Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

Подождите немного. Документ загружается.

1088 Part E Modeling and Simulation Methods

20 40 60 80 Al

C content (mol. %)

Al content (mol. %)

b) 1000°C a) 900°C

c) 1300°C

20 40 60 80Ni Al

C content (mol. %)

Al content (mol. %)

20 40 60 80Ni Al

C content (mol. %)

Al content (mol. %)

Fig. 20.27a–c Isothermal section diagram of the Ni

−

C system calculated at (a) 900

◦

C, (b) 1000

◦

C, and (c) 1300

◦

Part E 20.3

The CALPHAD Method 20.3 Prediction of Thermodynamic Properties of Compound Phases with First-principles Calculations 1089

0.20

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0 0.2 0.4 0.6 0.8 1

C content (mole fraction)

Ni content (mole fraction)

AlC

Fig. 20.28 Calculated miscibility gap in the Co

−

AlC

−

Al pseudoquaternary system at

1000

◦

1800

1600

1400

1200

1000

800

600

400

200

73 mol. % Co

10 20 30 40 50 60 70

Temperature T (°C)

Ni content (mol. %)

γ ++

Fig. 20.29 Vertical section diagram calculated at a constant C content of 4 mol % and an Al content of 23 mol %

Part E 20.3

1090 Part E Modeling and Simulation Methods

References

20.1 N. Saunders, A.P. Miodownik: CALPHAD (Calcula-

tion of Phase Diagrams): A Comprehensive Guide

(Elsevier, Oxford 1998)

20.2 F.R.deBoer,R.Boom,W.C.M.Mattens,

A.R. Miedema, A.K. Niessen: Cohesion in Metals,

Transition Metals Alloys (North-Holland, Amster-

dam 1988)

20.3 G. Ghosh, C. Kantner, G.B. Olson: Thermodynamic

modeling of the Pd–X (X = Ag, Co, Fe, Ni) systems,

J. Phase Equil. 20, 295–308 (1999)

20.4 M. Hillert, L.-I. Staffansson: The regular solution

model for stoichiometric phases and ionic melts,

Acta Chem. Scand. 24, 3618–3626 (1970)

20.5 W. Oelsen, F. Johannsen, A. Podgornik: Erzmetall

9, 459–469 (1956)

20.6 W. Kohn, L.J. Sham: Self-consistent equations in-

cluding exchange and correlation effects, Phys.

Rev. 140, A1133–1138 (1965)

20.7 P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka,

J. Luiz: WIEN2k, An Augmented Plane Wave and

Orbitals Program for Calculating Crystal Properties

(Karlheinz Schwarz, Tech. Universit t Wien, Vienna

2001)

20.8 J.P. Perdew, K. Burke, Y. Wang: Generalized gra-

dient approximation for the exchange-correlation

hole of a many-electron system, Phys. Rev. B 54,

16533–16539 (1996)

20.9 H. Ohtani, M. Hasebe: Thermodynamic analysis of

phase diagrams by incorporating ab initio ener-

getic calculations into CALPHAD approach, Bull. Iron

Steel Inst. Japan 9, 223–229 (2004)

20.10 J.W.D. Connolly, A.R. Williams: Density-functional

theory applied to phase transformations in

transition-metal alloys, Phys. Rev. B 27,5169–5172

(1983)

20.11 H. Ohtani, Y. Takeshita, M. Hasebe: Effect of the

order–disorder transition of the bcc structure on

the solubility of Be in the Fe–Be binary system,

Mater. Trans. 45, 1499–1506 (2004)

20.12 F.D. Murnaghan: The compressibility of media un-

der extreme pressures, Proc. Nat. Acad. Sci. USA 30,

244–247 (1944)

20.13 M.H.F. Sluiter, Y. Watanabe, D. de Fontaine,

Y. Kawazoe: First-principles calculation of the

pressure dependence of phase equilibria in

the Al–Li system, Phys. Rev. B 53,6137–6151

(1996)

20.14 T. Mohri, Y. Chen: First-principles investigation of

-disorder phase equilibrium in the Fe–Pt sys-

tem, Mater. Trans. 43, 2104–2109 (2002)

20.15 H. Ino: A pairwise interaction model for decom-

position and ordering processes in b.c.c binary

alloys and its application to the Fe–Be system, Acta

Metall. 26,827–834(1978)

20.16 T. Takayama, M.Y. Wey, T. Nishizawa: Effect of

magnetic transition on the solubility of alloying

elements in bcc iron and fcc cobalt, Trans. Japan

Inst. Met. 22, 315–325 (1981)

20.17 H. Ohtani, Y. Chen, M. Hasebe: Phase separation

of the B2 structure accompanied by an ordering in

Co–Al and Ni–Al binary systems, Mater. Trans. 45,

1489–1498 (2004)

20.18 H. Ohtani, M. Yamano, M. Hasebe: Thermody-

namic analysis of the Fe–Al–C ternary system by

incorporating ab initio energetic calculations into

the CALPHAD approach, ISIJ Intern. 44, 1738–1747

(2004)

20.19 R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser,

K.K. Kelley: Selected Values of the Thermody-

namic Properties of Binary Alloys (ASM, Metals Park

1973)

20.20 Y. Kimura, M. Takahashi, S. Miura, T. Suzuki,

Y. Mishima: Phase stability and relations of multi-

phase alloys based on B2 CoAl and E21 Co

AlC,

Intermet. 3, 413–425 (1995)

20.21 M. Palm, G. Inden: Experimental determination of

phase equilibria in the Fe–Al–C system, Intermet.

3, 443–454 (1995)

20.22 H. Ohtani, M. Yamano, M. Hasebe: Thermo-

dynamic analysis of the Co–Al–C and Ni–Al–C

systems by incorporating ab initio energetic cal-

culations into the CALPHAD approach, Comp.

Coupling Phase Diag. Thermochem. 28, 177–190

(2004)

Part E 20

1091

Phase Field A

21. Phase Field Approach

The term phase ﬁeld has recently become known

across many ﬁelds of materials science. The mean-

ing of phase ﬁeld is the spatial and temporal order

parameter ﬁeld deﬁned in a continuum-diffused

interface model. By using the phase ﬁeld order

parameters, many types of complex microstruc-

ture changes observed in materials science are

described effectively. This methodology has been

referred to as the phase ﬁeld method, phase ﬁeld

simulation, phase ﬁeld modeling, phase ﬁeld ap-

proach, etc. In this chapter, the basic concept and

theoretical background for the phase ﬁeld ap-

proach is explained in Sects. 21.1 and 21.2.The

overview of recent applications of the phase ﬁeld

method is demonstrated in Sects. 21.3 to 21.6.

Phase ﬁeld models have been successfully

applied to various materials processes including

solidiﬁcation, solid-state phase transformations

and microstructure changes. Using phase ﬁeld

methodology, one can deal with the evolution of

arbitrary morphologies and complex microstruc-

tures without explicitly tracking the positions of

interfaces. This approach can describe different

processes such as diffusion-controlled phase sep-

aration and diffusionless phase transition within

the same formulation. It is rather straightforward

to incorporate the effect of coherency and applied

stresses, as well as electrical and magnetic ﬁelds.

Since phase ﬁeld methodology can model

complex microstructure changes quantitatively, it

will be possible to search for the most desirable

microstructure using this method as a design sim-

ulation, i. e., through computer trial-and-error

testing. Therefore, the most effective strategy

for developing advanced materials is as follows.

First, we elucidate the mechanism of microstruc-

ture changes experimentally, then we model the

microstructure evolutions using the phase-ﬁeld

method based on the experimental results, and

21.1 Basic Concept

of the Phase-Field Method....................1092

21.2 Total Free Energy of Microstructure ........1093

21.2.1 Chemical Free Energy....................1093

21.2.2 Gradient Energy ...........................1095

21.2.3 Elastic Strain Energy .....................1097

21.2.4 Free Energy for Ferromagnetic

and Ferroelectric Phase Transition..1101

21.3 Solidiﬁcation........................................1102

21.3.1 Pure Metal ..................................1102

21.3.2 Alloy ...........................................1104

21.4 Diffusion-Controlled Phase

Transformation ....................................1105

21.4.1 Cahn–Hilliard Diffusion Equation ...1105

21.4.2 Spinodal Decomposition

and Ostwald Ripening ..................1107

21.5 Structural Phase Transformation ............1108

21.5.1 Martensitic Transformation............1108

21.5.2 Tweed-Like Structure

and Twin Domain Formations ........1108

21.5.3 Twin Domain Growth Under

External Stress and Magnetic Field . 1109

21.6 Microstructure Evolution .......................1110

21.6.1 Grain Growth

and Recrystallization ....................1110

21.6.2 Ferroelectric Domain Formation

with a Dipole–Dipole Interaction ...1111

21.6.3 Modeling Complex Nanogranular

Structure Formation .....................1112

21.6.4 Dislocation Dynamics....................1112

21.6.5 Crack Propagation ........................1113

References ..................................................1114

ﬁnally we search for the most desirable micro-

structure while simultaneously considering both

the simulation and experimental data.

Part E 21

1092 Part E Modeling and Simulation Methods

21.1 Basic Concept of the Phase-Field Method

Over the last two decades, the phase-ﬁeld ap-

proach [21.1–7] has been developed for many ﬁelds

in materials science as a powerful tool to simulate

and predict complex microstructure developments (for

example, solidiﬁcation [21.3, 8–12], spinodal decom-

position [21.13, 14], Ostwald ripening [21.15], crystal

growth and recrystallization [21.16–19], domain micro-

structure evolutions in ferroelectric materials [21.20,

21], martensitic transformation [21.22], dislocation dy-

namics [21.23–25], crack propagation [21.26], etc.).

An essence of the phase-ﬁeld method is based

on a diffuse-interface approach [21.6] proposed more

than a century ago by van der Waals [21.27]andin-

dependently almost a half-century ago by Cahn and

Hilliard [21.28, 29]. Because the phase-ﬁeld methodol-

ogy has an efﬁcient ability to model various types of

complex microstructure changes quantitatively, there is

large possibility to search for the most desirable micro-

structure using this method as a design simulation, i. e.,

trial-and-error testing on a computer.

Since the phase-ﬁeld method has developed over a

wide range within the ﬁeld of materials science, the

deﬁnition of the method has divided into two main

categories. One is the interface-tracking approach com-

monly used in solidiﬁcation simulation of dendrite

growth, and the other is the continuum diffuse-interface

model, which describes the temporal and spatial de-

velopments of the dynamics of inhomogeneous order

parameter ﬁelds during phase transformations.

In phase-ﬁeld modeling, the determination of order

parameters that are well deﬁned and called phase-

ﬁeld variables is the most important step. The order

parameters are, in general, classiﬁed into two types:

the conserved order parameters and the nonconserved

ones. Typical examples of the former and the latter are

the composition of a solute atom and the long-range

order parameter describing an order–disorder phase

transition, respectively. There is no systematic crite-

rion for selecting the order parameters, but it is usually

enough for the order parameters to describe the targeted

microstructure morphology and its total free energy,

quantitatively.

In order to simulate the temporal and spatial mi-

crostructure developments the total free energy of the

inhomogeneous microstructure should ﬁrst be evalu-

ated. The total free energy of microstructure G

sys

deﬁned by a sum of a chemical free energy density g

gradient energy density f

grad

, an elastic strain energy

density e

str

, a magnetic energy density f

mag

,andan

electric energy density f

ele

, so that the functional G

sys

is written as

sys



⎡

⎢

⎣

, s

)

grad

, s

) +e

str

, s

)

mag

, s

) + f

ele

, s

)

⎤

⎥

⎦

dr ,

(21.1)

where c

(r, t)ands

(r, t) are the order parameters of the

conserved and nonconserved ﬁelds, respectively, and

these are the function of spatial position r and time t.

The subscripts i and j of each order parameter are the

numbers by which it is determined if the order parame-

ter belongs to the same category. The order parameters,

such as c

(r, t)ands

(r, t), are commonly called the

phase-ﬁeld variables in the phase-ﬁeld method. Since

, f

grad

, e

str

, f

mag

,and f

ele

are expressed as a function

of order parameters c

(r, t)ands

(r, t), these parameters

interact with each other through the total free energy

during microstructure changes.

The temporal evolution of ﬁeld variables in a phase-

ﬁeld model can be obtained by solving (21.2)and(21.3)

including the dynamic coupling term of the time varia-

tion of the order parameters (see the second term of the

right-hand side of both equations)

∂c

(r, t)

∂t

=∇·





∇

δG

sys

δc

+ξ



∂s

(r, t)

∂t

, (21.2)

∂s

(r, t)

∂t

=−L



δG

sys

δs

+ξ



∂c

(r, t)

∂t

(21.3)

where the symbols M

and L

is the mobility concern-

ing the changes of the corresponding order parameters.

These are precisely described as a function of the order

parameters, but in many actual calculations the mobility

is often set constant or regarded as a function of temper-

ature. ξ

is a Gaussian thermal noise term for the order

parameter p (= c

or s

). The second terms in the right-

hand side of (21.2)and(21.3) represent the dynamic

coupling of each phase-ﬁeld variable, where K

and K

are the coupling coefﬁcients. These are often assumed

to be constant, or set to 0 when the dynamic interaction

between order parameters is ignored. The microstruc-

ture change, which this coupling term should introduce,

is actually limited to the case when a highly complex

morphology of microstructure is formed almost as far

from equilibrium, e.g., the dendrite growth, the fractal

pattern formation, and so on.

Part E 21.1

Phase Field Approach 21.2 Total Free Energy of Microstructure 1093

21.2 Total Free Energy of Microstructure

In order to simulate microstructure changes quantita-

tively, the precise evaluation of the total free energy of

the microstructure is required. In this section, the detail

of the equations used for estimating the total free energy

of microstructure is explained.

21.2.1 Chemical Free Energy

Several types of description for chemical free energy

for a homogeneous state (i. e., mean ﬁeld) have been

proposed for phase-ﬁeld simulations. They are mainly

classiﬁed into two categories. One is the expression

based on the polynomial expansion with respect to

the order parameters known as a Landau expansion

method [21.30], the other is the case where the well-

deﬁned chemical free energy curves for speciﬁc crystal

structures are connected to each other continuously us-

ing phase-ﬁeld variables. Some typical examples are

provided below.

Grain Growth

The chemical free energy density g

utilized for

a crystal grain growth in polycrystalline single-phase

materials is given by

, s

,...,s

)



i=1



−





i=1



j>i

, (21.4)

where s

= s

(r, t), (i = 1, 2,...,p) are the order

parameters describing the probability to ﬁnd the crys-

tal with its orientation designated by i at position r and

time t [21.16]. Each integer number i(= 1 ∼ p) corres-

ponds to the speciﬁc orientation of the crystal structure,

respectively. The probability of ﬁnding the grain with

orientation i at position r and time t is deﬁned by

(r, t)|. The symbols A, B,andC are the numerical

constants determined phenomenologically. If we only

employ i = 1, (21.4) reduces to

) =−

(21.5)

=−



1+s





1−s





Hence g

) =0 is satisﬁed at s

= 0, ±

√

2A/B,and

the function g

) has a maximum at s

= 0, a min-

imum at s

=±

√

A/B, which is determined from the

condition ∂g

/∂s

=0.

The last term in the right-hand side of (21.4)isthe

penalty term, which forbids the coexistence of crystals

with different orientation at the same position (because

the coefﬁcient C is deﬁned as C > 0).

As an extended application of (21.4) to the two-

phase mixture [21.31], the chemical free energy for a

polycrystal containing grains of different phase is writ-

ten as



c ;s

, s

,...,s

, s

,...,s



=−

(c −c

)

(c −c

)

(c −c

)

(c−c

)



i=1



−



c−c















i=1



−

(c−c

)













i=1



j=1









, (21.6)

where the composition ﬁeld c = c(r, t) is also consid-

ered, and c

and c

are the composition of α and

β phase, respectively; c

is deﬁned by c

= (c

)/2. Symbols A, B, C

, C

, U

, V

and W

are the phenomenologically determined numerical con-

stants. The phase-ﬁeld order parameters for different

crystal structures, α and β, are denoted by s

(r, t)

and s

(r, t), respectively.

Phase Separation Between Different Crystal

Structure Phases (Landau Expansion)

A typical example is the case that both a phase sepa-

ration and a structural phase transition from cubic to

tetragonal occur simultaneously [21.32]. The chemical

free energy function is expressed based on the Landau

expansion as

(c, s

, s

)

A(c −c

)

B(c −c

)





−



























+Ws

, (21.7)

Part E 21.2

1094 Part E Modeling and Simulation Methods

where s

(r, t) are the order parameters indicating

the probability of ﬁnding the tetragonal phase desig-

nated by i at position r and time t. Index i (=1, 2, and 3)

corresponds to the direction of the c-axis of the tetrago-

nal phase. (In the structural phase transition from cubic

to tetragonal, there are three orientation relations be-

tween cubic and tetragonal phases, i. e., three variants.)

The speciﬁc state s

=0 corresponds to cubic

phase. The order parameter c =c(r, t) is a solute com-

position at position r and time t. The symbols c

, c

, A,

B, C, D, U, V ,andW are constants.

Phase Separation Between Different Crystal

Structure Phases (Use of Thermodynamic

Database of Phase Diagrams)

The construction method of chemical free energy us-

ing a thermodynamic database of phase diagrams is

proposed in the phase-ﬁeld method for alloy solidiﬁca-

tion [21.1, 2, 9]. In this formulation, since the chemical

free energy is directly related to the real phase diagrams

and we can simulate phase transformations in accord

with the phase diagrams, this methodology is quite im-

portant and useful for practical engineering purposes,

and the details are as follows.

First we suppose, for instance, that the α and

β phase have different crystal structure in the A–B bi-

nary alloy system, and that the chemical free energy

curves for these two phases are denoted as g

(c, T )and

(c, T ), respectively, where c is the solute composition

of component B. The chemical free energy function

for each phase is easily obtained from the thermody-

namic database of phase diagrams such as ThermoCalc,

Pandat, etc. [21.33].

Next, we deﬁne the phase-ﬁeld variable s = s(r, t),

that is, the probability of ﬁnding the β phase at posi-

tion r and time t. Therefore, the cases s = 0ands = 1

correspond to the α single phase and the β one, respec-

tively. Utilizing the phase-ﬁeld variable s, we are able to

formulate the chemical free energy surface as a function

of c and s, which is expressed as

(c, s, T ) = g

(c, T ) [1 −h(s)] +g

(c, T )h(s)

α↔β

(T )g

(s) , (21.8)

where W

α↔β

(T ) is an energy barrier originating in the

latent heat during α ↔ β structural phase transition. In

the phase-ﬁeld method the function g(s) is usually de-

ﬁned by

g(s) ≡s(1 −s) .

(21.9)

As for the function h(s), two different deﬁnitions have

mainly been employed in the phase-ﬁeld method, that is

h(s) =



g(s



)ds





g(s



)ds



(3 −2s) (21.10)

and

h(s) =





)ds







)ds



(10−15s +6s

) . (21.11)

It seems that there is no signiﬁcant effect on the simula-

tion result as to whether (21.10)or(21.11) is employed.

Cleary recognized from (21.10)and(21.11), the func-

tion h(s) shows an S-shaped curve, the value of which

ranges from 0 to 1. The function g

(s)is0whens =0

and s = 1, and a maximum value at s = 0.5. Equa-

tion (21.8) constructs the energy surface connecting

the chemical free energy curves of the α and β phase

smoothly using function h(s). The physical meaning of

the third term in the right-hand side of (21.8)isthela-

tent heat induced with a structural phase transition. (The

detail for the formulation of a chemical free energy for

each single phase is explained in Chap. 20.)

It should be noted that the free energy surface

deﬁned by (21.8) is a conveniently determined sur-

face with no strict physical meaning. For instance, the

atomic structure corresponding to the state of s =0.5is

not deﬁned in the phase-ﬁeld method. (The same deﬁni-

tion of atomic arrangement may be possible for s =0.5,

but the explicit image for the atomic arrangement is

not needed because the phase-ﬁeld method is a contin-

uum model for microstructure calculation where only

an energy is evaluated depending on the local order

parameter values.)

The chemical free energy of the multicomponent

and multiphase system is expressed as an extended form

, c

,...,c

, s

,...,s

, T )



i=1

, c

,...,c

, T )h(s

)



i=1



j=1

↔φ

(T )g(s

, s

) , (21.12)

where N

and N

are the number of components and

phases considered. The function g

, c

,...,c

, T )

is a chemical free energy of φ

phase denoted by i.

In this case, since s

= s

(r, t) is a phase-ﬁeld vari-

able indicating the probability to ﬁnd the φ

phase at

position r and time t, the condition



i=1

(r, t) =1

Part E 21.2

Phase Field Approach 21.2 Total Free Energy of Microstructure 1095

must be satisﬁed. The function g(s

, s

)isdeﬁnedby

g(s

, s

) ≡(1−δ

and g(s

, s

) ≡(1−δ

for

a binary system and the other multicomponent system,

respectively, where δ

is Kronecker’s delta, i. e., δ

for i = j and δ

= 0fori = j. This distinction is to

avoid the appearance of a concave region on the free

energy curve constructed by order parameters s

21.2.2 Gradient Energy

A gradient energy F

grad

[21.29–34] is given by a gradi-

ent squire approximation for the order parameter proﬁle

in the phase-ﬁeld method, which is expressed as

grad







i=1

(

∇c

)



i=1

∇s



dr ,

(21.13)

where symbols κ

and κ

are the gradient energy coefﬁ-

cients.

It is sometimes treated that a gradient energy is

same as an interfacial energy. Strictly speaking, this is

not correct. The interfacial energy where the interface

is deﬁned by the diffused order parameter proﬁle not

only includes the gradient energy but also the chem-

ical free energy. Since the physical meaning of an

interfacial energy is the total free energy at the inter-

face, a gradient energy is different from an interfacial

energy.

Equation (21.13) is derived by means of the Tay-

lor expansion of a chemical free energy with respect to

the gradient and curvature of the order parameter proﬁle

and with the higher order terms removed. The key idea

is that the gradient and the curvature of the order param-

eter proﬁle are treated as invariant variables except for

the order parameter itself. The physical meaning of this

equation is explained as follows.

In order to make the argument simple, we only con-

sider the composition ﬁeld c(r) for the A–B binary

alloy system, and focus on c(r

) =0.5 at arbitrary po-

sition r

. When we only consider the chemical free

energy for the total free energy at r

, the total free

energy value is not changed even if the gradient and

curvature of the composition proﬁle at r

is altered.

But when the composition proﬁle is extremely sharp,

such that it is observed in the modulated structure

during spinodal decomposition, the number of A–A,

B–B, and A–B atom pair at r

will depend on the

shape of the composition proﬁle across the position r

This deviation is evaluated by considering the com-

position ﬁeld information c(r

±Δr) around r

,i.e.,

the deviation is described using the gradient and the

curvature of the composition proﬁle as an invariant vari-

able. Therefore, the composition gradient energy is the

excess energy induced from the spatial ﬂuctuation of

composition ﬁeld, and in this case, the chemical free

energy is the mean ﬁeld energy without composition

ﬂuctuation.

The Orientation Dependence

of Gradient Energy

The interface between phases with different crystal

structure and orientation is often considered in the

phase-ﬁeld method, and commonly, the orientation

dependence is introduced into the gradient energy co-

efﬁcient [21.1, 2]in(21.13). Therefore, the variation

calculation for the gradient energy is signiﬁcantly in-

ﬂuenced in this case. For simplicity, we consider the

composition ﬁeld of the A–B binary alloy system, and

a two-dimensional calculation is supposed. Under this

condition, the composition gradient energy coefﬁcient

is deﬁned by κ

≡ ε

(θ), where ε is a function of the

orientation θ. Then the composition gradient energy is

given as

grad



(θ)(∇c)

dr . (21.14)

We consider the phase separated microstructure, and let

n be a unit normal at the interface between precipitate

and matrix. The relations among n, θ,andc(r, t) are

given by

n =(n

, n

) =(cos θ,sin θ)

=−

∇c

=−



C, x

+C, y

(

C, x, C, y

)

∴ tan θ =

C, y

C, x

→ θ =tan

−1

C, y

C, x

(21.15)

where C, x and C, y are deﬁned by C, x ≡ ∂c/∂x

and C, y ≡∂c/∂y, respectively. It is worth noting that

(21.15) insists θ and c(r, t) are not independent. Since

the orientation of θ at the interface is a function of

a composition gradient, the composition gradient en-

ergy coefﬁcient κ

is also a function of a composition

gradient. Therefore, the variation calculation should

be performed in (21.14) not only for the gradient

squared part but also for the composition gradient en-

ergy coefﬁcient itself. The explicit form of the variation

Part E 21.2

1096 Part E Modeling and Simulation Methods

calculation is written in the form

δF

grad

δc

=−



∂ f

grad

∂C, x



−



∂ f

grad

∂C, y





εε



C, y −ε

C, x



−



εε



C, x +ε

C, y



=−ε

(

C, xx+C, yy

)

−





+εε





× (θ, yC, x −θ, xC, y)

−2εε



(

θ, xC, x +θ, yC, y

)

, (21.16)

where f

grad

≡ (1/2)ε

(θ)(∇c)

, ε



≡∂ε/∂θ, ε



≡ ∂

ε/

∂θ

, C, xx ≡ ∂

c/∂x

, C, yy ≡ ∂

c/∂y

, C, xy ≡ ∂

(∂x∂y), θ, x ≡ ∂θ/∂x,andθ, y ≡∂θ/∂y.

The diffusion potential concerning the composition

gradient energy has to be calculated by (21.16), when

the composition gradient energy coefﬁcient has an ori-

entation dependence. In the case when a facet interface

is observed, further modiﬁcation is required in (21.16),

the detail of this modiﬁcation is discussed by Eggleston

et al. [21.35].

Gradient Energy Expression

for Multicomponent and Multiphase Systems

We ﬁrst consider the composition ﬁeld for the N

component system (i = 0 ∼ N

). The composition gra-

dient energy in this case is obtained by

grad





i=0

(∇c

)





i=1



j=1

(δ

+1)(∇c

)(∇c

)dr ,

(21.17)

where the condition



i=0

=1 is used and we assume

to be a constant. δ

is Kronecker’s delta. As for the

ternary alloy system, (21.17) is reduced to

grad

=κ





(∇c

)

+(∇c

)

+(∇c

)(∇c

)



dr .

(21.18)

Note that the cross-term (∇c

)(∇c

) appears.

On the other hand, in the case for the multiphase

system where the number of phases is i =1 ∼ N

,the

phase-ﬁeld order parameter φ

, (0 ≤φ

≤ 1) is intro-

duced, and the gradient energy is given by

grad





i=1



j=1



∇φ

−φ

∇φ



dr ,



i=1

=1 . (21.19)

This is an irreducible expression for the gradient en-

ergy [21.36].

How to Derivate the Equation

of Composition Gradient Energy

As a typical example for the derivation of a gradient

energy equation, the formulation of the composition

gradient energy that was proposed in the spinodal theory

is explained [21.28,29]. We concretely derive the com-

position gradient energy equation for the A–B binary

alloy system and the solute composition of B compo-

nent is referred to c(r, t) that is a function of local

position r and time t. In this formulation, since we

employ the composition gradient ∇c and the curvature

∇

c as invariant variables, the chemical free energy is

expanded in Taylor series form as follows



c, ∇c, ∇



= g

(c, 0, 0) +K

(c)(∇c)

(c)



∇



(c)(∇c)

(c)



∇



(c)(∇c)



∇



+···

∼

(c, 0, 0) +K

(c)(∇c)

(c)



∇



(c)(∇c)

, (21.20)

where we ignored the higher order terms in the Taylor

series. K

tion of composition. A function g

(c, 0, 0) corresponds

the chemical free energy with no composition ﬂuctua-

tion (i. e., homogeneous single phase state). Expression

(21.20) is the chemical free energy of the inho-

mogeneous system including the information of the

composition proﬁle shape.

Here, we focus on the free energy at position x

the one dimensional composition proﬁle with arbitrary

shape. The chemical free energy at x

should be writ-

ten as g



c(x

), ∇c(x

), ∇

c(x

)



. When the coordinate

conversion of a ﬂip horizontal is performed to the func-

tion g

around x

, i. e., it corresponds to the case that the

point x

is looked at from back, the chemical free energy

at x

is given as g



c(x

), −∇c(x

), ∇

c(x

)



. Since en-

ergy must be invariant with the coordinate conversion

Part E 21.2

Phase Field Approach 21.2 Total Free Energy of Microstructure 1097

because energy is a scalar quantity, it is necessary to

satisfy the following relation



c(x

), ∇c(x

), ∇

c(x

)



= g



c(x

), −∇c(x

), ∇

c(x

)



. (21.21)

Substituting (21.20)into(21.21), we obtain

{c(x

), 0, 0}+K

{c(x

)}{∇c(x

)}

{c(x

)}



∇

c(x

)



{c(x

)}{∇c(x

)}

= g

{c(x

), 0, 0}−K

{c(x

)}{∇c(x

)}

{c(x

)}



∇

c(x

)



{c(x

)}{−∇c(x

)}

∴ K

{c(x

)}{∇c(x

)}=0 → K

{c(x

)}=0 .

Hence K

should be satisﬁed at any position x

. Therefore, the

chemical free energy for an inhomogeneous system has

to be written as



c, ∇c, ∇



= g

(c, 0, 0) +K

(c)



∇



(c)(∇c)

. (21.22)

Therefore, the excess term originating in the composi-

tion ﬂuctuation is expressed as

grad





(c)



∇



(c)(∇c)





(c)



∇



dV +



(c)(∇c)

dV .

(21.23)

grad

is a composition gradient energy. Furthermore,

the ﬁrst term of the right-hand side of (21.23) is trans-

formed as



(c)



∇





(∇K

) ·(∇c)dV



∂K

∂c

(∇c)

=−



∂K

∂c

(∇c)

dV , (21.24)

where the divergence theorem of Gauss,



f ∇·gdV =



f g·ndS −



∇ f ·g dV, is utilized and n is a unit

normal at the surface of the object. The reason the term

for the surface integral in (21.24) has been eliminated

is that the integrated value over the whole surface of

the object becomes 0 statistically. Therefore, (21.23)is

rewritten as

grad



(c)(∇

c)dV +



(c)(∇c)





(c)−

∂K

∂c



(∇c)

dV . (21.25)

Deﬁning κ(c)by

κ(c) = K

(c)−

∂K

∂c

, (21.26)

ﬁnally, we obtain

grad



κ(c)(∇c)

dV , (21.27)

where κ(c) is the so-called composition gradient en-

ergy coefﬁcient. From (21.26), strictly speaking, κ(c)is

a function of a composition, but κ(c) is often assumed

to be constant and given by κ = kΩd

, where Ω is an

atomic interchange energy, d is the interatomic distance,

and k is a phenomenological constant.

21.2.3 Elastic Strain Energy

In this section, the evaluation for the elastic strain en-

ergy of the coherent two-phase microstructure in A–B

binary alloy system is explained as a simple exam-

ple [21.37,38]. The steps for evaluating the elastic strain

energy is as follows

Step 1 The eigenstrain [21.38] ε

(r)isexpressedas

a function of the local composition c(r), where r is

a position vector. The subscripts i and j take the

integer values 1, 2, and 3.

Step 2 The total strain ε

(r) is divided into two parts:

(r) =

+δε

(r), where

is the spatial average

of ε

(r), and δε

(r) is the deviation from

. There-

fore, the following relations should be satisﬁed



(r)dr,



δε

(r)dr =0 .

Step 3 The equation to calculate the displacement, that

is initially an unknown quantity, is deduced from the

mechanical equilibrium equation using the ε

(r)as

a boundary condition, then δε

(r) is calculated from

(r) directly.

Step 4 The elastic strain energy is written as a function

of ε

(r)andε

(r).

Part E 21.2