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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

www.actamat-journals.com

*Corresponding author. Tel.: +1-814-8650389; fax: +1-814-

8670476.

E-mail address: zhu@cerse.psu.edu (J.Z. Zhu).

1359-6462/02/$ - see front matter ? 2002 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc.

PII: S1359-6462(02)00013-1

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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

0idealþ 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–

Laudauequationsfor

parameters fields. The morphological evolution

during the precipitation and coarsening can be

obtained by solving the four non-linear kinetic

[11,12]:

thelong-rangeorder

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

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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.

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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

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particles and plates or undergoing splitting to form

doublets or octets, which should occur only when

the particle size grows to a very large length scale.

In summary, we have developed a 3D phase-

field model to link with CALPHAD method

through the local free energy construction as a

function of field variables where most important

thermodynamic and kinetic parameters are taken

from either a database or independent experimen-

tal measurements. The compositional dependence

of atomic mobilities and elastic constants is taken

into account in solving the phase-field equations

and elastic equilibrium equations. The Ni3Al pre-

cipitate shape evolution in a disordered fcc matrix,

as illustrated from the simulations, depends on the

balance between the interfacial energy and the

elastic energy relaxation. Carrying out 3D simu-

lations in a larger system size is currently underway

to have a complete analysis of the precipitate

coarsening behavior in Ni-base superalloys.

Acknowledgements

The authors are grateful for financial support

from NASA under grant no. NCC3-920 and from

the National Science Foundation under grant nos.

DMR-0122638 and DMR-9983532. The simula-

tion was performed at the San Diego Supercom-

puter Center.

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