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Mean-field model for the growth and coarsening of stoichiometric precipitates at grain
boundaries
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2010 Modelling Simul. Mater. Sci. Eng. 18 015011
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IOP PUBLISHING
MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011 (19pp)
doi:10.1088/0965-0393/18/1/015011
Mean-field model for the growth and coarsening of
stoichiometric precipitates at grain boundaries
E Kozeschnik1,2,3,7 , J Svoboda1,4 , R Radis3,5 and F D Fischer1,6
1
Materials Center Leoben Forschungsgesellschaft mbH, 8700 Leoben, Austria
Institute of Materials Science and Technology, Vienna University of Technology, 1040 Vienna,
Austria
3 Christian Doppler Laboratory for Early Stages of Precipitation, Institute of Materials Science
and Technology, Vienna University of Technology, 1040 Vienna, Austria
4 Institute of Physics of Materials, Academy of Sciences of the Czech Republic, 616 62 Brno,
Czech Republic
5 Institute for Materials Science and Welding, Graz University of Technology, 8010 Graz, Austria
6 Institute of Mechanics, University of Leoben, 8700 Leoben, Austria
2
E-mail: [email protected], [email protected], [email protected] and
[email protected]
Received 13 July 2009, in final form 4 November 2009
Published 15 December 2009
Online at stacks.iop.org/MSMSE/18/015011
Abstract
In this paper, a model for growth and coarsening of precipitates at grain
boundaries is developed. The concept takes into account that the evolution
of grain boundary precipitates involves fast short-circuit diffusion along grain
boundaries as well as slow bulk diffusion of atoms from the grain interior
to the grain boundaries. The mathematical formalism is based on a meanfield approximation, utilizing the thermodynamic extremal principle. The
model is applied to the precipitation of aluminum nitrides in microalloyed
steel in austenite, where precipitation occurs predominately at the austenite
grain boundaries. It is shown that the kinetics of precipitation predicted by
the proposed model differs significantly from that calculated for randomly
distributed precipitates with spherical diffusion fields. Good agreement of
the numerical solution is found with experimental observations as well as
theoretical treatment of precipitate coarsening.
List of symbols
cMi
c
cMi
eq
cMi
7
mean concentration of component i in the matrix, i = 1, . . . , n
concentration of component i in the center of the grain, i = 1, . . . , n
equilibrium concentration of component i in the matrix, i = 1, . . . , n
Author to whom any correspondence should be addressed.
0965-0393/10/015011+19$30.00
© 2010 IOP Publishing Ltd
Printed in the UK
1
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
grain
cMi
cPi
Dgi
Dbi
δ
G
gP
γ
jki
Jki
n
N
ρk
R
Rg T
E Kozeschnik et al
initial concentration of component i in the matrix, i = 1, . . . , n
mean concentration of component i in precipitates, i = 1, . . . , n
tracer diffusion coefficient of component i in grain boundary, i = 1, . . . , n
tracer diffusion coefficient of component i in bulk, i = 1, . . . , n
thickness of the grain boundary
Gibbs energy
molar Gibbs energy of precipitate
specific precipitate/matrix interfacial energy
flux of component i in the grain boundary around the precipitate k, i = 1, . . . , n
flux of component i perpendicular to the grain boundary and corresponding to
the precipitate k, i = 1, . . . , n
radius of the precipitate diffusion zone at the grain boundary
number of components in the system
number of precipitates at the grain boundary of one grain
radius of precipitate—state parameters describing the system, k = 1, . . . , N
radius of grain
Rg gas constant, T temperature
molar volume
1. Introduction
In modeling the diffusion-controlled growth of randomly distributed precipitates, one
commonly assumes spherical diffusion fields around the precipitates. The precipitation
problem for individual particles can then be treated with classical solutions of the moving
boundary problem, which involves the long-range diffusion of solute atoms as well as the
simultaneous motion of the precipitate/matrix phase boundary. Corresponding analytical
models [1–5], as well as numerical solutions for general multi-component systems [6], are
well established and generally utilized.
If precipitation occurs at grain boundaries, the assumptions of a random distribution
of precipitates and spherical diffusion fields are no longer applicable. Grain boundaries
represent favorable nucleation sites as well as very efficient short-circuit diffusion
paths with diffusivities several orders of magnitude larger than in the bulk [7].
Figure 1 shows schematics of spherical diffusion fields in a random distribution of
precipitates and the diffusion geometry for an arrangement of precipitates along grain
boundaries.
A few models are available in the literature for simultaneous precipitation (nucleation
and growth) and short-circuit diffusion along grain boundaries. Early approaches have been
developed for ferrite nucleation at austenite grain boundaries in the Fe–C and Fe–C–X (see the
review paper by Lange et al [8]) and allotriomorph growth in the Al–Cu systems [9–11]. Of
particular interest is the work of Aaron and Aaronson [9], who solved the problem of precipitate
growth governed by diffusion through the bulk and along grain boundaries by the collectorplate mechanism. These authors solved the problem for the planar case, which represents a
reasonable approximation for the early stages of the growth process, and for a binary system.
In this case, an analytical solution for the lengthening and thickening rates of grain boundary
precipitates could be obtained. In this work, we, however, develop an approximate multicomponent solution to the problem for spherical geometry, which represents a reasonable
approximation also for the late stages, where the grain interior becomes depleted from solute
2
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
Figure 1. Schematic precipitate distributions and diffusion fields (shaded areas) for random
precipitation (left) and heterogeneous precipitation at grain boundaries (right) in 2D.
atoms and the decreasing volume of solute reservoir toward the center of the grain slows down
the final stages of the precipitation process.
The coarsening characteristics of grain boundary precipitates have been investigated
several times, both experimentally [12–15] and theoretically [16–20]. The investigations show
that the coarsening of grain boundary precipitates occurs with time exponents between 1/3 and
1/5, depending on temperature, precipitate volume fraction, grain size, etc. These results are
discussed and compared with our numerical results in the application section of this paper.
In [21], multi-component effects are incorporated, however, only in the nucleation stage.
The diffusion paths toward the grain interior, together with those along the grain boundary, are
not explicitly accounted for. The short-circuit diffusion characteristics are met by employing
a pre-factor to the activation energy for diffusion.
Alternatively, models have also been developed based on the Monte Carlo technique
[22–24], providing basic insight into the precipitation process. These techniques, however,
suffer the disadvantage of being computationally expensive.
Real progress in modeling a global diffusive process interacting with the formation of
precipitates in distinct zones as interfaces or grain boundaries has been presented in [25].
The authors combine the solution of the classical Fick equation with a pseudo-front tracking
method. This direct solution technique of the leading equations, meeting local and global mass
balances as constraints, is used in an incremental way to look for the growth of intergranular
precipitates at the expense of precipitate-free zones.
The goal of our paper is to develop a model for the growth and coarsening of intergranular
precipitates at the expense of a supersaturated matrix in grains leading to depleted zones near the
grain boundaries. We follow a variational approach describing the concentrations and fluxes of
the individual components by rather simple shape functions, multiplied by distinct parameters
as local concentrations or geometrical quantities as the precipitate radii. The advantage of this
concept lies in the derivation of the explicit evolution equations for the state parameters, as
e.g. the radii of the precipitates, for general multi-component, multi-particle and multi-phase
systems. As variational principle we use the thermodynamic extremum principle as outlined
in detail in section 2. It is assumed that the precipitates have a stoichiometric composition.
The present model is implemented in the software MatCalc [26] (version 5.30).
3
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
Figure 2. Schematic representation of the diffusion geometry for a single precipitate located on
the grain boundary: I. Diffusional zone in the bulk, II. Diffusional zone in the grain boundary.
2. The model
2.1. General aspects
The grain, where precipitation is assumed to occur on its boundary, is approximated by a sphere
with a representative grain radius R as depicted in figure 2. Consider an ensemble of spherical
precipitates, which are nearly equidistantly arranged along the grain boundary. We assume that
two parallel processes occur in the system: (i) growth of precipitates on account of components
dissolved in the matrix and (ii) coarsening of precipitates by means of redistribution of the
fixed total volume of the grain boundary precipitates amongst precipitates of different sizes. To
each of these parallel processes, the driving force due to the decrease in the total Gibbs energy
of the system and the corresponding dissipation function can be found. This is demonstrated
in the following sections. The most important quantities and their symbols are listed in the
‘List of symbols’.
2.2. Treatment of precipitate growth
2.2.1. Study of the diffusive processes. First, we introduce a diffusion zone being a spherical
cap over an arc length 2 (see figure 2). Since we assume that the grain radius R is sufficiently
larger than , we replace the cap by a circular disk and relate to R with the number N of
precipitates as
√
and
= 2R/ N .
(1)
π2 N = 4πR 2
We further assume spherical intergranular precipitates with the radius ρk , k = 1 . . . N, with
no diffusive flux inside the precipitates (stoichiometric precipitates). For the flux jki (ρk ) of a
component i, i = 1 . . . n, outside the precipitate and inside the diffusion zone, we assume that
the matter moving out of the precipitate with the surface area 4πρk2 flows into the diffusion
zone through a cylindrical mantle with the radius 2πρk δ, with δ being the thickness of the grain
boundary. We assume that atoms can be transported to/from the contact surface between the
4
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
precipitate and the grain boundary via sufficiently fast interface diffusion. Since the precipitate
radius varies with time as ρ̇k , the mass balance at the interface fulfills the jump condition (see,
e.g., [27])
eq
2πρk δ · jki (ρk ) = ρ̇k 4πρk2 (cMi − cPi ),
(2)
eq
cMi
with
being the equilibrium concentration of element i in the matrix and cPi being the
concentration of element i in the precipitate. Inside the diffusion layer, for the radial diffusive
flux jki (r) and with the polar coordinate r, and ρk < r < , we assume
eq
jki (r) = −2ρ̇k (cPi − cMi )
2 − r 2 ρk2
.
2 − ρk2 rδ
(3)
The flux jki (r) is compatible with jki (ρk ) at r = ρk and shows a constant divergence as
eq
div(jki (r)) = 4ρ̇k (cPi − cMi )
2
ρk2
1
.
2 δ
− ρk
(4)
One could now assume that the negative rate of concentration change of component i in the
diffusion zone must be identical to the divergence (4). However, in the case at hand, the
divergence is compensated by the influx from the bulk into the grain boundary acting as a
constant source density. As a consequence, the concentration of component i in the grain
eq
boundary is kept constant in time and space with the value cMi . No segregation is assumed to
occur within the grain boundary.
Following the treatment for equidistant precipitates and radial diffusion fields as presented
in [28], the total dissipation due to diffusion in the grain boundary in the diffusion zones
corresponding to the growth of all precipitates at the surface of the grain is given by
N
n 1 Rg T 2
Qg =
j 2π rδ dr.
(5)
eq
2 k=1 i=1 ρk cMi
Dgi ki
The factor 1/2 accounts for the fact that each precipitate corresponds to two grains. Dgi is
the tracer diffusion coefficient of component i in the grain boundary, T is the temperature and
Rg is the gas constant. With the assumption that the concentration of elements in the grain
eq
boundary is given by their equilibrium value cMi , the integral in equation (5) can be solved,
and the total dissipation in the grain boundary reads
Qg ≈ 4πRg T
N
k=1
αk ρk4 ρ̇k2
n
eq
(cPi − cMi )2
eq
cMi δDgi
i=1
with the coefficient αk as
ρ 2 ρ 2 2
3 ρk 2
k
k
αk () = ln
− +
1−
1−
.
ρk
4
(6)
(7)
Secondly, we deal with the diffusion process inside the grain. At the time t = 0 we assume
for the total volume of precipitates V = 0 and the concentration of component i in the whole
grain
3
grain is cMi . For t > 0 the volume of the precipitates is given by V = 4π /3 · N
k=1 ρk .
We assume that the concentration ci of component i in the grain varies linearly in the
interval R − di u R, with u being the distance from the center of the grain and di the
width of the interval (see figure 3) as
ci =
u + di − R eq R − u grain
cMi +
cMi .
di
di
(8)
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
Figure 3. Geometry and concentration profile for bulk diffusion of element i from the grain center
toward the grain boundary.
The value of di is given from mass conservation as
R
1
eq
grain
V (cPi − cMi ) = 4π
(cMi − ci )u2 du.
2
R−di
(9)
Again, the factor 1/2 in equation (9) appears due to the fact that each precipitate corresponds
to two grains. Performing the integral in equation (9) leads to a cubic equation for di yielding
the solution
2R 2 4R
βi
di =
−
.
(10)
+
3
3βi
3
The coefficients βi are detailed in appendix A.
The flux Jki of component i in the grain, perpendicular to the grain boundary and
corresponding to the precipitate k, is supposed to vary linearly from its zero value at the
position u = R − di and to reach its extremal value Jki∗ at the position u = R. For u R − di ,
the flux Jki is supposed to be zero. The value of Jki∗ can be determined from the condition that
no matter is collected/deposited at the grain boundary during the precipitate growth process.
Thus, we obtain
1
eq
π2 Jki∗ = − 4πρk2 ρ̇k (cPi − cMi ).
(11)
2
Again, the factor 1/2 in equation (11) is introduced to account for the fact that each precipitate
corresponds to two grains. Then, the flux Jki in the region R − di u R is approximated
by the linear relation
eq
2ρk2 ρ̇k (cPi − cMi )(u − R + di )
,
(12)
2 di
which fulfills both the global and, approximately, also the local mass balance, see appendix B.
The total dissipation due to diffusion in the grain is then given by
N n R
R g T 2 π 2 u2
Qb =
J
du,
(13)
c D ki R 2
k=1 i=1 R−di i bi
Jki = −
6
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
yielding with equations (8) and (12)
Qb =
n
N eq
πRg T ρk4 ρ̇k2 di (cPi − cMi )2
eq
k=1 i=1
grain
3Dbi 2 R 2 (cMi − cMi )5
4
i =
i ,
grain
eq
il (cMi )4−l (cMi )l .
(14)
l=0
The coefficients il are detailed in appendix C.
The total dissipation in the system Qg due to growth of the precipitates is given with
equations (6) and (14) as Qg = Qg + Qb .
2.2.2. Derivation of the evolution equations. To be able to calculate rigorously the total
Gibbs energy G and the corresponding driving forces in the system, the profiles of the chemical
potentials in the matrix must be determined and the respective integrals must be evaluated. In a
first approximation, the total Gibbs energy G is expressed in mean values of the concentrations
cMi , which are evaluated from the global mass balance as
N
4π 3 grain
2π 3
ρ (cPi − cMi ),
R (cMi − cMi ) =
3
3 k=1 k
yielding
cMi =
grain
2R 3 cMi
−
N
ρk3 cPi
k=1
2R −
3
N
(15)
ρk3
grain
grain
≈ cMi + (cMi − cPi ) ·
k=1
N
ρk3
. (16)
2R 3
k=1
From the conservation law for each component in a closed system we obtain with mi as the
fixed number of moles of component i in the system as
N
N
4π
1 3
2π
3
R −
ρk cMi = mi −
ρk3 .
(17)
cPi
3
2 k=1
3
k=1
The total Gibbs energy G of the system is given by
N
N
2πgP 3
4π
G=
ρk + 2πγs
ρk2 +
3P k=1
3
k=1
n
N
1 3 ρ
cMi µi ,
R −
2 k=1 k i=1
3
(18)
with gP being the molar Gibbs energy, P the molar volume of the precipitate, γs the
specific matrix/precipitate interfacial energy and µi the chemical potentials expressed in cMi ,
i = 1, . . . n. The Gibbs energy contribution of the grain boundary as well as any elastic strain
energy contributions from volumetric misfit are neglected. Substituting equation (17) into
equation (18) yields
N
N
N
n
2πgP 3
2π
G=
mi −
ρ + 2πγs
ρk2 +
ρk3 µi .
(19)
cPi
3P k=1 k
3
k=1
i=1
k=1
The evolution equations for the growth of precipitates can then be obtained from the
thermodynamic extremal principle, according to [29, 30], as
1 ∂Qg
∂G
=−
,
2 ∂ ρ̇k
∂ρk
k = 1, . . . N.
(20)
After inserting for Qg , equations (6) and (14) and for G, equation (19), and using the Gibbs–
Duhem relation in the calculation of the partial derivatives ∂G/∂ρk , equations (20) can be
7
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
resolved with respect to ρ̇k for di R as
n
gP
2γs
−
i=1 cPi µi −
P
ρk
g
,
ρ̇k =
n
eq 2
eq
n (cPi − cMi )
di i (cPi − cMi )2
1
2
2Rg T ρk αk i=1
+
eq
eq
grain
122 R 2 i=1 Dbi (cMi
cMi δDgi
− cMi )5
k = 1, . . . N.
(21)
g
The superscript g in ρ̇k denotes the contribution from growth to the evolution of the radius
grain
c
of k. When di has reached R, di must be set to R and cMi must be replaced by cMi
calculated
in appendix C.
2.3. Treatment of precipitate coarsening
During coarsening, the precipitates communicate via diffusion along the grain boundaries.
This process can be considered as independent, occurring simultaneously with the growth
process described in the previous section. One may accept the assumptions on the geometry
of diffusive fluxes being the same as those for the growth of precipitates. In this section
a superscript c refers to coarsening. It is, however, reasonable to choose the radius of the
diffusive zone in the grain boundary to be 2 instead of to ensure sufficient overlapping of
diffusional zones. With βk = αk (2) (equation (7)) the dissipation in the grain boundary Qc
can be calculated from equation (6) as
n
N
eq
(cPi − cMi )2
βk ρk4 ρ̇k2
,
(22)
Qc ≈ 4πRg T
eq
cMi δDgi
k=1
i=1
N
N
3
2
with the constraint 4π
k=1 ρk = const or
k=1 4πρk ρ̇k = 0.
3
Only the second term of the Gibbs energy G from equation (19) contributes to the driving
force for coarsening (see [31]). The governing equation is thus given, with λ being a Lagrange
multiplier, as
1 ∂Qc
∂G
+ λ4πρk2 = −
,
k = 1 . . . N.
(23)
2 ∂ ρ̇k
∂ρk
Partial differentiation of Qc and G yields with equation (23)
N
−1
−γS − λρk
l=1 (ρl · βl )
ρ̇kc =
and
λ
=
−γ
.
(24)
s N
eq 2
n (cPi − cMi )
(βl )−1
3
l=1
Rg T ρk βk i=1
eq
cMi δDgi
The Lagrange multiplier λ follows after insertion of ρ̇kc in the constraint.
Equation (24) can then be rewritten as
N
eq 2
n −1
cPi − cMi
1
l=1 (ρl · βl )
c
2
ρ̇k = γs − + N
Rg T ρ k β k
.
(25)
eq
−1
ρk
cMi δDgi
l=1 (βl )
i=1
The superscript c in ρ̇kc denotes the contribution from coarsening to the evolution of the radius
of k. The critical radius ρcrit is given according to ρ̇kc = 0 as
N
N
−1
(βl )
(ρl · βl )−1 .
ρcrit =
l=1
l=1
As both processes, growth and coarsening, are considered simultaneous and independent, the
g
total rate of change in the size of each precipitate is given by simple addition as ρ̇k = ρ̇k + ρ̇kc .
The actual values of all parameters must be recalculated before each integration step in
the time integration procedure.
8
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
3. Application to AlN precipitation in steel
In this section, the model characteristics are investigated on the example of aluminum nitride
precipitation in microalloyed steel. The iron-rich corner of the Fe–Al–N system in the fcc
+ AlN phase field of the Fe–Al–N phase diagram represents a classical example, where
nucleation of precipitates occurs predominantly at grain boundaries and is almost entirely
suppressed elsewhere [32]. This model system has, therefore, been selected for investigation
of the features and characteristics of the present model. The analysis is carried out in the form
of a parameter study as well as through comparison with selected experimental information
available from the literature.
3.1. Thermodynamic and kinetic data
The thermodynamic properties of the Fe–Al–N system have been assessed by Hillert and
Jonsson [33]. These authors observe good agreement between computed results and
experimental data over a wide range of compositions of the ternary phase diagram. The
thermodynamic parameters obtained in this study represent the basis for the following
computations.
In a series of preliminary simulations, however, unsatisfactory predictions of the solution
temperatures of AlN measured in microalloyed steel have been observed by the present authors.
(This possible discrepancy has already been admitted by Hillert and Jonsson [33] in their
discussion.) A reassessment of the available experimental data by the authors with focus on
the solubilities of AlN in the Fe-rich corner of the Fe–Al–N phase diagram [34] has delivered
a modified expression for the Gibbs energy of the AlN phase, which reads
SER
GAlN − GSER
= −262982 + 63T
Al − GN
(26)
in units of J mol−1 , with T being temperature and the superscript SER denoting the standard
element reference state [35]. The computations presented subsequently make use of this
revised expression.
The diffusional mobilities of Fe and N needed in the simulations are taken from the
works of Jönsson [36, 37]. The diffusion coefficient of Al in Fe is taken to be five times
the self-diffusion coefficient of Fe [38]. The simulations are carried out with the software
MatCalc [26, 39]. In the simulations, the entire precipitation process is divided into small
isothermal time increments, evaluating the evolution equations for nucleation and growth.
The entire precipitation process is obtained by numerical integration in an iterative procedure.
3.2. Growth of equi-sized precipitates
First, the growth characteristics of individual precipitates are explored utilizing an ensemble
of identical particles located at the grain boundaries of unit volume of polycrystalline material
with given grain radius. In this setup, the kinetics of a single particle is representative for the
kinetics of the whole system.
The first set of simulations is initialized with a constant number of precipitates (N0 =
1018 m−3 ) with supercritical nucleation radius according to classical nucleation theory [40].
The corresponding model for multi-component systems is summarized in a previous work [41].
The precipitates are assumed to be homogeneously distributed over the entire grain boundary
area. The simulations are carried out for different austenite grain radii between R = 1 µm
and R = 250 µm at a temperature T = 1000 ◦ C. The composition of the system is taken to be
0.05 wt% Al, 0.005 wt% N, balance Fe, which are typical values for microalloyed steel. It is
9
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
Figure 4. Evolution of phase fraction and radius of a constant number of identical AlN precipitates
for different grain radii. Al = 0.05 wt%, N = 0.005 wt%, T = 1000 ◦ C, Dgi = 10 000Dbi .
further assumed that the diffusion of elements in the grain boundaries is 104 times faster than
diffusion in the matrix (this parameter will be further investigated later).
Figure 4 shows the calculated growth kinetics of a constant number of equi-sized AlN
precipitates (N0 = 1018 m−3 ) using the new grain boundary diffusion geometry (GBDG)
formalism at hand. The left plot displays the evolution of the phase fraction of AlN precipitates
versus time, whereas the right plot shows the evolution of the precipitate radius. In accordance
with expectation, the growth kinetics of grain boundary precipitates is faster the smaller R is.
The leftmost curve in the phase fraction plot is computed for a grain radius of R = 1 µm. The
corresponding particle density at the grain boundary is 8 precipitates per grain with a mean
diffusion field radius of approximately 700 nm. The precipitate radius evolution shows a
rapid increase in the early stages of the growth process and an almost constant growth rate
until soft impingement hinders further growth. If R is increased from R = 1 µm to 9 µm,
while holding the total number of precipitates constant, the precipitate density at the grain
boundary increases from 8 precipitates per grain to a value of 5200. This increase leads to a
significant decrease in the mean diffusion field width from 700 to 250 nm and, together with
the increased diffusion distance from the grain center R, to a slower growth rate in the final
stages of the growth process. This effect is even more pronounced, if R is further increased to
R = 250 µm, where the decreasing growth rate is already visible for medium size precipitates.
For the largest R, a precipitate density of 108 precipitates per grain is observed with a mean
diffusion field radius of ∼ 50 nm. In the phase fraction evolution shown in the left plot of
figure 4, the different growth kinetics with increasing grain radius and precipitate density is
clearly reflected in a change in the slope of the phase fraction versus time curves. This becomes
most obvious in comparing also with the classical parabolic growth behavior observed in the
random-distribution spherical diffusion field geometry (RSDG), as described in a previous
paper [28]. This curve is shown as a dashed line for comparison.
The second set of simulations is carried out for variable grain radius and constant
precipitate density of 10 000 precipitates per grain. This constraint is maintained by proper
adaptation of the total number of precipitates with variable grain radius. The results are
summarized in figure 5. The left image displays the evolution of the phase fraction versus
time, showing that the time taken to reach full precipitation is strongly retarded in larger
grains. Since the diffusion field radius is approximately constant in this simulation setup
10
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
Figure 5. Evolution of phase fraction and radius computed in GBDG for constant precipitate density
of 10 000 precipitates per grain. Al = 0.05 wt%, N = 0.005 wt%, T = 1000 ◦ C, Dgi = 10 000Dbi .
(one precipitate always shares the same amount of grain boundary area), this effect is mainly
due to the increased diffusion distances from the grain center to the grain boundary as well as
the fact that the precipitates grow to considerably larger size (mass conservation). For the six
values R = 1, 3, 9, 27, 100 and 250 µm used in figure 5, the corresponding number densities
are 2 × 1021 , 7 × 1019 , 3 × 1018 , 1 × 1017 , 2 × 1015 and 1.5 × 1014 m−3 . These results reflect
the well-known fact that grain boundary precipitation (viewed in terms of phase fraction) can
be effectively controlled by the parent phase grain radius.
3.3. Evolution of non-equi-sized precipitates
All simulations start with a pre-defined initial size distribution of precipitates (Gaussian
distribution) and a total precipitate phase fraction of 1/1000 of the equilibrium value. The
simulations are carried out for a composition of 0.1 wt% Al, 0.01 wt% N, balance Fe and a
temperature T = 1000 ◦ C. Figure 6 summarizes the results of the simulations for various grain
radii between R = 1 and R = 1000 µm. The major difference given by these simulation
conditions is the average number of precipitates per grain, which varies from approximately
105 precipitates per grain for R = 1 µm to 1015 precipitates per grain for R = 1000 µm.
Investigating the evolution of the phase fraction for this simulation setup in figure 6, similar
characteristics are observed as in the simulations shown in figure 5. The evolution of the
radius, however, is more complex since it involves the interaction of precipitates with different
sizes. The curves for the smallest grain radius of R = 1 µm as well as the curve for the RSDG
treatment exhibit a clear distinction of the growth stage from the coarsening regime. The two
regions are separated by a plateau, indicating the transient region after completion of growth
and before the onset of coarsening. For grain radius R = 10 µm and more, this transient
region disappears due to a significant overlapping of precipitate growth and coarsening by
short-circuit diffusion in the grain boundary. This observation is discussed in the following
section.
Figure 7 displays two typical precipitate distributions observed at different stages of
coarsening. Since the number of precipitates per grain changes continuously during coarsening,
no stationary precipitate distribution is achieved for the entire simulation time. Instead, for
a sufficiently long time, a distribution corresponding to classical Ostwald ripening of grain
boundary precipitates [16–20] is observed, with a time exponent of approximately 1/4.2–1/4.0
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Figure 6. Phase fraction, mean radius and number density of AlN precipitates as a function of
grain radius (R = 1, 10, 100, 1000 µm). The dashed line represents simulation with random
precipitate distribution and radial diffusion fields. Al = 0.1 wt%, N = 0.01 wt%, T = 1000 ◦ C,
Dgi = 10 000Dbi .
for the evolution of the mean precipitate radius. This is in good accordance with theory,
predicting a value of 1/4 for purely grain boundary diffusion driven coarsening. The left plot
in figure 7 shows a typical numerical distribution obtained with the present model compared
with the theoretical precipitate distribution for the limit of zero phase fraction (equation (24)
in [18]). Higher values of the time exponent up to 1/3 are predicted [19, 20], if solute transfer
occurs by volume and grain boundary diffusion simultaneously. This occurs typically at
higher temperatures. Lower values down to 1/5 are observed experimentally [12–14] and
theoretically [18], if the diffusional transport occurs via pipe diffusion through a dislocation
network of, e.g., a low-angle grain boundary. In the simulations, the time exponent for the
evolution of the number density is observed with approximately 1/1.4.
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Figure 7. Typical precipitate distributions observed during coarsening. Left image: t 1/4 -coarsening
distribution. Right image: classical t 1/3 -LSW distribution.
For approximately two orders of magnitude in time, the distribution changes its shape and
closely resembles the classical LSW distribution [44, 45] for coarsening of randomly distributed
precipitates. This region is characterized by a ‘hump’ in the mean radius and the total number
of precipitates curve, as clearly visible in figure 6. The right plot in figure 7 compares a
corresponding distribution with the theoretical LSW distribution. In our simulations, the
particular region, where this progression occurs, is always found at the position, where the
precipitate density adopts values of approximately 104 –105 precipitates per grain.
3.4. Influence of grain boundary and bulk diffusion
In this section, the growth kinetics and interactions of an ensemble of grain boundary
precipitates of different sizes are investigated. By changing the ratio between the diffusional
mobilities of all elements in the bulk and the grain boundaries, Dgi /Dbi , the important influence
of this quantity on the overall precipitation kinetics is explored. In contrast to the numerical
analyses in the previous sections, the present simulations are carried out without pre-existing
nuclei. The formation of new precipitates follows a numerical procedure implemented in the
MatCalc software [26] based on classical nucleation theory. The simulations are performed
with a maximum of 500 size classes. A constant grain radius of R = 250 µm is used, and the
ratio between grain boundary and bulk diffusional mobility of all elements Dgi /Dbi is varied
between 1 and 106 . If the diffusivities in the grain boundary and the bulk are assumed to be
similar, this would correspond to temperatures close to the melting point. If they are assumed
to be different by several orders of magnitude, this would correspond to situations commonly
observed at relatively low temperatures, when the bulk diffusion is already sluggish and most
diffusional transport occurs along short-circuit diffusion paths such as dislocations and grain
boundaries [7]. The parameter study is performed including the nucleation simulation [28, 39]
with a temperature-dependent interfacial energy engaging a recently developed generalized
n-nearest-neighbor broken-bond model [42] and a size correction factor, taking into account
the interfacial curvature of the small nuclei [43].
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Figure 8. Phase fraction, mean radius and number density of AlN precipitates as a function of the
ratio between grain boundary and bulk diffusion coefficient Dgi /Dbi .
Figure 8 shows the evolution of the AlN phase fraction, the mean radius and the precipitate
density for various values of Dgi /Dbi while leaving the values of Dbi unchanged. The
maximum precipitate density is limited to a value of approximately 1020 m−3 according to
the saturation of nucleation sites. The slowest precipitation kinetics is observed for the lowest
values of Dgi /Dbi , which is easily attributed to the slow transport velocity of atoms toward
the precipitate. With increasing ratio Dgi /Dbi , this transport is speeded up together with the
entire precipitation process. In the diagram for the phase fraction of AlN versus time, the
kinetics of precipitation is increased significantly, if the grain boundary diffusivity is increased
starting at low values. Interestingly, at a given point, this increase comes to a stop, and only
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the shape of the diagram changes. The faster the diffusion in the grain boundary is, the flatter
is the slope of the phase fraction increase. This observation can be understood on the basis
of the evolution of the other precipitation parameters shown in figure 8. The curves for the
precipitate mean radius and number density are continuously shifted to shorter times with
increasing ratio Dgi /Dbi . This observation is attributed to the fact that the nucleation process
of precipitates is controlled mainly by the diffusional mobility of elements within the grain
boundary. With increasing Dgi /Dbi , the transport of atoms inside the grain boundary becomes
faster, and growth of precipitates speeds up. An increase in Dgi /Dbi thus simultaneously shifts
the start time of nucleation and the curves for mean radius and number density.
On closer investigation the evolution of the precipitate mean radius, we find that the
growth of precipitates occurs approximately with a slope of 1/2 in the case of a low ratio of
Dgi /Dbi . This corresponds to the well-known parabolic growth laws for diffusion-controlled
transformations. If the grain boundary diffusivity becomes faster, the coarsening contribution
to ρ̇k starts to dominate over the growth contribution to ρ̇k , and the slope of the precipitate
mean radius evolution starts to decrease to a value of approximately 1/3 to 1/4 for values
of Dgi /Dbi > 10. A comparable decrease in the time exponent for the precipitate mean
radius evolution is also observed in the case of classical precipitate coarsening [16, 20, 44, 45].
The higher the ratio Dgi /Dbi is, the earlier occurs coarsening inside the grain boundary and
the smaller becomes the contribution from diffusional transport through the bulk to the grain
boundaries. This fact is reflected in the phase fraction curves for higher ratio Dgi /Dbi . The
curves are not shifted to shorter times. Instead, the slope of the curves changes, once a critical
value of Dgi /Dbi is reached. In this case, growth and coarsening occur simultaneously, with
increasing dominance of the coarsening contribution with increasing ratio Dgi /Dbi . It should
be emphasized that this latter effect of change of slope occurs in addition to the effect described
in the previous section based on the variation of the precipitate diffusion zone radius in the
grain boundary.
3.5. Time–temperature-precipitation (TTP) diagram of AlN
Using the present model with the software MatCalc [26], the TTP diagram for AlN precipitation
in austenite can be constructed. The grain boundary diffusion ratio Dgi /Dbi is taken from
the data given by Balluffi [7] and varies between 10 000 at 1200 ◦ C and 235 000 at 850 ◦ C.
Otherwise the same simulation conditions apply as used in the previous sections.
Figure 9 shows that the TTP plot calculated with the GBDG model deviates significantly
from the TTP plot evaluated with the RSDG model. A grain radius R = 50 µm and a dislocation
density of 1012 m−2 have been assumed in these calculations. Although both models predict
approximately the same upper temperature limit for the start of AlN precipitation, the nose of
the C-curve for the grain boundary diffusion model is shifted to higher temperature by almost
100 ◦ C. At lower temperatures, the model with radial diffusion fields predicts faster kinetics
by several orders of magnitude in time for the end of precipitation. This effect is attributed
to the long diffusion distances from the grain interior to the grain boundary precipitates for
completion of the precipitation process.
3.6. Comparison with experiment
The predictions of the present grain boundary precipitation model are compared with
experimental data [46] on the precipitation kinetics of aluminum nitride at austenite grain
boundaries, performed on steel with 0.058 wt% Al, 0.0058 wt% N (alloy 1) and 0.079 wt%
Al, 0.0072 wt% N (alloy 2). After solution treatment at 1260 ◦ C for 15 min, alloy 1 has been
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Figure 9. Calculated TTP diagram for AlN precipitation in austenite. Dashed lines from original
treatment with radial diffusion fields.
Figure 10. Comparison of simulations with experimental data [46]. Solid lines: GBDG. Dashed
lines: random distribution, spherical diffusion fields.
isothermally treated at 927 ◦ C, alloy 2 at 982 ◦ C. Austenite grain diameters of 100 µm are
assumed for alloy 1 and 50 µm for alloy 2. Figure 10 compares the experimental points [46]
with simulations based on the new grain boundary precipitation model. Very good agreement
is observed for the precipitation kinetics in both alloys, if using the GBDG model. This is
particularly important for the slope of the phase fraction curves (solid lines). For comparison,
the figure also shows simulation results obtained with the RSDG model. It is obvious that
the increase in the phase fraction is severely overestimated by the RSDG model with random
precipitate distribution and spherical diffusion fields.
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4. Summary
Evolution equations are developed for the growth and coarsening of stoichiometric precipitates
located at grain boundaries, demonstrated as the GBDG model, taking into account the fast
diffusion along grain boundaries and the comparably slow diffusion inside the grains. The
thermodynamic extremal principle is utilized to relate the time evolution of the total Gibbs
free energy and the dissipative terms for growth and coarsening of the precipitates.
It is shown that the model reproduces the well-known effect of grain size on the
grain boundary precipitation kinetics, where large grain sizes significantly retard the overall
precipitation process due to large inner grain diffusion distances. Moreover, depending on
the ratio of grain boundary diffusion to bulk diffusion, simultaneous growth and coarsening
of precipitates can occur. In the case of high grain boundary diffusivity, the initial stage of
precipitation is accelerated by fast nucleation kinetics. However, the phase fraction increase
is simultaneously significantly slower compared with the random-distribution case. The
precipitate size distributions produced by the evolution equations are in good accordance with
the Ostwald ripening theory for grain boundary precipitation. Only for special combinations
of microstructural parameters, distributions close to LSW are observed. During coarsening,
the evolution of the precipitate mean radius with time is determined by a power law with an
exponent of 1/4.2 to 1/4.0, again in accordance with theory. The number density evolves with
an exponent of approximately 1/1.4.
A comparison with experimental data on AlN precipitation at austenite grain boundaries
shows very good agreement, where the original treatment, employing the RSDG model,
with random precipitate distribution and spherical diffusion fields, strongly overestimates the
precipitation kinetics and fails to reproduce the general shape of the phase fraction versus time
curves.
Acknowledgment
Financial support by the Austrian Federal Government (in particular from the
Bundesministerium für Verkehr, Innovation und Technologie and the Bundesministerium für
Wirtschaft und Arbeit) and the Styrian Provincial Government, represented by Österreichische
Forschungsförderungsgesellschaft mbH and by Steirische Wirtschaftsförderungsgesellschaft
mbH, within the research activities of the K2 Competence Center on ‘Integrated Research
in Materials, Processing and Product Engineering’, operated by the Materials Center Leoben
Forschung GmbH in the framework of the Austrian COMET Competence Center Program is
gratefully acknowledged.
Appendix A
The coefficients βi in equation (10) follow as
eq
grain
eq
grain
3
[−44R 3 (cMi − cMi ) − 162Ki + 18Li ](cMi − cMi )2
βi =
eq
grain
cMi − cMi
with the abbreviations Ki , Li
V
eq
eq
grain
eq
grain
Ki =
(cPi − cMi ),
Li = 6R 6 (cMi − cMi )2 + 44R 3 Ki (cMi − cMi ) + 81Ki2 .
8π
The distance di can be approximated for di /R 1 by a truncated Taylor series with two terms,
eq
eq
grain
using the variable vi = (cPi − cMi )V /[(cMi − cMi )R 3 ] as di /R ≈ −0.07958vi + 0.00422vi2 .
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015011
E Kozeschnik et al
Since di /R can be expressed as a polynomial function of vi , finally, ḋi can be assumed to be
proportional to V̇ .
Appendix B
The divergence of Jki , equation (12), can be linearized with respect to u/R ≈ 1 as
eq 2ρ 2 ρ̇k (cPi − cMi )
R − di
divJki = − k
3
−
2
2 di
u
eq
2ρ 2 ρ̇k (cPi − cMi ) 4di
u
di
≈− k
−
1
+
2
1
−
.
2 di
R
R
R
The local mass balance reads as
ċi = −divJki ,
and ci is a linear function in u/R. Since both relations are linear in u/R, the relation for Jki
also fulfills the local mass balance, atleast approximately.
Appendix C
The coefficients il with reference to equation (14) read as
grain
eq
i0 = [18R 2 + 8Rdi − di2 − 12R 2 ln(cMi /cMi )],
grain
eq
i1 = [−60R 2 + 4Rdi + 8di2 + 24R(R − di ) ln(cMi /cMi )],
grain
eq
i2 = [72R 2 − 36Rdi − 12(R − di )2 ln(cMi /cMi )],
i3 = [−36R 2 + 28Rdi − 8di2 ],
i4 = [6R 2 − 4Rdi + di2 ].
During growth of the grain boundary precipitates, the value of di increases and can reach the
value R. Then, the value of di must be fixed to R, and the concentration of component i in
grain
c
the center of the grain cMi
, having the initial value cMi , starts to decrease. In analogy to
equation (8), we write
u eq R − u c
c +
c ,
R Mi
R Mi
yielding with equation (8) and (R − di ) = 0
ci =
grain
eq
c
cMi
= 4cMi − 3cMi −
12Ki
.
R3
grain
c
Concerning the dissipation Qb , the values of cMi and di must be replaced by cMi
and R,
respectively, in equations (13) and (14). The growth of the precipitates stops if one of the
c
values of cMi
reaches zero.
c
It should be emphasized, here, that the values of cMi
can change dynamically, if
precipitation at the grain boundary occurs in competition with precipitation inside the
grain. Appropriately controlling this quantity and coupling it to the bulk precipitation offers
interesting possibilities in precipitation simulation; however, this will be a subject of another
paper.
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E Kozeschnik et al
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