which is based on a vacancy diffusion mechanism on a rigid bcc crystal lattice, also referred to as the
Rigid Lattice Method (RLM) [4]. Although in nature Cu has fcc structure, it is well known that Cu clusters
with sizes smaller than 2 nm are coherently embedded on α-Fe lattice sites [13,14], justifying the RLM. The
KMC simulation used in this study was first described by Soisson et al. [4]. A detailed description of the here
presented KMC method can be found in [4,5]. A size of L = 128 lattice constants as starting configuration
yields N = 2L 3 = 4, 194, 304 lattice sites and a cubic box with an edge length of 36.6 nm. The box surfaces
possess normals in [100], [010] and [001] directions, respectively. Periodic boundary conditions are set in
all directions to approximate single crystalline bulk behaviour. Specific amounts of Fe atoms are replaced
randomly by Cu atoms to obtain Fe-Cu solid solutions with 1 wt.% Cu and 2 wt.% Cu, respectively. Diffusion proceeds via a thermally activated vacancy mechanism where a vacancy (vac) and a neighbouring atom
(Fe and Cu) exchange their positions. The jump rates are given by
ΔE Fe,vac
ΔE Cu,vac
ΓFe,vac = νFe exp −
and
ΓCu,vac = νCu exp −
,
(1)
kB T
kB T
where νFe and νCu denote attempt frequencies and ΔE Fe,vac and ΔE Cu,vac the activation energies that depend
on the local atom configuration with the corresponding chemical binding interactions due to first and second
nearest neighbour pair interactions and the energy at the saddle point between atom and vacancy. A detailed
description can be found in [4,5,18]. The attempt frequencies are estimated using the diffusion ‘constants’ of
the pure metals. The annealing temperature varies between 250 and 500 ◦ C in order to ensure supersaturated
solutions. For each neighbour of the vacancy, the jump frequencies Γ1 , . . . , Γ8 are calculated. By applying a
rejection-free residence time algorithm [4], one of these eight possible jumps is selected and performed. The
Activation energies for nucleation and growth from KMC simulations
Table 1 Material data from experiments, ab initio (AI) and CALPHAD calculations for Fe and Cu applied in the kinetic
Monte-Carlo (KMC) simulations
E coh, Fe = 4.28 eV
[4,16]
AI
[17,16]
E coh,
Cu = 3.49 eV
E coh, Cu = 4.08 eV
[18]
F
[19]
E vac,
Fe = 1.95 eV
F
E vac,
[10,19]
Cu = 0.88 eV
M
E vac, Fe = 0.68 eV
[16,17]
M
[10]
E vac,
Cu = 0.57 eV
a = 0.287 nm
[20]
ωFe-Cu = −0.545 eV
[10]
D0Fe = 2.01 · 10−4 m2 s−1 [21]
D0Cu = 2.16 · 10−4 m2 s−1 [3]
νFe = 2.44 · 1015 s−1 (D0Fe /a 2 )
Cohesive energy Fe
Cohesive energy Cu (AI)
Cohesive energy Cu
Vacancy formation Fe
Vacancy formation Cu
Vacancy migration Fe
Vacancy migration Cu
Lattice constant
Mixing energy
Diffusion constant Fe
Diffusion constant Cu
Attempt frequency Fe
νCu = 2.62 · 1015 s−1 (D0Cu /a 2 )
Attempt frequency Cu
Table 2 General conditions of the kinetic Monte-Carlo (KMC) simulations
Lattice constants
Box edge length
Number of atoms
Number of vacancies
Cu concentrations
Annealing temperatures
Boundary conditions
Number of KMC steps
128
36.3 nm
4,194,303
1
1 wt.%, 2 wt.%
250–500 ◦ C
Periodic in x, y, z
1011
weighted random jumps are repeated over 1011 times during the simulation of precipitation. The time scale is
adjusted according to the number of Monte-Carlo steps and the vacancy concentration, that is,
cvac,sim
treal =
(2)
tMC ,
cvac,theo
where cvac,sim and cvac,theo denote vacancy concentrations calculated according to [10,19] where the former
corresponds to the vacancy concentration in the simulation and the latter is a theoretical estimation of the
equilibrium vacancy concentration based on the minimisation of the free enthalpy [5,22]. tMC is given by the
residence time for a jump
tMC
⎛
⎞−1
8
=⎝
Γj⎠ .
(3)
j=1
As reported in [23], the calculated time scale depends sensitively on the used energies, which are listed among
further material data in Table 1. Nevertheless, the calculated times are directly proportional to the estimated
real time. Table 2 summarises the parameters used for the KMC simulations.
3 Simulation results
Two starting configurations, with 1 wt.% Cu and 2 wt.% Cu, respectively, were thermally aged at constant
temperatures ranging from 250–500 ◦ C. In total, over 1011 KMC steps are performed during each simulation,
yielding the formation of small Cu clusters. During the simulation, the Cu concentration within the solid
solution (matrix m) is calculated and the result is compared to the concentration at equilibrium, which is given
by
ΔSnc
ωFe-Cu
cm (t → ∞) ∼
exp
,
(4)
·
exp
=
kB
kB T
D. Molnar et al.
a
b
Fig. 1 Degree of advancement of precipitation for 1 wt.% Cu (a) and 2 wt.% Cu (b) in α-Fe: Higher copper concentrations yield
a faster precipitation process. The annealing temperature is varied from 250 ◦ C (blue) to 500 ◦ C (orange) in 50 ◦ C steps (colour
figure online)
where the non-configurational entropy ΔSnc is set to 1k B according to [10]. The ratio [4]
ξ(t) =
cm (t = 0) − cm (t)
cm (t = 0) − cm (t → ∞)
(5)
can be seen as the degree of advancement of precipitation (Fig. 1), cm (t = 0) being the starting concentration of
Cu atoms which are completely surrounded by Fe atoms. cm (t) is the remaining Cu concentration in the matrix
at time t. As soon as a solved Cu atom approaches another Cu atom or a Cu cluster, it is no longer considered
as solved within the matrix. Hence, it contributes to the degree of advancement ξ(t). At the beginning of the
simulation, that is, for a solid solution, ξ(t = 0) equals zero. During the simulation, the amount of Cu within
the Fe matrix decreases due to precipitation. As a consequence, ξ(t) increases, saturating at a value of one
where all remaining dissolved Cu atoms are in thermal equilibrium.
The degree of advancement can be approximated by the function
n t
ξ (t) ≈ 1 − exp −
,
(6)
τ
which is a well-known result of JMAK (Johnson, Mehl, Avrami and Kolmogorov) theory yielding S-shaped
curves when using a logarithmic time axis and thus satisfying fits, especially for the lowest temperatures in
Fig. 1a, b. [6–8]. However, the assumptions made within JMAK theory such as
(i) random nucleation and growth within an infinite matrix,
(ii) a high undercooling or supersaturation
are often not completely satisfied in situations where applied. In the KMC simulations presented here, the
assumptions are satisfied from the beginning of the simulation, where nucleation can be considered being
random while the supersaturation is high enough. Yet, both assumptions weaken during precipitation as there
is no further nucleation at precipitate sites and the supersaturation decreases due to the precipitation. Nevertheless, JMAK theory is considered being appropriate at early precipitation stages and lower temperatures and
will be applied in this paper.
Considering Fig. 1a, b for higher temperatures, deviations from the expected precipitation behaviour
become visible in the form of double S-shaped curves with intermediate stages. A similar behaviour can be
found in [10], affirming the necessity of a detailed analysis on the origin of these deviations which will follow
in the next section.
4 Critical cluster size dependence
The graphs for the degree of advancement ξ(t) shown in Fig. 1 are obtained by considering Cu dimers,
that is, atom pairs, to form the smallest possible clusters. Additionally, clusters which are pre-existing due
to the random distribution (mostly dimers and some trimers) do not contribute to the degree of advancement. Although these assumptions may be intuitively correct, the assumed precipitate consisting of two Cu
atoms is line shaped. Already a slightly bigger amount of atoms can form clusters that much more likely
can be considered as particles; for example, 4 atoms forming a tetrahedron or 6 atoms forming a bi-pyramid.
Activation energies for nucleation and growth from KMC simulations
a
b
Fig. 2 Dependence of the transformed fraction (a) and the number of particles (b) on the number of atoms that are considered to
form a critical cluster, that is, the smallest cluster size which contributes to the transformed fraction f (t) for the case of 2 wt.%
Cu and an annealing temperature of 500 ◦ C
In the following, the minimum number of neighbouring atoms, which are assumed to form a particle and thus
contribute to the transformed fraction, will be denoted as the critical cluster size n cut-off . It is not our intention
at this point to relate n cut-off to the critical radius of a stable nucleus, which can be derived in the framework
of classical nucleation theory. At later stages of precipitation, nucleation becomes more unlikely to occur due
to the reduced supersaturation in the matrix. Therefore, the choice of n cut-off mainly affects the beginning of
precipitation, that is, nucleation and growth.
Figure 2a shows the dependence of the transformed fraction of 2 wt.% Cu at 500 ◦ C on the critical cluster
size n cutt-off . In contrast to the degree of advancement in Eq. (6), the transformed fraction is obtained here by
taking
n prec
,
(7)
f (t) =
n prec,eq
n prec being the number of atoms in precipitates that contain n cut-off or more atoms. n prec,eq is the number of
atoms expected in precipitates at thermal equilibrium. As the total number of Cu atoms in the simulation sample
is known, n prec,eq is calculated from the equilibrium Cu concentration of the Fe matrix applying Eq. (4) for the
corresponding annealing temperature. For n cut-off = 2, the curve is similar to the corresponding one (2 wt.%,
500 ◦ C) in Fig. 1 except for the fact that in Fig. 2 dimers, trimers, etc., which are existing at the beginning of
the transformation process are not considered being solved within the matrix. Hence, they already contribute
to the transformed fraction and f (t = 0) = 0. For the case of 2 wt.% Cu, this means that at the beginning
of the simulation, there exist certain amounts of Cu dimers, trimers, etc., due to the statistical distribution of
the Cu atoms. As n cut-off increases, more and more Cu atoms have to form a cluster in order to contribute
to the transformed fraction. Hence, the transformed fraction as well as the number of particles (see Fig. 2b)
decreases.
Remarkably, with increasing n cut-off , the deviation from the expected S-shaped behaviour, which appears
to be an intermediate state (see Fig. 1b at 500 ◦ C) as already discussed in Sect. 3, vanishes and the curves
approximate the expected JMAK S-shape (see Fig. 2a). Hence, it can be assumed that particles consisting
of only a few atoms (dimers, trimers, etc.) cause the considerable deviations. Furthermore, they appear more
pronounced at high temperatures and low concentrations where the undercoolings (or the supersaturations)
are smaller. One should be aware that for these conditions, the assumptions made in JMAK can no longer
be considered as fully satisfied due to the low driving forces for the transformation [9], that is, for further
precipitation.
The critical radius of a precipitated Cu particle is proportional to the reciprocal undercooling ΔT [24].
Relating the radius of a particle of critical size to the corresponding number of atoms yields
C
.
(8)
ΔT 3
The undercooling ΔT is given by the difference between the temperature below which the Fe-Cu solid solution
will become supersaturated and the annealing temperature of the simulation experiment. Once the constant C
is known, the critical cluster size at any temperature may be calculated from Eq. (8). In the following section,
we will take C as small as possible satisfying the condition that the double S-shaped curves that are visible in
Fig. 1 disappear. The values calculated by Eq. (8) are rounded to integer values.
n cut-off =
D. Molnar et al.
5 JMAK kinetics
In order to derive the growth exponents n in Eq. (6), the effective activation energies for nucleation and
growth Q eff as well as the activation energies for nucleation Q N and growth Q G , we will follow the procedure described in detail in [9,25]. According to this procedure, the above-mentioned quantities can be derived
without the recourse to any specific kinetic model for nucleation. The analysis of the phase transformation
will show the dependence of the growth exponent and the activation energies for nucleation and growth on the
impingement model, which will be discussed in the following.
In general, the precipitation process can be divided into three stages:
(i) Nucleation: atoms form clusters of which some will grow and some will disappear due to a permanent
addition and dissolving of Cu atoms.
(ii) Growth: By agglomerating more Cu atoms from the surrounding iron matrix than releasing Cu atoms
into it, there is a net growth of the clusters. Thus, the Cu concentration in the matrix decreases.
(iii) Impingement: growing clusters contact each other and inhibit further growth into the contact direction.
Ostwald ripening also has to be taken into account, i.e., the coarsening of larger particles at the expense
of smaller ones without net Cu concentration change within the matrix.
5.1 Hard impingement
With increasing time of transformation, particles of supercritical size grow and may impinge on each other.
The extended transformed volume V e is calculated by the volume of all growing nuclei supposing particles
may overlap and continue to grow through surrounding neighbouring particles. Evidently, the real transformed
volume V t , which takes the overlapping and blocking of neighbouring particles into account, must be smaller
than the extended transformed volume V e . If transformation time increases by dt, the increase of the real transformed volume is considered to be only a part of the (larger) increase of the extended transformed volume.
This part is as large as the untransformed fraction (1 − f ) where f = V t /V, V being the sample volume
[6–8].
Hence,
dV t = (1 − f )dV e .
(9)
dV e
df
=
,
1− f
V
(10)
Ve
f = 1 − exp −
,
V
(11)
After rearranging Eq. (9) to
integration yields
which is similar to Eq. (6). For isothermal annealing,
temperature dependence [25]
where
Ve
V
in Eq. (11) can be expressed with an Arrhenius-type
Ve
= (kt)n(t) ,
V
(12)
Q eff (t)
.
k = k0 (t) exp −
RT
(13)
In Eq. (13), k0 (t) and R denote the pre-exponential factor and the gas constant, respectively. Q eff , k0 and n are
allowed to vary in time as we do not presume a kinetic model. For the case of pre-existing nuclei or continuous
nucleation, the growth exponent would be constant at n = 1.5 or n = 2.5, respectively. For mixed nucleation
modes, the values of n are expected to lie between 1.5 and 2.5. With Eqs. (13), (11) and (12) can be expressed
as
ln (−ln (1 − f )) = n ln (kt) .
(14)
Activation energies for nucleation and growth from KMC simulations
a
b
c
d
Fig. 3 Here, the 2 wt.% Cu results are shown. With n cut-off = 5 at 250 ◦ C and applying Eq. (8), the transformed fraction curves
in (a) are obtained. The slope in (b) equals the JMAK growth exponent n. As it may vary in time, it is calculated piecewise
yielding values between 0.40 and 1.80, which is lower than the expected range between 1.5 and 2.5. Assuming hard impingement,
the slope in (c) yields the effective activation energies Q eff lying between 206.8 and 227.2 kJ/mol. Fitting the data points in
(d) with Eq. (16), the average activation energies of nucleation and growth can be estimated as Q N = 206.7 ± 0.8 kJ/mol and
Q G = 214.0 ± 1.6 kJ/mol, respectively
Hence, the slope in Fig. 3b depicts the growth exponents of the transformed fraction curves in Fig. 3a, which
are obtained from the 2 wt.% Cu simulation data (see Fig. 1b) by choosing the critical cluster size as described
in Sect. 4. The growth exponents are approximated by fitting piecewise regression lines. Starting from values
between n = 0.95 and n = 1.80, the growth exponent decreases during transformation, a behaviour which
can be explained by taking the Avrami nucleation as the nucleation model [25]. However, the overall absolute
values of n do not fit into the expected range between 1.5 and 2.5. This aspect will be discussed further in the
following section.
By taking times t f1 , t f2 between two stages of transformation f 1 , f 2 at different annealing temperatures,
the effective activation energy Q eff can be determined as [9]
Q eff = R
d
ln(t f2 − t f1 ) ,
d(1/T )
(15)
that is, by determining the slope in Fig. 3c. The crosses in the figure correspond to changes of the transformed
fraction of f 2 − f 1 = 0.1 starting at f = 0.1. The fitted curves show perfectly linear behaviour and the slopes
yield effective activation energies of Q eff = 206.8–227.2 kJ/mol and Q eff = 214.3–229.9 kJ/mol for 1 wt.%
Cu and 2 wt.% Cu, respectively.
The values of the growth exponent n and the effective activation energies can be plotted in one graph as
shown in Fig. 3d. According to [25], the effective activation energy Q eff , for a wide range of nucleation and
growth modes with Arrhenius temperature dependence, can be expressed as
d
d
m QG + n − m Q N
Q eff =
,
(16)
n
where Q N and Q G are the activation energies for nucleation and growth, respectively. The constant parameters
d and m are given by the dimensionality of growth (d = 3) and diffusion controlled growth as the kinetic
growth model (m = 2) [25]. The dashed lines in Fig. 3d represent the fit curves of Eq. (16) at different
temperatures. In contrast to Q eff , Q N and Q G do not depend on time and temperature. An averaging over all
D. Molnar et al.
temperatures yield Q N = 188.3 ± 2.0 kJ/mol and Q G = 205.0 ± 4.5 kJ/mol for the activation energies for
nucleation and growth for 1 wt.% Cu, respectively. For 2 wt.% Cu, the activation energies for nucleation and
growth yield Q N = 206.7 ± 0.8 kJ/mol and Q G = 214.0 ± 1.6 kJ/mol, respectively. According to [15,25],
the activation energies for nucleation and growth can be compared to the activation energy of diffusion Q D ,
although the quantities are not equal. Within the KMC simulations, Q D is used as
F
M
+ E V,X
,
Q D,X = E V,X
(17)
where X ∈ {Fe, Cu}. The resulting activation energies for diffusion are Q D,Cu = 139.9 kJ/mol and Q D,Fe =
253.8 kJ/mol for Cu and Fe, respectively. An experimentally obtained value of Q D = 284 kJ/mol for the
diffusion of Cu in Fe can be found in [21]. The values obtained for the activation energies for nucleation and
growth assuming hard impingement are in good agreement with the calculated and the experimentally obtained
values for the activation energies for diffusion.
5.2 Severe impingement due to Ostwald ripening
In the previous section, the growth exponent obtained by the assumption of JMAK-like behaviour yielded
lower values than expected. This may be due to the fact that the correction for hard impingement does not
suffice when dealing with transformations that are considerably accompanied by Ostwald ripening where some
particles will tend to grow at the expense of smaller particles, thereby decreasing their total amount of surface
area. A net movement of nuclei towards each other to form larger particles could be observed by visualisation
of the precipitation simulations. The convergence of stable nuclei by Ostwald ripening in comparison with
transformations in which nuclei remain rather fixed to their original nucleation sites is depicted in Fig. 4. It
can be seen in Fig. 4c that a cluster mobility may yield an additional overlap of the precipitates. Once particles
reach the critical size, the extended volume will be calculated by their growth independent of the presence of
surrounding neighbouring particles. If particles approach each other to form a larger particle, severe overlap
of the now touching particles results in an enormous increase in the extended transformed volume, while there
will only be a slight increase in the real transformed volume. Considerable Ostwald ripening as observed in
the performed KMC simulations leads to deviations from the classical JMAK kinetics and can be accounted
for by a severe impingement correction by introducing an impingement parameter ε into Eq. (9):
dV t = (1 − f )ε dV e ,
(18)
Fig. 4 Starting from configuration (a), precipitates may grow while their center stays at the original nucleation positions. The
overlap is accounted for by assuming hard impingement (b; see Sect. 5). Additional cluster mobility may lead to more severe
impingement (c), which is accounted for by the correction factor ε in Eq. (18)
Activation energies for nucleation and growth from KMC simulations
a
b
Fig. 5 Taking severe impingement into account, the correction parameter ε affects the slopes in Fig. 3b and thus the growth
exponents n. The resulting temperature averaged activation energies are shown in (a) with the corresponding error bars where
n cut-off = 5 at 250 ◦ C is held fixed. The error bars equal one standard deviation. Calculating the growth exponents by fitting
regression lines (similar to Fig. 3c), they lie in a different range for each ε having a maximum n = n max and a minimum n = n min .
These two quantities are shown in (b) as limits for all obtained growth exponents (green circles between n min and n max ) for the
corresponding ε (colour figure online)
where ε > 1. This approach for the impingement correction stems from the proposed correction of anisotropically growing particles [26,27], which takes a more severe impingement into account as compared to Eq. (9).
Errors which may occur due to the selection of the wrong model, that is, hard or severe impingement, have
also been discussed in [28].
The difference between the real transformed volume according to Eqs. (9) and (18) gets more pronounced
with increasing ε. From Eq. (18), the transformed fraction can be derived as
1
V e 1−ε
f = 1 − 1 + (ε − 1)
,
V
(19)
with the initial condition that f (t = 0) = 0, which is satisfied by choosing n cut-off as described in Sect. 4.
Similar to the procedure in the previous section, the activation energies Q eff , Q N and Q G can be obtained.
However, ε is an additional parameter accounting for severe impingement. By varying ε from ε = 1 to ε = 4,
different activation energies for nucleation and growth can be obtained (see Fig. 5a). The corresponding maximum and minimum values of the growth exponents (similar to Fig. 3a) are shown in Fig. 5b for the case of
2 wt.% Cu. The dashed lines correspond to the limits for continuous nucleation (n = 1.5) and pre-existing
nuclei (n = 2.5), respectively. For small ε, Q G is larger than Q N . By increasing ε, Q N exceeds Q G . Nevertheless, neither Q N nor Q G deviate immoderately and they yield values of Q N = 163.6–267.3 kJ/mol and
Q G = 206.9–227.1 kJ/mol, which are still comparable to the experimentally determined activation energies
for diffusion of Cu in Fe. The main reason here for varying ε is to obtain growth exponents which lie in
the expected range (1.5–2.5). While the minimum value n min only very slowly tends towards n = 1.5, the
maximum value n max increases rapidly. A possible trade-off can be found for the condition of n min being as
big as possible, while n max ≤ 2.5. This is the case for ε = 3.15 yielding Q N = 261.4 ± 5.1 kJ/mol and
Q G = 215.7 ± 7.7 kJ/mol. Furthermore, plotting the transformed fraction f (see Eq. (19)) similar to Fig. 3b
(JMAK kinetics) yields almost straight lines for ε = 3.15 (see Fig. 6a). From then on, increasing ε further
results in curved lines again (see Fig. 6b). For 1 wt.% Cu, similar considerations yield ε = 2.25 and thus
Q N = 227.6 ± 15.6 kJ/mol and Q G = 214.4 ± 6.3 kJ/mol. However, the trade-off applied here has no direct
physical justification. Therefore, the results have to be considered with some care.
6 Summary and concluding remarks
In this study, transformed fraction data obtained by KMC thermal ageing simulations of 1 wt.% Cu and 2 wt.%
Cu in α-Fe have been analysed in order to
(i) reveal the dependence of the transformed fraction on the critical cluster size, i.e. on the minimum number
of Cu atoms which form a cluster that contributes to the transformed fraction.
(ii) obtain thermodynamical quantities which can serve as input parameters for larger length scale simulation
methods.
D. Molnar et al.
a
b
Fig. 6 The 2 wt.% Cu results are shown for the case of severe impingement with impingement factors ε = 3.15 (a) and ε = 4.00
(b). With increasing ε, a concave behaviour (see Fig. 3c, which corresponds to the ε = 1 case) changes via almost straight lines
(ε ≈ 3.15) to a convex behaviour (ε > 3.15)
Table 3 Summary of the results
Concentration [wt.% Cu]
Model
Q [kJ/mol]
ε
1
1
1
1
1
2
2
2
2
2
–
HI
HI
SI
SI
–
HI
HI
SI
SI
Q eff = 206.8 − 227.2
Q N = 188.3 ± 2.0
Q G = 205 ± 4.51
Q N = 227.6 ± 15.6
Q G = 214.4 ± 6.3
Q eff = 214.3 − 229.9
Q N = 206.7 ± 0.8
Q G = 214.0 ± 1.6
Q N = 261.4 ± 5.1
Q G = 215.7 ± 7.7
–
–
–
2.25
2.25
–
–
–
3.15
3.15
The effective activation energy Q eff is model independent. The activation energies for nucleation Q N and growth Q G depend on
whether hard impingement (HI) or severe impingement (SI) is assumed. The values can be compared to the activation energy of
diffusion Q D
By increasing the critical cluster size n cut-off starting with n cut-off = 2 (the value which is assumed to be
typically applied in KMC simulations), considerable deviations from S-shaped JMAK behaviour disappear.
Hence, before fitting a kinetic model to transformed fraction data, the number of atoms to form a cluster has
to be chosen appropriately as it will considerably change the results when calculating the growth exponents.
Assuming JMAK-like behaviour with hard and severe impingement (see Sects. 5.1 and 5.2, respectively),
effective activation energies and activation energies for nucleation and growth can be derived.
The results are summarised in Table 3 and are in good agreement with the activation energy of diffusion
obtained elsewhere [21]. However, the results have to be taken with some care, as they sensitively depend on
the applied parameters, that is, on the critical cluster size n cut-off and the JMAK correction factor ε as well
as on the impingement model. Additional research will be necessary in order to obtain a physically based
explanation for the correct choice of n cut-off .
In this study, no complete kinetic model has been assumed. If so, for example, assuming Avrami or mixed
nucleation [25], the transformed fraction curves for all temperatures would have to be fitted with one set of
parameters simultaneously. Then, the best fit would suggest the nucleation and growth mechanism from a
thermodynamic point of view. This will be the task of future research.
Nevertheless, we have shown how sensitive transformed fraction curves depend on the critical cluster
size and how they reshape to JMAK-like S-shaped curves when n cut-off is increased. Furthermore, we have
adapted an experiment analysis approach [15] to analyse computer simulation-based KMC data. As a result,
thermodynamical quantities have been obtained, which may serve as input parameters for larger length scale
simulations reducing the number of parameters that have to be obtained by experimental studies.
Acknowledgments The authors D. Molnar, C. Niedermeier and S. Schmauder would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the
University of Stuttgart. The authors P. Binkele, A. Mora and S. Schmauder would like to thank the German Research Foundation
(DFG) for financial support of the project SCHM746/101-1.
Activation energies for nucleation and growth from KMC simulations
References
1. Willer, D., Zies, G., Kuppler, D., Föhl, J., Katerbau, K.-H.: Service-Induced Changes of the Properties of Copper-Containing
Ferritic Pressure-Vessel and Piping Steels. Final Report, MPA Stuttgart (2001)
2. Pan, F., Ruoff, H., Willer, D., Katerbau, K.-H.: Investigation on the Microstructure of a Pressure Vessel/Pipe Steel 15
NiCuMoNb 5 (WB36). MPA Report, Stuttgart (1996)
3. Schick, M., Wiedemann, J., Willer, D.: Untersuchungen zur sicherheitstechnischen Bewertung von geschweissten Komponenten aus Werkstoff 15 NiCuMoNb 5 (WB36) im Hinblick auf die Zähigkeitsabnahme unter Betriebsbeanspruchung,
Technischer Bericht, MPA Stuttgart (1997)
4. Soisson, F., Barbu, A., Martin, G.: Monte Carlo simulations of copper precipitation in dilute iron-copper alloys during
thermal ageing and under electron irradiation. Acta Mater. 44, 3789–3800 (1996)
5. Binkele, P.: Atomistische Modellierung und Computersimulation der Ostwald-Reifung von Ausscheidungen beim Einsatz
von kupferhaltigen Stählen. PhD thesis, University of Stuttgart (2006)
6. Johnson, W.A., Mehl, R.F.: Reaction kinetics in processes of nucleation and growth. Trans. Am. Inst. Min. Metal. Pet.
Eng. 135, 416–442 (1939)
7. Avrami, M.: Kinetics of phase change I—general theory. J. Chem. Phys. 7, 1103–1112 (1939)
8. Kolmogorov, A.E.: Ivz. Akad. Nauk SSR Ser. Fiz. Mat. Nauk 1, 355 (1937)
9. Mittemeijer, E.J.: Fundamentals of Materials Science. p. 442 Springer, Heidelberg (2011)
10. Soisson, F., Fu, C.-C.: Cu-precipitation kinetics in α-Fe from atomistic simulations: vacancy-trapping effects and Cu-cluster
mobility. Phys. Rev. B 76, 214102 (2007)
11. Claudio, D., Gonzalez-Hernandez, J., Licea, O., Laine, B., Prokhorov, E., Trapaga, G.: An analytical model to represent
crystallization kinetics in materials with metastable phase formation. J. Non-Cryst. Solids 352, 51–55 (2007)
12. Robson, J.D., Bhadeshia, H.K.D.H.: Modelling precipitation sequences in power plant steels Part 1—kinetic theory. Mater.
Sci. Technol. 13, 631 (1997)
13. Pizzini, S., Roberts, K.J., Phythian, W.J., English, C.A., Greaves, G.N.: A fluorescence EXAFS study of the structure of
copper-rich precipitates in Fe-Cu and Fe-Cu-Ni alloys. Philos. Mag. Lett. 61, 223–229 (1990)
14. Othen, P.J., Jenkins, M.L., Smith, G.D.W.: High-resolution electron-microscopy studies of the structure of Cu precipitates
in α-Fe. Philos. Mag. A 70, 1–24 (1994)
15. Bauer, R., Rheingans, B.F., Mittemeijer, E.J.: The kinetics of the precipitation of Co from supersaturated Cu-Co alloy. Metall.
Mater. Trans. A. doi:10.1007/s11661-010-0594-7 (2010)
16. Vincent, E., Becquart, C.S., Domain, C.: Atomic kinetic Monte Carlo model based on ab initio data: Simulation of microstructural evolution under irradiation of dilute Fe-CuNiMnSi alloys. Nucl. Instrum. Methods Phys. Res. B 255, 78–84 (2007)
17. Vincent, E., Becquart, C.S., Domain, C.: Solute interaction with point defects in α-Fe during thermal ageing: a combined
ab initio and atomic kinetic Monte Carlo approach. J. Nucl. Mater. 351, 88–99 (2006)
18. Molnar, D., Binkele, P., Hocker, S., Schmauder, S.: Atomistic multiscale simulations on the anisotropic tensile behaviour
of copper-alloyed alpha-iron at different states of thermal ageing. Philos. Mag. 92, 586–607 (2012)
19. Soisson, F., Fu, C.-C.: Energetic landscapes and diffusion properties in FeCu alloys. Solid State Phenom. 129, 31–39 (2007)
20. Kittel, C.: Introduction to Solid State Physics. Wiley, New York (1976)
21. Mehrer, H.: Landolt-Börnstein, numerical data and functional relationships in science and technology, new series
III. Springer, Heidelberg (1991)
22. Bergmann, L., Schaefer, C., Raith, W.: Bergmann, Schaefer, Lehrbuch der experimentalphysik, band 6, Festkörper. Walter
de Gruyter, Berlin (1992)
23. Schmauder, S., Binkele, P.: Atomistic computer simulation of the formation of Cu-precipitates in steels. Comput. Mater.
Sci. 24, 42–53 (2002)
24. Porter, D.A., Easterling, K.E.: Phase Transformations in Metals and Alloys, 2nd edn. Taylor and Francis, London (1992)
25. Liu, F., Sommer, F., Bos, C., Mittemeijer, E.J.: Analysis of solid state phase transformation kinetics: models and recipes. Int.
Mater. Rev. 52(4), 193–212 (2007)
26. Starink, M.J.: On the meaning of the impingement parameter in kinetic equations for nucleation and growth reactions.
J. Mater. Sci. 36, 4433 (2001)
27. Kooi, B.J.: Monte Carlo simulations of phase transformations caused by nucleation and subsequent anisotropic growth:
extension of the Johnson-Mehl-Avrami-Kolmogorov theory. Phys. Rev. B 70(22), 224108 (2004)
28. Wang, J., et al.: On discussion of the applicability of local Avrami exponent: errors and solutions. Mater. Lett. 63, 1153–1155
(2009)