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

Подождите немного. Документ загружается.

1108 Part E Modeling and Simulation Methods

[001]

(110)

[110]

a) 219s b) 548s c) 1.64 ks

100 nm

d) 5.48 ks e) 32.9 ks f) 110 ks

Fig. 21.6a–f Two-dimensional simulation of the spinodal decom-

position of the ZrO

-8 mol % YO

1.5

partially stabilized zirconia at

1773 K. The calculation is performed on the two-dimensional plane

of (110). The completely black and white regions indicate the com-

position of 16 and 0 mol % Y, respectively. Numerical values in the

ﬁgure are the aging time

that a modulated structure, i. e., the microstructure with

ﬁne Mo-rich zones along 100 directions, is produced

by the mechanism of spinodal decomposition. Since the

volume fraction of precipitates becomes higher, the en-

counter of precipitates happens so that the rod-shaped

particles are temporarily produced in many places, as

seen in Fig. 21.4d. However, with further aging particle-

splitting can be observed in Fig. 21.4f. This suggests

that uniformity of particle size caused by elastic in-

teractions becomes more energetically dominant than

the usual particle coarsening controlled by the inter-

facial energy. Figure 21.4d–f clearly shows that the

microstructure is self-regulated by the elastic interac-

tion energy so as to produce a periodic microstructure

with uniform particle size. The identical microstruc-

tures have experimentally been recognized in many real

alloy systems such as Ni-base superalloys.

Figure 21.5 shows three-dimensional simulation of

spinodal decomposition of Fe-40 at. % Cr alloy aged at

773 K. The precipitate is represented by an isocom-

position surface of 40 at. % Cr. With aging, the ﬁne

microstructure coarsens according to the Ostwald ripen-

ing mechanism that is driven by the reduction of the

interfacial energy.

Figure 21.6 demonstrates the two-dimensional

simulation of the spinodal decomposition of the

ZrO

-8 mol % YO

1.5

partially stabilized zirconia at

1773 K [21.42]. The calculation is performed on the

two-dimensional plane of (110). The completely black

and white regions indicate the composition of 16 and

0 mol % Y, respectively. The orientation of the mod-

ulated microstructure is mainly controlled by elastic

strain energy. Since the phase-ﬁeld simulation is a con-

tinuum model, this simulation method is directory

applicable for the phase transformation in ceramics and

polymer materials.

21.5 Structural Phase Transformation

21.5.1 Martensitic Transformation

Figure 21.7 shows the simulation results of the cu-

bic → trigonal martensitic transformation in AuCd

alloys calculated by Wang et al. [21.22]. For the cubic

→ trigonal transformation, four lattice-correspondence

variants are considered, i. e., four types of martensite

orientation variants, with trigonal axes along the [111],

[

111], [

11],and[1

11] directions of the parent phase,

are numbered 1, 2, 3, and 4, and are distinguished by

shades of gray. The phase-ﬁeld order parameter used

in the simulation is the possibility of ﬁnding the trig-

onal martensite phase with variant i (i = 1, 2, 3, 4) at

position r and time t. The symbols τ and ζ indicate

the reduced time and the ratio of the typical trans-

formation strain energy to the chemical driving force,

respectively.

The microstructures for ζ =1andζ =5atτ =30

are presented in Fig. 21.7a and b, respectively. The ef-

fect of the contribution of elastic strain energy to the

total free energy is demonstrated. Since the system with

a higher value of ζ coarsens faster, Fig. 21.7a,b actually

corresponds to the coarsening process of the domain

structures. The microstructure is a herringbone type

structure with coarsened domains. The schematic illus-

tration of the arrangement of the domains in Fig. 21.7d

is represented in Fig. 21.7c, which coincides with that

observed in experiment of this alloy system [21.43].

21.5.2 Tweed-Like Structure

and Twin Domain Formations

Figure 21.8 shows the calculated microstructure

changes of the cubic (c) → tetragonal (t) structural

Part E 21.5

Phase Field Approach 21.5 Structural Phase Transformation 1109

411

a) b)

c) d)

Fig. 21.7a–d Simulated three-dimensional microstructures

in a single crystal computational volume faceted by the

{100} planes: (a) ζ = 1atτ = 30; (b) ζ = 5atτ = 30.

shown in (d), where the numbers 1, 2, 3, and 4 represent

the four types of variant domains (after [21.22])

phase transformation in Ni

MnGa during isother-

mal aging, where the ﬁnal equilibrium state is a

single t phase and the chemical driving force for

c → t phase transition is assumed to be ΔG

c→t

−40 J/mol [21.44]. This driving force seems to be rela-

tively small compared to that for the normal martensitic

transformation, because we supposed the calculation

condition of aging temperature to be just below the

c →t phase transition temperature. This calculation is

performed on the two-dimensional plane (001). The ini-

tial condition of the microstructure is a single c phase

and the grayscale of the ﬁgure means the value of

(r, t), so the white part is t

phase (e.g., t phase with

variant p is written as t

phase), and the black part cor-

responds to the t

phase. But only for Fig. 21.8a, cubic

phase is also included in the black part. Numerical val-

ues in the ﬁgure are the dimensionless aging time (the

timeunitisdescribedass



not s).

At the initial stage of phase transformation, the

tweed-like structure with 110 striation appears as

shown in Fig. 21.8a,b. From Fig. 21.8c–e, t-phase do-

mains grow gradually during aging, then a lamellar-

shaped twin microstructure emerges (Fig. 21.8e) and the

straight twin domain with narrow width begins to shrink

b) 50s' c) 100s'

d) 200s'

e) 400s' f) 1ks'

300 nm

a) 20s'

Fig. 21.8a–f Two-dimensional simulation result of the cubic →

tetragonal structural phase transformation in Ni

MnGa during

isothermal aging calculated on the two-dimensional plane of (001).

The white part is the t

phase, e.g., the t phase with variant p is

written as t

phase, and black part corresponds to the t

phase

(ΔG

c→t

=−40 J/mol). Numerical values in the ﬁgure are the di-

mensionless aging time

by twin boundary motion (Fig. 21.8f). The morpholog-

ical changes of the microstructure are in very good

agreement with the martensitic transformation induced

by the weak ﬁrst order structural phase transition that is

observed, for instance, in the Fe-Pd system experimen-

tally.

21.5.3 Twin Domain Growth Under External

Stress and Magnetic Field

Figure 21.9 shows the twin microstructure develop-

ments under external stress and magnetic ﬁelds [21.44].

The microstructure in Fig. 21.8f is employed as the ini-

tial condition for the simulation of Fig. 21.9,andthe

white and black parts in the ﬁgure correspond to the

phase and t

phase, respectively. The microstruc-

ture changes from Fig. 21.9a–c demonstrate the twin

domain growth with twin boundary motion under the

condition of 1 MPa compressive applied stress along

the vertical direction. It is recognized that the t

phase

(black part) preferentially grows upwards. Because the

c-axis of the unit cell for t

phase coincides with the

direction along the y-axis (vertical direction) and the

tetragonality of the t phase is less than unity, the com-

pressive force makes the t

phase more stable. On

the other hand, the microstructure changes under the

external magnetic ﬁeld along the x-axis (horizontal di-

Part E 21.5

1110 Part E Modeling and Simulation Methods

300 nm

a) 1ks' b) 5ks' c) 10 ks'

d) 20ks'

e) 40ks'

f) 80 ks'

Fig. 21.9a–f The twin microstructure developments under

external stress and magnetic ﬁelds. The microstructure of

Fig. 21.8f is employed as the initial condition for the sim-

ulation of Fig. 21.9,andthewhite and black parts in the

ﬁgure correspond to t

phase and t

phase, respectively.

(ΔG

c→t

=−40 J/mol). Numerical values in the ﬁgure are

the dimensionless aging time

rection) without applied stress are demonstrated from

Fig. 21.9c–f. In this case, the t

phase (white part)

grows conversely. This is because t

phase has the c-

axis, that is, the easy axis for magnetization, along

the x-axis (horizontal direction), therefore, the mag-

netic energy of the t

phase is lower than that of

phase.

21.6 Microstructure Evolution

21.6.1 Grain Growth and Recrystallization

In crystal grain growth dynamics, several phase-ﬁeld

models have been proposed with the ﬁrst being pro-

posed by Chen and Yang [21.16], in which the grains of

different crystallographic orientations are represented

by a set of nonconserved order parameter ﬁelds (mul-

tiorder parameter model). The temporal development of

a) b)

d)c)

the order parameter ﬁeld is calculated by the Allen–

Cahn equation [21.34].

On the other hand, an interesting phase-ﬁeld model

was recently proposed for studying the crystalline grain

growth [21.17, 18]. It uses two order parameters to

describe a polycrystal structure; one represents the crys-

talline order and the other indicates the value of the

local orientation of the crystal. Whereas this model can

also simulate grain rotation, which is absent from the

multiorder parameter models, this order parameter is

undeﬁned in a disordered liquid state.

A typical example of the three-dimensional grain

growth simulation by Krill and Chen [21.16]isshown

in Fig. 21.10. It was obtained using a multiorder pa-

rameter free energy model with 25 order parameters.

With increasing simulation time, the grains are elimi-

nated by a mechanism of the grain boundary migration,

and owing to the conservation of total volume, the aver-

age grain size increases steadily. The square of average

grain size increases proportional to time, that is, ex-

pected for curvature-driven grain boundary migration.

Fig. 21.10a–d A typical grain evolution process obtained

from a three-dimensional phase-ﬁeld simulation of grain

growth assuming isotropic grain boundary energy and

isotropic boundary mobility (after [21.16]). The white lines

show the grain boundary. With increasing simulation time

from

(a)–(d), the grains are eliminated by the mechanism

of grain boundary migration and, owing to the conservation

of total volume, the average grain size increases steadily

Part E 21.6

Phase Field Approach 21.6 Microstructure Evolution 1111

a)a)

120s' 140s' 160s' 180s' 200s'

200s' 300s' 400s' 500s' 600s'

Fig. 21.11a,b Two-dimensional simulation of the recrystallization process. The microstructure changes of the upper and

lower layer correspond to the case that the initial strain in the polycrystal is large and small, respectively. The gray

parts are primary grain, and the white grains correspond to recrystallized ones. Numerical values in the ﬁgure are the

dimensionless aging time

Figure 21.11 shows a two-dimensional simulation

of the recrystallization process. The microstructure

changes of upper and lower layer correspond to the case

that the initial strain in the polycrystal is large and small,

respectively. When the initial polycrystal is highly de-

formed, recrystallization frequently occurs in the place

of grain growth, on the other hand, with less deforma-

tion, grain growth takes place sparsely.

21.6.2 Ferroelectric Domain Formation

with a Dipole–Dipole Interaction

Figure 21.12 shows the temporal evolution of a di-

electric domain structure without an external electric

ﬁeld. At the beginning of the evolution, a Gaussian

random ﬂuctuation was introduced to initiate the po-

larization evolution process. The shade regions imply

the polarization domain and the orientation of polariza-

tion moment is indicated by the arrows in Fig. 21.12f.

After t = 4s



(the symbol s



means the dimensionless

time unit), the polarization is still in the nucleation state.

Figure 21.12b illustrates the polarization distribution

after t = 12 s



, where the 90 and 180

◦

domain walls ap-

pear. Figure 21.12b–f demonstrate the temporal domain

structure evolution, where the morphology of domain

structure is changed by domain wall motion and the

coarsening of domains is progressed.

a) 4s' b) 12s' c) 40s'

d) 100s' e) 200 s' f) 400 s'

50 µm

Fig. 21.12a–f Temporal evolution of the domain structure

development in dielectric materials without an external

electric ﬁeld. The shaded regions imply the polarization

domain and the orientation of the polarization moment is

indicated by arrows in (f). Numerical values in the ﬁgure

are the dimensionless aging time

Figure 21.13a–f shows the temporal evolution

of polarization switching when an external electric

ﬁeld is applied along the x-direction (rightward di-

rection). Comparing Figs. 21.12 and 21.13 indicates

that the applied electric ﬁeld drives the domain walls

to align their polarization moment along the applied

ﬁeld.

Part E 21.6

1112 Part E Modeling and Simulation Methods

a) 0s' b) 10 s' c) 30s'

d) 40s' e) 50s' f) 70s'

50 µm

Fig. 21.13a–f Temporal evolution of the domain structure

development in dielectric materials under external electric

ﬁeld in the x-direction. Note that the applied electric ﬁeld

drives the domain walls to align their polarization moment

along to the applied ﬁeld. Numerical values in the ﬁgure

are the dimensionless aging time

21.6.3 Modeling Complex Nanogranular

Structure Formation

As a typical example of modeling complex microstruc-

ture evolution using the phase-ﬁeld method, the FePt

nanogranular microstructure formation during sputter-

ing and the order–disorder phase transition of FePt

nanoparticle in nanogranular thin ﬁlms during sub-

sequent annealing are demonstrated [21.45]. FePt

nanogranular structure is well known as the candidate

for the next generation of high-density recording media

because of its large magnetocrystalline anisotropy.

The simulation is performed on the nanostructure

formation in FePt nanogranular thin ﬁlm during sput-

tering and a subsequent phase transformation of FePt

nanoparticles from A1 to L1

structure with isothermal

aging.

Figure 21.14 shows the two-dimensional simulation

result for the FePt nanogranular structure formation and

a) 100s' b) 200 s' c) 500s'

e) 1ks' f) 2ks' g) 3ks'

d) 700s'

h) 4ks'

20nm

the ordering of FePt nanoparticles at 923 K [21.45]. The

composition of the FePt phase is set at Fe-45 at. %Pt.

The black and white regions with droplet shape are FePt

nanoparticles, and the degree of white means the degree

of L1

order. Therefore, a black particle is an FePt(A1)

disordered phase, and a white particle is an FePt(L1

)

ordered phase. The matrix (gray part) is an amorphous

alumina phase. Numerical values in the ﬁgure are nor-

malized aging time.

At the initial stage (as-spattered stage), as shown in

Fig. 21.14a and b, the FePt phase takes a disordered

state. With the aging progress, it is understood from

Fig. 21.14c–e that ordering from A1 to L1

proceeds

during coarsening of the FePt phase. It is noteworthy

that the speed of ordering for a particle with small size

seems to delay (see the left arrow in Fig. 21.14e). The

antiphase boundary is also recognized in the FePt par-

ticle (see the right arrow in Fig. 21.14e). It is quite

interesting that the degree of order decreases as the par-

ticle size becomes small (see the arrowed particle in

Fig. 21.14g). This size dependence on ordering of FePt

phase has already been observed experimentally by

Takahashi et al. [21.46]. These microstructure changes

agree well with features of FePt nanogranular structure

formation experimentally observed.

We would like to emphasize that once the mor-

phological shape of a microstructure is quantitatively

obtained using phase-ﬁeld modeling, it is possible to

calculate magnetic properties such as a magnetic hys-

teresis based on the micromagnetics simulation [21.47,

48] using the calculated microstructure data as bound-

ary conditions.

21.6.4 Dislocation Dynamics

A quite advanced phase-ﬁeld modeling of the disloca-

tion dynamics has been proposed by Wang et al. [21.26].

In order to describe a multidislocation system using

a set of order parameters, they employed discontin-

uous relative displacements between the two lattice

Fig. 21.14a–h Two-dimensional simulation for the FePt

nanogranular structure formation and the ordering of FePt

(Fe-45 at. % Pt) nanoparticles at 923 K. The black and

white regions with droplet shape are FePt nanoparticles,

and the degree of white means the degree of L1

order.

Therefore, a black particle is FePt(A1) disordered phase,

and a white particle is FePt(L1

) ordered phase, respec-

tively. The matrix (gray part) is the amorphous-alumina

phase. Numerical values in the ﬁgure are the dimenionless

aging time

Part E 21.6

Phase Field Approach 21.6 Microstructure Evolution 1113

planes below and above the slip plane, measured in

units of Burgers’ vectors, as a phase-ﬁeld order param-

eters. The temporal evolution of the order parameter

ﬁelds is obtained by solving the governing equation

for nonconserved order parameters. This model not

only takes into account the stress-induced long-range

interactions among individual dislocations but also

automatically incorporates the effects of short-range

interactions, such as multiplication and annihilation be-

tween dislocations.

Figure 21.15 shows a three-dimensional simulation

of Frank–Read source calculated by Wang et al. The pe-

riodic boundary condition is used and the rectangular

shaped loop is introduced as the pinned source segments

initially. In terms of inclusion formalism, a succes-

sion of dislocation generation through the Frank–Read

mechanism should be interpreted as sympathetic nucle-

ation of new thin lamellar inclusions.

Figure 21.16 shows an example from a phase-ﬁeld

simulation of plastic deformation of a model alloy un-

der a uniaxial tensile stress imposed along vertical

direction. The temporal dislocation generation during

deformation was provided by randomly placed Frank–

Read sources in the system. Figure 21.16a,b shows

the dislocation microstructure at points a and b on

the stress–strain curve. This model can be extended to

the polycrystal and to systems where phase transfor-

mations and dislocation dynamics are simultaneously

evolved.

Fig. 21.15 Three-dimensional phase-ﬁeld simulation of

a Frank–Read source under a periodic boundary condition,

where the rectangular loop (thin plate inclusion) serves as

the pinned source segments (after [21.23])

/10

–4

/10

–3

a) b)

01234

Fig. 21.16a,b The stress–strain curve obtained from a three-

dimensional phase-ﬁeld simulation of an fcc crystal under

uniaxial loading (after [21.23]). The ﬁgures (a) and (b)

show the dislocation microstructure at points a and b on

the stress–strain curve

21.6.5 Crack Propagation

The phase-ﬁeld simulation is also applicable to mul-

ticrack systems in polycrystalline materials if elastic

isotropy is assumed [21.26]. In this case, the grain

rotation does not change the components of the elas-

tic modulus, and thus the polycrystal can be treated

as an elastically homogeneous medium. However, the

grain rotation in the polycrystal changes the crack

propagation paths because it rotates the direction of

permitted cleavage planes. Since the cleavage planes

have different orientations in different grains, the crack

picks up new energetically favored cleavage planes

when it crosses the grain boundaries. The phase-ﬁeld

model explicitly takes into account this effect as well

as the elastic coupling between cracks in different

grains.

Figure 21.17 shows the phase-ﬁeld simulation of

crack propagation in a two-dimensional polycrystal cal-

culated by Wang et al. [21.26]. The initial polycrystal

Part E 21.6

1114 Part E Modeling and Simulation Methods

is generated by the Voronoi tessellations under a peri-

odic boundary constraint. The simulation sequence is

described brieﬂy as follows: (1) 40 grain seeds are ran-

domly positioned in the 512× 256 computational cell;

Fig. 21.17a–f Phase-ﬁeld simulation of crack propagation

in two-dimensional polycrystal. Under uniaxial stress in

the y-direction (vertical direction of the ﬁgure), the pre-

existing crack (a) starts to grow. When it crosses the grain

boundary, the crack changes its original propagation path

and picks up the energetically favored cleavage planes in

the neighboring grains (b)–(f). The resultant meandering

path is shown in (f) (after [21.26])

(2) each point in the space belongs to the grain whose

seed is the nearest to this point; and (3) a rotation angle

is randomly assigned to each grain. The generated poly-

crystalline structure composed of 40 randomly oriented

grains is shown as the dotted lines in Fig. 21.17.Two

cleavage planes, (100) and (010), are assumed for each

grain.

Uniaxial stress is applied to the polycrystal in

the y-direction (vertical direction), and as shown in

Fig. 21.17a, a preexisting crack is introduced in one

grain. The simulation results are shown in Fig. 21.17a–f.

When the crack crosses through the grain boundary

and enters the neighboring grain, the energetically fa-

vored cleavage plane is automatically picked up. This

results in the meandering cleavage path, as shown in

Fig. 21.17f.

References

21.1 R. Kobayashi: Modeling and numerical simulations

of dendritic crystal growth, Physica D 63,410–423

(1993)

21.2 A.A. Wheeler, W.J. Boettinger, G.B. McFadden:

Phase-ﬁeld model for isothermal phase transitions

in binary alloys, Phys. Rev. A 45, 7424–7439 (1992)

21.3 W.J. Boettinger, J.A. Warren, C. Beckermann,

A. Karma: Phase-ﬁeld simulation of solidiﬁcation,

Annu.Rev.Mater.Res.A32,163–194(2002)

21.4 L.-Q. Chen: Phase-ﬁeld models for microstructure

evolution, Annu. Rev. Mater. Res. 32, 113–140 (2002)

21.5 M. Ode, S.G. Kim, T. Suzuki: Recent Advances in

the phase-ﬁeld model for solidiﬁcation, Iron Steel

Inst.Jpn.Int.41, 1076–1082 (2001)

21.6 H. Emmerich: The Diffuse Interface Approach in

Materials Science (Springer, Berlin, Heidelberg

2003)

21.7 D. Raabe: Computational Materials Science (Wiley-

VCH, Weinheim 1998)

21.8 J.J. Robinson (Ed.): Solidiﬁcation and microstruc-

ture,J.Min.Met.Mater.Soc.56(4), 16–68 (2004)

21.9 J.A. Warren, W.J. Boettinger: Prediction of dendritic

growth and microsegregation patterns in a binary

alloy using the phase-ﬁeld method, Acta Metall.

Mater. 43, 689–703 (1995)

21.10 A. Karma, W.-J. Rappel: Quantitative phase-ﬁeld

modeling of dendritic growth in two and three

dimensions, Phys. Rev. E 57, 4323–4349 (1998)

21.11 S.G. Kim, W.T. Kim, T. Suzuki: Phase-ﬁeld model

for binary alloys, Phys Rev. E 60, 7186–7197 (1999)

21.12 T.Suzuki,M.Ode,S.G.Kim:Phase-ﬁeldmodelof

dendritic growth, J. Cryst. Growth 237–239,125–131

(2002)

21.13 T. Miyazaki, T. Koyama: Computer simulations of

the phase transformation in real alloy systems

based on the phase ﬁeld method, Mater. Sci. Eng.

A 312, 38–49 (2001)

21.14 T. Koyama: Computer simulation of phase decom-

position in two dimensions based on a discrete

type non-linear diffusion equation, 39,169–178

(1998)

21.15 J.Z. Zhu, T. Wang, A.J. Ardell, S.H. Zhou, Z.K. Liu,

L.Q. Chen: Three-dimensional phase-ﬁeld simula-

tions of coarsening kinetics of γ



particles in binary

Ni-Al alloys, Acta Mater. 52, 2837–2845 (2004)

21.16 C.E. Krill III, L.-Q. Chen: Computer simulation of

3-D grain growth using a phase-ﬁeld model, Acta

Mater. 50, 3059–3075 (2002)

21.17 A.E. Lobkovsky, J.A. Warren: Phase-ﬁeld model of

crystal grains, J. Cryst. Growth 225, 282–288 (2001)

Part E 21

Phase Field Approach References 1115

21.18 J.A. Warren, R. Kobayashi, W.C. Carter: Modeling

grain boundaries using a phase-ﬁeld technique,

J. Cryst. Growth 211, 18–20 (2000)

21.19 T. Miyazaki: Recent developments and the future

of computational science on microstructure forma-

tion, Mater. Trans. 43, 1266–1272 (2002)

21.20 J. Wang, S.-Q. Shi, L.-Q. Chen, Y. Li, T.-Y. Zhang:

Phase-ﬁeld simulations of ferroelectric/ferroelastic

polarization switching, Acta Mater. 52,749–764

(2004)

21.21 Y.L. Li, S.Y. Hu, Z.K. Liu, L.-Q. Chen: Effect of sub-

strate constraint on the stability and evolution of

ferroelectric domain structures in thin ﬁlms, Acta

Mater. 50, 395–411 (2002)

21.22 Y.M. Jin, A. Artemev, A.G. Khachaturyan: Three-

dimensional phase ﬁeld model of low-symmetry

martensitic transformation in polycrystal: Simula-

tion of ζ



martensite in AuCd alloys, Acta Mater. 49,

2309–2320 (2001)

21.23 Y.U.Wang,Y.M.Jin,A.M.Cuitino,A.G.Khachatu-

ryan: Nanoscale phase ﬁeld microelasticity theory

of dislocations: Model and 3D simulations, Acta

Mater. 49, 1847–1857 (2001)

21.24 D. Rodney, Y. Le Bouar, A. Finel: Phase ﬁeld meth-

ods and dislocations, Acta Mater. 51, 17–30 (2003)

21.25 S.Y. Hu, L.-Q. Chen: A phase-ﬁeld model for

evolving microstructures with strong elastic inho-

mogeneity, Acta Mater. 49, 1879–1890 (2001)

21.26 Y.U. Wang, Y.M. Jin, A.G. Khachaturyan: Phase ﬁeld

microelasticity theory and simulation of multiple

voids and cracks in single crystals and polycrystals

under applied stress, J. Appl. Phys. 91, 6435–6451

(2002)

21.27 J.S. Rowlinson: Translation of J.D. van der Waals’

“The thermodynamic theory of capillarity under

the hypothesis of a continuous variation of den-

sity”, J. Stat. Phys. 20, 197–244 (1979)

21.28 W.C. Carter, W.C. Johnson (Eds.): The Selected Works

of J.W. Cahn (Min.Met.Mater.Soc.,Warrendale

1998)

21.29 H.I. Aaronson (Ed.): Phase Transformation (ASM,

Metals Park 1970) p. 497

21.30 C. Kittel, H. Kroemer: Thermal Physics (Freeman,

New York 1980)

21.31 D. Fan, L.-Q. Chen: Topological evolution dur-

ing coupled grain growth and Ostwald ripening

in volume-conserved 2-D two-phase polycrystals,

Acta Mater. 45, 4145–4154 (1997)

21.32 D. Fan, L.-Q. Chen: Possibility of spinodal de-

composition in yttria-partially stabilized zirconia

(ZrO

-Y

) system – A theoretical investigation,

J. Am. Ceram. Soc. 78, 1680–1686 (1995)

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

tion of Phase Diagrams): A Comprehensive Guide

(Pergamon, New York 1998)

21.34 S.M. Allen, J.W. Cahn: A microscopic theory for an-

tiphase boundary motion and its application to

antiphase domain coarsening, Acta Metall. Mater.

27, 1085–1095 (1979)

21.35 J.J. Eggleston, G.B. McFadden, P.W. Voorhees: A

phase-ﬁeld model for highly anisotropic interfa-

cial energy, Physica D 150, 91–103 (2001)

21.36 I. Steinbach, F. Pezzolla, B. Nestler, M. Seeselberg,

R. Prieler, G.J. Schmitz, J.L.L. Rezende: A phase ﬁeld

concept for multiphase systems, Physica D 94,135–

147 (1996)

21.37 A. Khachaturyan: Theory of Structural Transforma-

tions in Solids (Wiley, New York 1983)

21.38 T. Mura: Micromechanics of Defects in Solids,2nd

edn. (Kluwer, Dordrecht 1991)

21.39 L.-Q. Chen, J. Shen: Applications of semi-implicit

Fourier-spectral method to phase ﬁeld equations,

Comput. Phys. Commun. 108, 147–158 (1998)

21.40 P.H. Leo, J.S. Lowenngrub, H.J. Jou: A diffuse

interface model for microstructural evolution in

elastically stressed solids, Acta Mater. 46, 2113–2130

(1998)

21.41 M.E. Glicksman: Diffusion in Solids (Wiley, New York

2000)

21.42 M. Doi, T. Koyama, T. Kozakai: Experimental and

theoretical investigation of the phase decompo-

sition in ZrO

-YO

1.5

system, Proc. Fourth Pac. Rim

Int. Conf. Adv. Mater. Process (PRICM 4), Honolulu

2001, ed. by S. Hanada, Z. Zhong, S.W. Nam, R.N.

Wright (The Japan Institute of Metals, Sendai 2001)

741–744

21.43 K. Otsuka, C.M. Wayman (Eds.): Shape Memory Ma-

terials (Cambridge Univ. Press, Cambridge 1998)

21.44 T. Koyama, H. Onodera: Phase-ﬁeld simulation of

microstructure changes in Ni

MnGa ferromagnetic

alloy under external stress and magnetic ﬁelds,

Mater. Trans. JIM 44, 2503–2508 (2003)

21.45 T. Koyama, H. Onodera: Modeling of microstructure

changes in FePt nano-granular thin ﬁlms using the

phase-ﬁeld method, Mater. Trans. JIM 44,1523–

1528 (2003)

21.46 Y.K. Takahashi, T. Koyama, M. Ohnuma, T. Ohkubo,

K. Hono: Size dependence of ordering in FePt

nanoparticles, J. Appl. Phys. 95, 2690–2696 (2004)

21.47 A. Hubert, R. Schafer: Magnetic Domains (Springer,

Berlin, Heidelberg 1998)

21.48 H. Kronmüller, M. Fähnle: Micromagnetism and

the Microstructure of Ferromagnetic Solids (Cam-

bridge Univ. Press, Cambridge 2003)

Part E 21

1117

Monte Carlo Si

22. Monte Carlo Simulation

In the general overview on materials and their

characteristics, outlined in Sect. 1.3, it has been

stated that materials and their characteristics

result from the processing of matter.Thus,con-

densed matter physics is one of the fundamentals

for the understanding of materials. The Monte

Carlo Method, which is a powerful method in this

respect, is presented in this ﬁnal chapter of the

Handbook’s Part E on Modelling and Simulation

Methods as follows:

First, the principles of this simulation technique

are introduced

•

Monte Carlo Method: the fundamentals

•

Improved Monte Carlo algorithms

•

Quantum Monte Carlo Method.

Second, the application of the Monte Carlo

Method is explained in considering selected areas

of materials science.

•

Electronic correlations: antiferromagnetism

•

Perfect conductance of electricity: super-

conductivity

•

Vortex states in condensed matter physics

•

Quantum critical phenomena.

22.1 Fundamentals of the Monte Carlo Method1117

22.1.1 Boltzmann Weight........................1118

22.1.2 Monte Carlo Technique .................1118

22.1.3 Random Numbers ........................1119

22.1.4 Finite-Size Effects ........................1120

22.1.5 Nonequilibrium Relaxation

Method.......................................1121

22.2 Improved Algorithms ............................1121

22.2.1 Reweighting Algorithms ................1121

22.2.2 Cluster Algorithm and Extensions ...1122

22.2.3 Hybrid Monte Carlo Method ...........1123

22.2.4 Simulated Annealing

and Extensions ............................1124

22.2.5 Replica Monte Carlo......................1125

22.3 Quantum Monte Carlo Method ...............1126

22.3.1 Suzuki–Trotter Formalism..............1126

22.3.2 World-Line Approach....................1126

22.3.3 Cluster Algorithm .........................1127

22.3.4 Continuous-Time Algorithm...........1128

22.3.5 Worm Algorithm...........................1129

22.3.6 Auxiliary Field Approach ...............1130

22.3.7 Projector Monte Carlo Method........1130

22.3.8 Negative-Sign Problem .................1132

22.3.9 Other Exact Methods.....................1133

22.4 Bicritical Phenomena in O(5) Model ........1133

22.4.1 Hamiltonian ................................1134

22.4.2 Phase Diagram.............................1134

22.4.3 Scaling Theory .............................1135

22.5 Superconductivity Vortex State...............1137

22.5.1 Model Hamiltonian ......................1138

22.5.2 First Order Melting........................1138

22.5.3 Continuous Melting: B||ab Plane ....1141

22.6 Effects of Randomness in Vortex States...1143

22.6.1 Point-Like Defects ........................1143

22.6.2 Columnar Defects .........................1144

22.7 Quantum Critical Phenomena ................1146

22.7.1 Quantum Spin Chain.....................1146

22.7.2 Mott Transition ............................1147

22.7.3 Phase Separation .........................1148

References ..................................................1149

22.1 Fundamentals of the Monte Carlo Method

In material science, we are always facing systems

composed of a huge number of atoms or molecules, typ-

ically of order 10

. Knowing the interactions among

the particles, one can, in principle, predict the individual

trajectories from a given initial conﬁguration, speciﬁed

by the positions and the velocities of the particles (for

quantum systems the initial wave functions), and thus

know the destination of the system. However, even with

the most advanced supercomputer, the system one can

treat in this way is very much limited. On the other

hand, in most cases, we are interested in the total, aver-

aged behaviors of the system, such as density of gas

Part E 22