Linking phase-field model to CALPHAD: application to precipitate shape evolution in Ni-base alloys

Abstract

A three-dimensional phase-field model is proposed with the thermodynamic and kinetic parameters directly extracted from existing databases using the CALPHAD method. We modelled the γ′ precipitate microstructure evolution in a Ni-base alloy, particularly a single precipitate morphology at different sizes using independently assessed thermodynamic, kinetic and structural parameters.

Full text

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Linking phase-field model to CALPHAD: application
to precipitate shape evolution in Ni-base alloys
J.Z. Zhu*, Z.K. Liu, V. Vaithyanathan, L.Q. Chen
Department of Materials Science and Engineering, The Pennsylvania State University, 119 Steidle Building, University Park,
PA 16802-5005, USA
Received 9 October 2001; accepted 5 December 2001
Abstract
A three-dimensional phase-field model is proposed with the thermodynamic and kinetic parameters directly ex-
tracted from existing databases using the CALPHAD method. We modelled the c0precipitate microstructure evolution
in a Ni-base alloy, particularly a single precipitate morphology at different sizes using independently assessed ther-
modynamic, kinetic and structural parameters. ? 2002 Published by Elsevier Science Ltd. on behalf of Acta Materialia
Inc.
Keywords: Phase field; CALPHAD; Ni-base alloys; Microstructure; Computer simulation
1. Introduction
The phase-field approach has found increasing
applications in modelling phase transformation
and microstructural evolution in solids [1]. One of
its main advantages is that the temporal evolution
of any arbitrary microstructures can be predicted
without any a priori assumptions about their evo-
lution path. For example, it has been used to ex-
plain many of the morphological evolutions in
coherent solids including Ni-base superalloys [2–5].
Despite the tremendous success of phase-field
modelling in predicting many of the experimen-
tally observed microstructures in solids, additional
progress is required in order to apply it to predict
the microstructure evolution in real multi-compo-
nent alloy systems. For example, there exists no
systematic approach for obtaining the thermody-
namic and kinetic input to the phase-field models
although a number of efforts have been reported in
connecting phase-field models with existing ther-
modynamic and kinetic databases. Furthermore,
the dependence of kinetic parameters such as dif-
fusional mobility of atoms and of mechanical
properties such as elastic constants on the compo-
sition is generally ignored whereas in real systems
they are almost always composition dependent.
Finally, existing phase-field simulations have been
largely confined to two-dimension and the exten-
sion to three-dimensional (3D) systems requires
efficient numerical algorithms.
The main purpose of this paper is to describe our
initial attempt to develop a 3D phase-field model
for modelling the microstructure evolution in Ni-
basesuperalloys. The localfree energyasafunction
Scripta Materialia 46 (2002) 401–406
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*Corresponding author. Tel.: +1-814-8650389; fax: +1-814-
8670476.
E-mail address: zhu@cerse.psu.edu (J.Z. Zhu).
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of composition and order parameters is directly
constructed using the CALPHAD method. Effi-
cient iterative method and semi-implicit spectral
methods are developed for solving the inhomoge-
neous elastic equilibrium equation and the field
equations which take into account the composition
dependence of atomic mobilities. In particular, we
examine the morphological evolutions of a single
precipitate as well as of a multi-precipitate system
at a typical service temperature of 1300 K.
2. Model description
The phase-field model describes a microstruc-
ture by using a set of field variables. In Ni-base
alloys, the (c0þ c) two-phase microstructure can
be described by a composition field variable cðrÞ,
and a three-component long-range order parame-
ter field giðrÞ (i ¼ 1;2;3). The composition field
describes the local compositional distribution of
all the species in a microstructure, whereas the
long-range order parameter field describes the four
types of possible L12ordered domains related by a
lattice translation and distinguishes the structural
difference between ordered precipitates and the
disordered matrix related by the L12ordering.
Within the phase-field description, the total
free energy of a microstructure, including the local
chemical free energy, elastic energy and interfacial
energy, can be written as a function of all field
variables. The precipitation of c0phase is driven by
the local free energy difference between the disor-
dered fcc c phase and the equilibrium two-phase
mixture of the ordered c0particles contained in the
disordered c matrix. In most of the existing phase-
field descriptions of the local chemical free energy,
a Landau-type polynomial form is constructed as
only an approximation [2–5]. Most parameters in
the polynomials, which have no physical mean-
ings, have to be determined empirically with trial
and error. In this paper we show that the local
chemical free energy can be directly constructed
using the CALPHAD method [6,7] and thus relies
on existing thermodynamic databases to supply
the necessary thermodynamic input.
In the CALPHAD method, the Gibbs energy of
individual phases is modelled from their crystal-
lography information, and the model parameters
are optimized through coupling of phase equilib-
rium and thermochemical data obtained from
experimental measurements and first-principles
calculations. To consider the disordered and or-
dered fcc phases in the Ni–Al system using a single
Gibbs energy function of composition and tem-
perature, a four-sublattice model is used with Ni
and Al soluble in each sublattice [8]. The mole
fraction of Ni and Al in each sublattice can then
be represented by the three-component order pa-
rameter gi. The Gibbs energy of the two-phase
system including both ordered and disordered
structures can be written in terms of the compo-
sition and the order parameter field as,
gðc;g1;g2;g3Þ ¼ gdisðcÞ þ gorderðc;g1;g2;g3Þ
? gorderðc;gi¼ 0Þ
gdisand gorderare the Gibbs free energies of the
disordered phase and ordered phase respectively
which can be given by
ð1Þ
gdis¼ gdis
gorder¼ gorder
where
0þ Dgdis
idealþ Dgdis
þ Dgorder
xsþ Dgdis
mag
0 idealþ Dgorder
xs
ð2Þ
gdis
Dgdis
Dgdis
0¼ cgAl
ideal¼ RT½clnðcÞ þ ð1 ? cÞlnð1 ? cÞ?
xs¼ cð1 ? cÞ½L0þ L1ð2c ? 1Þ þ L2ð2c ? 1Þ2
þ L3ð2c ? 1Þ3?
Dgdis
¼ 6U1c2X
Dgorder
ðRT=4Þf½cð1 þ g1þ g2þ g3Þ?ln½cð1 þ g1þ g2þ g3Þ?
þ ½1 ? cð1 þ g1þ g2þ g3Þ?ln½1 ? cð1 þ g1þ g2þ g3Þ?
þ ½cð1 ? g1? g2þ g3Þ?ln½cð1 ? g1? g2þ g3Þ?
þ ½1 ? cð1 ? g1? g2þ g3Þ?ln½1 ? cð1 ? g1? g2þ g3Þ?
þ ½cð1 ? g1þ g2? g3Þ?ln½cð1 ? g1þ g2? g3Þ?
þ ½1 ? cð1 ? g1þ g2? g3Þ?ln½1 ? cð1 ? g1þ g2? g3Þ?
þ ½cð1 þ g1? g2? g3Þ?ln½cð1 þ g1? g2? g3Þ?
þ ½1 ? cð1 þ g1? g2? g3Þ?ln½1 ? cð1 þ g1? g2? g3Þ?g
Dgorder
xs
¼ ?2U1c2X
?
0þ ð1 ? cÞgNi
0
mag¼ RT lnðb þ 1ÞfðsÞ
gorder
0
g2
i
ideal¼
g2
iþ 12U4ð1 ? 2cÞc2
i? 48U4c3g1g2g3
X
g2
ð3Þ
402
J.Z. Zhu et al. / Scripta Materialia 46 (2002) 401–406

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All the parameters and functions in Eq. (3) such as
gAl
obtained from CALPHAD database as a function
of temperature [8].
At a constant temperature and composition, the
local free energy as a function of composition for
the precipitate phase can be determined by mini-
mizing the free energy with respect to the order
parameters, i.e. gðc;gi0ðcÞÞ. The corresponding lo-
cal free energy for the disordered matrix phase can
be obtained by setting all the three components of
the order parameter to zero, i.e. gðc;gi¼ 0Þ. From
the two free energy curves as a function of com-
position, the equilibrium compositions of the
two-phase mixtures at a given temperature can be
obtained by constructing a common tangent. Fig. 1
plots the chemical free energy as a function of
composition at 1300 K where one curve describes
the disordered c phase and another describes the
ordered c0phase. The equilibrium composition
values of the two phases at the stress-free state at
1300 K are determined to be 0.165 and 0.230, re-
spectively. The maximum driving force Dgmax at
1300 K is approximately 125 J/mol as seen from
Fig. 1.
In the diffuse-interface phase-field model, the
interfacial energy between the precipitate and
the matrix is introduced through the gradients of
0, gNi
0, L0, L1, L2, L3, b, fðsÞ, U1, and U4can be
the field variables. Denote fðc;giÞ as the Helm-
holtz free energy density converted from the Gibbs
free energy density gðc;giÞ. A general form for the
total stress-free chemical free energy can be written
as [9]
Fc¼
Z
V
fðc;giÞ
"
þa
2ðrcÞ2
þ
X
3
p¼1
bijðpÞ
2
rigprjgp
#
dV
ð4Þ
where ri¼ o=oxi is the ith component of the
vector operator r and xiis the ith component of
the spatial coordinate vector r. In Eq. (4), a and
bijðpÞ are gradient energy coefficients, which con-
trol the diffuse-interface thickness, need to be
determined from the experimentally measured in-
terfacial energy and anti-phase boundary (APB)
energy. The gradient energy coefficient for the
orientation field variables bijðpÞ is written in a
tensor form so as to incorporate the interfacial
energy anisotropy. For simplicity isotropic inter-
facial energy is assumed in this paper. The elastic
strain energy has an important contribution to the
total free energy. We have developed an efficient
iterative method for solving an elastically aniso-
tropic coherent system with elastic inhomogeneity
to compute the elastic energy Eel [10]. The total
free energy F is the sum of the incoherent free
energy Fc and the coherent elastic energy Eel,
i.e. F ¼ Fcþ Eel. The temporal evolution of field
variables is described by time-dependent kinetic
field equations––namely, the Cahn–Hilliard equa-
tions for the composition field and the Ginzburg–
Laudauequations for
parameters fields. The morphological evolution
during the precipitation and coarsening can be
obtained by solving the four non-linear kinetic
[11,12]:
the long-range order
ocðr;tÞ
ot
¼ r ? Mðr;tÞr
dF
dcðr;tÞ
??
þ nðr;tÞð5Þ
ogiðr;tÞ
ot
¼ ?K
dF
dgiðr;tÞþ fiðr;tÞð6Þ
where i ¼ 1;2;3. c is the composition of Al. K
and M are kinetic coefficients for the two field
Fig. 1. Calculated chemical free energy as a function of com-
position for both the disordered and ordered phase at 1300 K.
J.Z. Zhu et al. / Scripta Materialia 46 (2002) 401–406
403

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variables. Mðr;tÞ describes a composition-depen-
dent mobility which can be written as M ¼ cð1 ? cÞ
½cMNiþ ð1 ? cÞMAl?. MNi and MAl are the atomic
mobilities of Ni and Al which can be calculated
from kinetic database at a given temperature [13].
nðr;tÞ and fiðr;tÞ are noise terms which generate
fluctuations in the composition and structural or-
der parameter fields. We employ an accurate and
efficient semi-implicit Fourier spectral method for
solving the two kinetic equations [14].
3. Results and discussion
Our simulations are performed on a cubic
domain with periodic boundary conditions. The
gradient coefficients a and bijare determined from
measured c0matrix interfacial free energy and
the APB energy on {100}. To avoid some huge or
very small numbers involved in the computa-
tion, the parameters are scaled by reduced time
t?¼ KtDgmax and r?¼ r=l where l represents the
length scale of the system [4]. The typical input
parameters used for the two field equations at
1300 K are unit grid size Dx ¼ 1 nm, scaled time
step Dt?¼ 0:02, compositional gradient coefficient
a ¼ 2:524 ? 10?9J/m, order parameter gradient
coefficients bij¼ 7:29 ? 10?12J/m, scaled average
mobility ? M M?¼? M M=ðKl2Þ ¼ 0:007937, and the lat-
tice mismatch d ¼ 0:00294 [15]. The elastic cons-
tants of the two phases are estimated from Refs.
[16,17] as Cc
Cc
GPa, and Cc0
To investigate the shape evolution of the c0
precipitate, we first studied a single particle evo-
lution where the interaction among precipitates is
ignored. Under an aging temperature of 1300 K,
a spherical c0particle with the equilibrium com-
position cc0 ¼ 0:230 and order parameter values
gi¼ 0:948288 was initially embedded in the c
matrix (cc¼ 0:165, gi¼ 0). The shape evolution of
the particle will then be mainly determined by the
balance between the interfacial energy and elastic
energy relaxation, which are included in the vari-
ational derivatives in the kinetic equations (5) and
(6). Fig. 2(a) shows the shape evolution of a single
particle with a radius of 16 nm in a 48 ? 48 ? 48
11¼ 195:8 GPa, Cc
11¼ 194:64 GPa, Cc0
44¼ 91:9 GPa.
12¼ 144:0 GPa,
12¼ 141:8
44¼ 89:6 GPa, Cc0
lattice at three different times. It can be seen that
the particle remains a near-spherical shape at its
length scale (?30 nm in Fig. 2(a)). When the
particle is small in size, the interfacial energy will
dominate in the total energy and thus equilibrium
particle shape will be that of a minimal interfacial
energy, which is spherical due to the isotropic in-
terfacial energy.
An increase in particle size requires more
number of points used for the particle in a dis-
cretized lattice. To remain a constant particle vol-
ume percentage where the boundary effects can be
neglected, a larger system size is therefore needed
for a larger particle size. However, an increase of
the system size will increase the computational
time and memory dramatically, especially when a
3D system is considered. To simulate the effects
of particle size on the equilibrium shape without
having to run simulations in a large system size, we
introduce a dimensionless parameter size L to
characterize the ratio of the elastic energy to the
interfacial energy [18,19]. Assuming?l l as the aver-
age length scale of the particle,?k k as the effective
elastic constant to characterize the average stiff-
ness, ? e e as the average elastic strain, csas the surface
energy density, L is defined by
L ¼
?k k? e e2?l l3
cs?l l2¼
?l l
l0
ð7Þ
Fig. 2. Shape evolution of a single c0particle at different length
scales.
404
J.Z. Zhu et al. / Scripta Materialia 46 (2002) 401–406

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where l0¼ cs=ð?k k? e e2Þ can be regarded as a kind of
material constant that has a dimension of an
length (?10 nm for Ni–Al at 1300 K). The real
average length scale?l l will be proportional to the
dimensionless parameter L, and therefore, differ-
ent real length scale can be reached by different L
values. The parameter L can be changed by the
ratio of total elastic energy to the interfacial
energy. For example, double elastic energy Eelwill
double L, which will achieve very similar effect of
double the average length scale?l l. Fig. 2(b) and (c)
illustrate the morphologies of the c0particle
evolving as a function of time where L ¼ 3 and
L ¼ 5 respectively with their corresponding real
length scales are shown. The particle gradually
becomes cuboidal that is more or less cubic in
shape with rounded edges and corners. It acquires
slightly curved convex interfaces parallel to the
elastic soft f100g directions. As the interfacial
energy is proportional to the interfacial area and
the elastic energy is proportional to the particle
volume, the elastic energy part of the total energy
will dominate at large particle size, which drives
the particle shape change from spherical to cub-
oidal in an elastically anisotropic system. In Fig.
2(b) and (c), the corresponding real particles sizes
are approximately 80 nm (Fig. 2(b)) and 110 nm
(Fig. 2(c)). As seen from Fig. 2, the equilibrium
precipitate volume fraction decreases with in-
creasing L. It decreases from 15% in Fig. 2(a)
(L ¼ 1) to 6% in Fig. 2(c) (L ¼ 5), indicating the
ratio of elastic energy to interfacial energy has
important influence on the equilibrium composi-
tions of the two phases. We find that the computed
equilibrium compositions cc and cc0 of the two
phases have changed from 0.167 and 0.230 for
L ¼ 1 to 0.173 and 0.222 for L ¼ 5, resulting in the
decrease of the equilibrium particle volume frac-
tion.
To represent a real 3D Ni–Al system we need to
investigate the multi-particle interaction effects
which play an important role in determining the
morphology of particles as well as their spatial
arrangement. Fig. 3 shows an example of micro-
structural evolution in a lattice with 128 ? 128?
128 grid points at an equilibrium particle volume
fraction ?35%. Initially the system is prepared in a
high-temperature homogeneous initial state where
the composition deviation from the average value
is only caused by fluctuation. The nucleation,
growth and coarsening of c0particles are evident in
Fig. 3, driven by the decrease of the total free
energy. During the nucleation and growth periods
where the particle size is small, the particles are
spherical (Fig. 3(a) and (b)). The average domain
size increases at later times accompanying with the
particle shape changing to cuboidal. Particle
coarsening by the coalescence of neighboring do-
mains are observed.
For a single particle in Ni–Al alloy system
aging at 1300 K, it can be predicted from Fig. 2
that the shape evolution from spherical to cuboi-
dal occurs in the length scale of approximately 80
nm. However, in a multi-particle system with high
volume fraction where the elastic interaction en-
ergies cannot be neglected, such a shape change
occurs even at smaller particle size 15–20 nm as
shown in Fig. 3(d). The predicted length scale is in
good agreement with experimental observations
[20]. With the length scale and thermodynamic and
kinetic parameters considered in our work, we did
not observe c0developing to concave cuboidal
Fig. 3. Morphological evolution from 3D simulations showing the nucleation, growth and coarsening of c0precipitates at 1300 K.
J.Z. Zhu et al. / Scripta Materialia 46 (2002) 401–406
405