ma-

Ag

TH,

ginee

vised

line 1

A phase-eld method, based on a Gibbs energy functional, is formulated for c! a transformation in FeC. The derived phase-

The transformation of austenite to ferrite upon

growth rates that essentially agree with the experimen-

tally observed ones. At higher undercoolings ferrite ra-

ther grows with a plate-like so-called acicular or

Widmanstatten morphology [1]. On the broad sides theaustenite/ferrite phase interface is partly coherent and

[2]. Analytical solutions based on carbon diusion

For the transition from allotriomorphic to plate-like

growth at least two dierent opinions have been ex-

pressed. Townsend and Kirkaldy [4] suggested that

plates would develop from grain-boundary allot-riomorphs by a morphological instability of a similar

type as discussed by Mullins and Sekerka [5] during

solidication. On the other hand, Aaronson et al. [6]

5406*cooling is one of the most studied and well documented

subjects in physical metallurgy. Nevertheless, some as-

pects, which are technologically important as well as of

fundamental interest, still remain less well understood

and they often lead to controversies. It is generally ac-

cepted that at low undercooling more or less equiaxed

and rather coarse ferrite particles form along austenitegrain boundaries, so-called allotriomorphic ferrite. They

grow with a rate controlled by carbon diusion in aus-

tenite. Calculations based on carbon diusion and local

equilibrium at the austenite/ferrite phase interface yield

control and the Ivantsov solution seem capable of rep-

resenting the experimentally observed growth rates if

local equilibrium is assumed and proper account is ta-

ken for the eect of interfacial energy at the curved tip.

At even higher undercooling the ferrite growth turns

partitionless, i.e. there is no redistribution of carbon,

and results in a characteristic blocky microstructure, so-called massive transformation. The transition to parti-

tionless transformation was recently analyzed by the

present authors by means of the phase-eld method and

solute drag modeling [3].eld model reproduces the following important types of phase transitions: from C diusion controlled growth through Wid-

manstatten microstructures to massive growth without partitioning of C. Applying thermodynamic functions assessed by theCalphad technique and diusional mobilities available in the literature, we study two-dimensional growth of ferrite side plates

emanating from an austenite grain boundary. The morphology of the ferrite precipitates is dened by a highly anisotropic interfacial

energy. As large values of anisotropy lead to an ill-posed phase-eld equation we present a regularization method capable of cir-

cumvent non-dierentiable domains of interfacial energy.

2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Phase-eld method; Widmanstatten plates; Diusion; Morphological instability

1. Introduction already a long time ago the KS orientation relationship

between austenite and ferrite was reported by Mehl et al.On the formation of WidFeC phase

Irina Loginova a,*, John

a Department of Mechanics, Kb Department of Materials Science and En

Received 10 November 2003; received in re

Available on

Abstract

Acta Materialia 52 (2004) 405Corresponding author. Tel.: +46-879-068-71; fax: +46-879-698-50.

E-mail address: irina@mech.kth.se (I. Loginova).

1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. Adoi:10.1016/j.actamat.2004.05.033nstatten ferrite in binaryeld approach

ren b, Gustav Amberg a

S-100 44 Stockholm, Sweden

ring, KTH, S-100 44 Stockholm, Sweden

form 10 May 2004; accepted 11 May 2004

9 June 2004

3

www.actamat-journals.comsuggest that a plate on a grain boundary allotriomorph

ll rights reserved.

additional variable, the phase-eld, is introduced for thesole purpose of avoiding explicit tracking of the position

4056 I. Loginova et al. / Acta Materiof the evolving phase boundary. The derivation of the

phase-eld equations is based on thermodynamic prin-

ciples; the coecients of the equations are chosen to

match the corresponding parameters in the conventional

sharp-interface equations through asymptotic analysis.

The phase-eld method has also been successfully ap-

plied when predicting microstructures during solid-statetransformations [25] as well as other solid-state processes

such as grain growth and coarsening [26], facet formation

[19,20], multicomponent interdiusion [27], etc.

However, to the authors knowledge, the formation

of Widmanstatten ferrite in binary FeC has not yetattracted the attention of the phase-eld community.

As mentioned, Loginova et al. [3] derived a phase-eld

model for c! a transformation in binary FeC. Theyperformed one-dimensional simulations and demon-

strated a transition between diusion controlled and

massive growth. A transition from diusion controlled

to a massive transformation is predicted when the

temperature falls below the T0 line and close toa=a c phase boundary. In the present study we ex-tend their phase-eld approach [3] from one to two

dimensions in order to investigate growth of Wid-manstatten plates emanating from an austenite grainboundary. We introduce a highly anisotropic non-

dierentiable interfacial energy and describe its regu-

larization and numerical treatment.

2. Phase-eld model

The phase-eld formulation of the isobarothermal

c! a transformation is based on the Gibbs energyfunctional

G ZX

Gm/; uC; T Vm

2

2jr/j2

dX; 1

where Gm denotes the Gibbs energy per mole of substitu-tional atoms and Vm is the molar volume of substitutionalatoms and will be approximated as constant, / is thephase-eld variable varying smoothly between 0 in ferriteforms by a nucleation process, so-called sympathetic

nucleation.

In this paper we shall investigate the morphological

instability hypothesis for plate-like growth by means of

the phase-eld approach which is particularly suitablefor modeling pattern formation during phase transfor-

mations. It has been very successful when studying the

morphological instability during crystallization of a li-

quid and the subsequent dendritic growth, see for ex-

ample [12,13]. In this approach the interface between the

phases is treated as a region of nite width having

gradual variation of the dierent state variables. Anand 1 in austenite. The temperature T is assumed to beconstant due to the rapid heat conduction. The u-fraction

uC is dened from normal mole fraction of C, xC as

uC xC1 xC : 2

The parameter is related to the interfacial energy rand the interface thickness d by means of 2 3 2p rd[3]. In the case of Widmanstatten plates the coherentbroad sides should have quite a low interfacial energy

whereas the more or less incoherent tip would have a

much higher interfacial energy. The interfacial energy

thus is highly anisotropic, i.e. it depends strongly on the

orientation of the phase boundary

r r0gh; 3where h arctan/y=/x approximates the angle be-tween the interface normal and the x axis. r0 is themaximum interfacial energy and is an input parameter.

The anisotropy function 0 < gh6 1 will be discussedin the following section. The thickness of the interface

also varies due to anisotropy. In general we expect a

coherent interface to be much thinner than an incoher-

ent interface. For the perfectly coherent interface the

thickness would vanish. For simplicity we have chosento represent the anisotropy in interface thickness with

the same function, i.e.

d d0gh; 4where d0 is the thickness of the incoherent interface andis taken as an input parameter. For the anisotropic case

we thus have 2 3 2p r0d0gh2.The molar Gibbs energy Gm is postulated as a func-

tion of the phase-eld variable

Gm 1 p/Gam p/Gcm g/W ; 5where

g/ /21 /2; 6

p/ /310 15/ 6/2 7and the choice of the parameter W was described in [3].Here we have assumed that it is constant and equal to

the value of the incoherent interface, i.e. W 6r0Vm=

2

pd0. Gam and G

cm denote the normal Gibbs en-

ergy of the a and c phases and are taken from the as-sessment of Gustafson [8]. The complete expressions are

given in Appendix A. The evolution of the non-con-served phase-eld variable is governed by the Cahn

Allen equation [7]

_/M/dGd/M/ 1Vm

oGmo/

2r2/ o

ox0h

o/oy

ooy

0ho/ox

:

8The kinetic parameter M/ is related to the interfacialmobility M as M/ 0:235M=d [3]. Taking the derivative

alia 52 (2004) 40554063of Gm with respect to / gives us the phase-eld equation

such very thin interfaces, which implies not only the use

of highly dense grids but also the solution of very sti

I. Loginova et al. / Acta Materi_/ M/ 2r2/

oox

0ho/oy

ooy

0ho/ox

M/ p0/Gam GcmVm

g0/ W

Vm

: 9

The evolution of the concentration eld is governed by

the normal diusion equation. Assuming approximately

constant molar volume and introducing u-fractions the

normal diusion equation can be written as

_uCVm

r JC: 10The diusional ux of carbon JC is given by the Onsagerlinear law of irreversible thermodynamics

JC L00r dGduC

: 11

If the gradient terms in Eq. (1) are neglected, we nd

dG=duC oGm=ouC, which is the normal chemical po-tential lC of carbon. Eq. (11) may be expanded in termsof concentration and phase-eld gradients

JC 1Vm DCruC L00r o

2GmouCo/

r/: 12The rst term corresponds to the normal Ficks law and

we may thus identify the normal diusion coecient of

C as

DC VmL00 o2Gmou2C

: 13The second-order derivative corresponds to Darkens

thermodynamic factor and the parameter L00 is related tothe diusional mobility MC by means of

L00 uCVm

yvaMC; 14where yva denotes the fraction of vacant interstitials, i.e.1 uC for c and 1 uC=3 for a. For a given C contentthe fraction of vacancies would thus depend on the

character of the phase, i.e. it will depend on the phase-

eld variable. We have postulated that

uCyva 1 p/uC1 uC=3 p/uC1 uC:15

Taking into account that the diusional mobility in the

two phases could dier by several orders of magnitude,

we have chosen the following combination:

MC MaC1p/M cCp/: 16The mobilities of carbon in a and c as functions oftemperature and uC are taken from Agren [9,10] andpresented in Appendix B. Finally, substituting Eqs.

(11)(16) into Eq. (10), we obtain the diusion equation

_uC r MaC1p/M cCp/1

p/uC1 uC=3

p/uC1 uC 1Vmo2Gmou2C

ruC

o2Gm

ouCo/r/

:17equations (since the properties of the system dier sig-

nicantly in the two directions), is extremely time-con-

suming. Applying the sophisticated computational

technique described in Section 4 and using high per-

formance computers we managed to perform simula-

tions with as thin interfaces as 2 nm, which is aremarkable achievement compared to simulations of

dendritic growth, where d0 is typically 10100 timeslarger than its physical value.

In the case of dilute binary alloy solidication,

Almgren [15] demonstrated that large values of interface

thicknesses give rise to non-physical eects such as

interface stretching and solute diusion. Karma and co-

workers [16,17] proposed that introducing a phenome-nological antitrapping current eliminates these articial

eects even for mesoscale values of the interface thick-

ness. Lan and Shih [18] demonstrated numerically that

the presence of an antitrapping term recovers realistic

morphologies and driving forces of dendritic growth

with d0 70 times larger than in reality. As in the presentstudy the interface thickness is set to values close to the

physical value, the eects of interface stretching andsurface diusion have little impact on the obtained re-

sults. However, introducing an antitrapping term into

our model might be a future issue.

3. Anisotropy of the interfacial energy

As mentioned, Widmanstatten growth is character-ized by a strong anisotropy in the interfacial properties.

Such strong anisotropy of the kinetic coecient [20] and

the interfacial energy [19] were recently studied in a case

of faceted solidication. The facets are formed when the

anisotropy function has a narrow minimum or a cusp in

a certain direction. In solidication the amorphous li-

quid is usually isotropic and directions only need to be

expressed relative the lattice of the growing crystal. Inthe case of a solidsolid transformation the situation is

more complex because the anisotropy depends on theThe presented phase-eld model is based on a Calphad

type of thermodynamical description and uses realistic

functions of the interface mobility and diusion coe-

cients. Consequently, the model is capable to produce

quantitative results if the interface thickness d0 is set tophysically realistic values which are in the regime of 0.5

1.5 nm. Applying such thin interfaces is only computa-

tionally feasible for planar fronts in one dimension. The

present authors used d0 1 nm to study transition to amassive transformation in 1D [3] and obtained results

which are in satisfactory quantitative agreement with

the sharp-interface solute-drag model by Odqvist [14].

However, in two dimensions, the numerical treatment of

alia 52 (2004) 40554063 4057relative orientation of the crystalline lattices of the two

phases as well as the orientation of the phase interface

itself. For a two-dimensional case we thus need two

parameters to represent the orientation of the interface.

The parameter h has already been introduced as theangle between the interface normal and the x axis. Wenow consider cases where the orientation relation be-

tween ferrite and austenite is such that it is always

possible to have a good crystallographic t along an

interfacial plane having the angle h0 with the x axis. If aplate with coherent sides develops we expect it to grow

with that angle toward the x axis. We have thus used amodied anisotropy function presented in [19]

gh 11 c 1 cjcosh h0j: 18

In the above expression c denes the amplitude of theanisotropy. It should be emphasized that with this

choice of the anisotropy function, see Eq. (3), the

maximum interfacial energy that represents the inco-

herent part of the interface is simply r0, while the vari-ation of c only aects the minimum interfacial energyrepresenting the coherent part, rmin r0=1 c. For

the present calculations, the smallest and largest grid

4058 I. Loginova et al. / Acta Materithe simulations to be presented later we have rather

arbitrarily taken r0 1 Jm2.One important aspect of applying strong anisotropy

is to check for what values of c the term g g00 in theanisotropic extension of the GibbsThomson relation

remains positive [21]. Given this choice of the anisotropy

function, g g00 1=1 c > 0 for any non-negative c.This is true everywhere, except at the cuspsh h0 np=2, where the rst derivative g0h is dis-continuous. A way to circumvent this problem is to

smooth the cusps by replacing gh with a smooth

2 1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

0

Fig. 1. The solid plot represents the anisotropy function g calculatedwith c 10. The dashed line shows the corresponding regularizedfunction gr. A large angle ~h p=10 was used in order to visualize the

smoothing of the cusps.resolution was 0.25l and 8l, respectively. The choice of

numerical parameters was veried by comparison with

the results of 1D calculations [3] in the case of diusion

controlled and massive growth.In the present simulations, the initial state of the

system was homogeneous austenite, except for a thin

layer of ferrite on the bottom of the domain. The initial

composition of C in ferrite was always ua1C 0:001,whereas initial uc1C and the temperature diered (seeFig. 2). Fig. 2 shows the FeC phase diagram with im-

posed operating points for dierent types of phase

transformations, discussed in the following sections. Inorder to observe growth of the precipitates, the phasefunction where h h0 is close to np=2. The regularizedanisotropy function grh is dened as follows:grh

1

1 c

1BA sinh h0; p=26h h06 p=2 ~h;1 ccosh h0; p=2 ~h< h h0 < p=2 ~h;1BA sinh h0; p=2 ~h6h h06p=2

8>:

19with A c cos~h= sin~h and B c= sin~h, where~h p=200 is a smoothening angle. The anisotropyfunction and its regularization is illustrated in Fig. 1.

4. Numerical issues

For convenience, the governing equations, Eqs. (9)

and (17), are transformed into dimensionless form.

Length and time have been scaled with a reference

length l 0:9d0 and the diusion time l2=RTMaC, re-spectively. The non-dimensionalized equations are

solved by the nite element method on adaptive un-structured grids. A rst-order semi-implicit time scheme

is used for the diusion equation, while the phase-eld

equation demonstrates very sti properties and needs to

be solved with fully implicit time-stepping. The resulting

system of non-linear equations is solved iteratively by

the NewtonRaphson method. The complete Fortran/C/

C++ code was generated automatically by the symbolic

computational tool femLego [23]. The discretizedproblem was solved in parallel, typically on eight pro-

cessors with the dynamic load balancing performed after

every grid renement [24].

As it is characteristic for the phase transformations,

the variation of uC and / are highly localized over thephase interface. The width of interface is much smaller

than the other length scales in the system, which makes

the use of mesh adaptivity benecial. The mesh distri-bution follows the evolution of the interface: the phase

boundary region has the highest resolution, while the

rest of the domain is discretized with large triangles. In

alia 52 (2004) 40554063boundary was perturbed from a planar shape with a

and c 20 is illustrated in Fig. 4, where a time sequenceof the phase boundary dened as / 0:5 is presented.In agreement with experimental observations, e.g.

aterisingle sinusoidal wave of length 12l and amplitude 6l.

Zero-ux boundary conditions were imposed for both

variables.

5. Formation of Widmanstatten plates

5.1. Initiation

In all simulations a thin layer of ferrite was initially

put along the x axis which we take as the prior austenitegrain boundary. First, we study the initiation of a single

Widmanstatten plate with h0 p=2, i.e. it will growperpendicularly to the grain boundary. The simulations

presented in this section were performed for T 993 Kand an alloy content of uc1C 0:01, i.e. 0.22 mass% C

0 0.005 0.01 0.015980

1000

1020

1040

1060

1080

1100

1120

1140

1160

tem

pera

ture

, K

u fraction C

A B

C

Fig. 2. FeC phase diagram. The superimposed dashed line shows

transition to partitionless transformation [3]. The points A, B and C

specify the operating points for massive transformation, Wid-

manstatten plates and diusion controlled growth, respectively.

I. Loginova et al. / Acta M(point B in Fig. 2). The results from a large number of

simulations will now be summarized. First it should be

emphasized that the anisotropies which will now be

considered, 06 c6 100 are much stronger than usuallyconsidered during dendritic solidication where c is 3 or4 orders of magnitude lower. The high anisotropy turns

out to be the key to understand the initiation of Wid-

manstatten growth. Our simulations demonstrate thatWidmanstatten plates will only develop if the interfacialenergy of the coherent sides, i.e. r0=1 c, is below acritical value, i.e. the anisotropy amplitude c should belarger than ccritical. If c < ccritical, the initial perturbationdecays and we observe a classical diusion controlled

phase transformation with a planar interface, i.e. the

grain boundary allotriomorph. It was also found, that

the critical value of c depends on d0, which is treated asan input parameter in the present phase-eld formula-

tion. We vary d0 from 10 nm down to 2 nm and forevery case we nd the smallest value of ccritical abovewhich the Widmanstatten morphology is the stablegrowth mode. Fig. 3 shows that 1=1 ccritical dependslinearly on d0. Some discrepancy from the linear be-havior of the data can be explained by the fact, that we

used only integer values of c to dene ccritical. The pa-rameter ccritical varies from 43 to 10 for d0 2 tod0 10 nm, respectively.

5.2. Characteristics of growth

The growth of a Widmanstatten plate for d0 5 nm

0 2 4 6 8 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

, nm

1/ (1

+)

Fig. 3. The maximal interfacial energy of coherent sides for Wid-

manstatten growth as a function of d0. States above the line representallotriomorphic growth and states below the line Widmanstatten

growth.

alia 52 (2004) 40554063 4059[28,29], the tip of the plate grows with a constant ve-

locity, while the sides grow parabolically. After a short

transition period, the plate propagates with a steady-

state interface shape which can be divided into threedistinct parts: a circular tip, planar sides diverging at a

small angle and a bottom part, where the sides are

parallel to each other. Except for the tip this shape is

rather realistic. In micrographs the observed tip is much

sharper than simulated, see Fig. 13.

The phase-eld and diusion elds for the nal time

are presented in Fig. 5. One observes the build-up of C

in austenite at the sides and the increase of the diusionlength downwards from the tip. The distribution of the

elds at the tip, Fig. 6, deserves a special consideration.

The large variation in the interface width (the width of

the transition layer in the phase-eld variable) can be

explained by the fact that the interface width varies with

orientation and is proportional to the anisotropy func-

tion gh [21]. We can dene the tip radius of the plate asthe one of the isoline / 0:5, however it would give us avalue for the tip radius as 3 4d0, see Fig. 8, which is

smaller than the interface width in the direction of

growth. This makes it inadequate to talk about the tip

radius as such, but rather consider the tip as being sharp

which is consistent with metallographic observations.

One may then seriously question the applicability of theGibbsThomson relation to Widmanstatten growth andthe whole classical Zener-like theory where the tip radius

plays the essential role. In the classical approach the

Ivantsov solution yields the growth rate of the tip as

inversely proportional to the tip radius rather than a

unique rate and tip radius. When account is taken for

the eect of interfacial energy by means of the Gibbs

Thomson relation one nds a critical radius below whichthe tip cannot grow and a radius at which the tip grows

with a maximum rate. Usually one then assumes that the

tip would grow with this maximum rate and the corre-

sponding tip radius, the Zener maximum growth-rate

hypothesis.

In our phase eld calculation there is no need for such

Fig. 5. Distributions of the phase-eld to the left and concentration to

the right. The distributions correspond to the time evolution in Fig. 4.

Fig. 4. Time evolution of a Widmanstatten plate obtained with d0 5nm and c 20. The domain size is 0.8 lm 2 lm.

Fig. 6. Distributions of the phase-eld (left) and concentration (right)

at the plates tip. The axis are given in d0. The colormaps are identicalto the ones in Fig. 5.

4060 I. Loginova et al. / Acta Materialia 52 (2004) 40554063a hypothesis. As soon as the interface thickness d0 andthe anisotropy are known the growth rate may be de-termined by the simulations. In Fig. 7 the tip velocity is

given as a function of anisotropy for d0 5 nm, which istoo large a value to be really realistic. Anyhow, we can

read, for example, that the anisotropy of 0.05 would

yield a growth rate around 0:4 103 m s1 and fromFig. 8 that the tip radius would be 5d0. This growth rateshould be compared with the experimentally reported

[28] for a similar C content but a lower temperature, i.e.973 K, which is 0:2 103 m s1. On the other hand, it isevident from Fig. 3 that for a realistic interface thickness

of d0 1 nm we should have r0=1 c < 0:015 in or-der to observe Widmanstatten growth. Such high an-isotropy would yield a growth rate one order of

magnitude large than observed, see Fig. 7.

0.025 0.03 0.035 0.04 0.045 0.05 0.0550.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 x 103

1/(1+)

tip v

eloc

ity, m

/s

Fig. 7. The tip velocity as a function of the interfacial energy obtainedfor d0 5 nm, T 993 K and uc1C 0:01.

We investigated the dependency of the tip velocity on

tilted with respect to the grain boundary by h0 p=3.One notice that perturbations having larger wavelength

start growing faster, while those of small wavelength may

0.025 0.03 0.035 0.04 0.045 0.05 0.0552

2.5

3

3.5

4

4.5

5

5.5

6

1/(1+)

tip ra

dius

in

Fig. 8. The tip radius as a function of the interfacial energy obtained

for d0 5 nm, T 993 K and initial uc1C 0:01.

I. Loginova et al. / Acta Materithe interface thickness and the interfacial energy of co-

herent sides. First, we xed d0 as 5 nm and varied1=1 c. We found that the tip velocity decays withincrease of 1=1 c, Fig. 7, while the tip radius (denedfor the isoline / 0:5) increases, Fig. 8. Second, we xedthree values of 1=1 c allowing d0 to vary. The di-mensionless velocity Vl=RTMac shown in Fig. 9 reduces asd0 approaches realistic physical values and increases withthe decrease of the interfacial energy. For all the cases of

1=1 c, the dependence is linear. It is interesting toobserve that extrapolation of the data to smaller values

of d0 gives a negative velocity if the value of 1=1 c isgreater than 1=1 ccritical for those d0. This indicatesthat as realistic values for d0 are considered, i.e. in the2 4 6 8 100.05

0

0.05

0.1

0.15

0.2

0.25

0.3

, nm

dim

ensi

onle

ss ti

p ve

loci

ty

Fig. 9. Variations of dimensionless tip velocity Vl=RTMaC with the in-terface thickness d0. The top, middle and bottom lines are obtainedfrom the simulations with 1=1 c equal to 0.021, 0.032 and 0.38,respectively. These are the critical values for the growth of Wid-

manstatten plates for d0 equal to 2, 3 and 4 nm, respectively.order of 1 nm or less, then Widmanstatten plates canonly grow if the anisotropy is large enough, i.e. c 100.

5.3. Growth of colonies

Additionally, we simulated the growth of a colony of

Widmanstatten plates emanating from an austenite grainboundary. In order to initiate the growth of the precip-

itates, the phase boundary was initially disturbed by a

combination of sinusoidal waves. The time sequence of

the growth is presented in Fig. 10. The precipitates are

Fig. 10. Colony of Widmanstatten plates. Concentration distribution is

calculated for d0 10 nm, c 10 in a box 2 lm 1 lm.alia 52 (2004) 40554063 4061decay or grow signicantly behind the others.

As a comparison, Fig. 13 shows Widmanstatten fer-rite plates that have developed from prior austenite

grain boundaries in a low-alloy steel, white areas. The

austenite matrix has subsequently transformed topearlite upon cooling. One observes that though the

simulated plates look very realistically, their sides are

too smooth compared to the experimental plates. This is

probably due to purely deterministic nature of the

model. One can expect that modeling heat uctuations

in the system in a similar way it was done for dendritic

growth [22] would reproduce even better the experi-

mentally observed Widmanstatten morphologies.

6. Transition between diusion controlled and massive

transformation

As it was shown in [3], depending on the initial con-

tent of C in austenite, the c! a phase transformation

can be either diusion controlled or massive. The latter

occurs if the initial uc1C falls close to the a=a c phaseboundary. The massive transformation is partitionless,

i.e. it does not involve any change of composition, thus a

long-range diusion is unnecessary. The time sequenceof the concentration distribution presented in Fig. 11

was obtained for T 993 K and uc1C 0:002 (point Ain Fig. 2). As one observes, the initial perturbation of

the interface does not develop into a Widmanstattenplate, but rather decays so that the interface becomes

at. The massive growth occurs with a constant growth

rate until all of austenite is transformed into ferrite. The

concentration prole in the vertical direction comprisesa traveling spike which is spread over a distance of 5d0.

A completely dierent behavior is found for an alloy

with uc1C 0:01 and T 1050 K (point C in Fig. 2). Theconcentration elds given in Fig. 12 again demonstrate

the disappearance of the initial disturbance. However, in

this case, the excess carbon is build-up ahead of the

interface and we observe diusion-controlled growth.

7. Conclusions

possible or not. For the supersaturation, i.e. temper-

ature and C content, considered here a realistic

thickness of the incoherent interface, i.e. somewhat

lower than 1 nm, shows that c must be greater than100 in order for Widmanstatten plates to grow. Itseems likely that higher supersaturation would require

lower c for Widmanstatten growth. This is the subjectof further research. If Widmanstatten growth occurs,larger values of c give sharper tips and higher tipvelocities. A lower anisotropy value would make the

tip more blunt and yield a lower tip velocity. The tip

radius, upon which the classical Ivantsov-based theory

is built, vary approximately as proportional to an-isotropy and for c 100 it is less than 1 nm, i.e. it isof atomic dimensions.

Our simulations thus indicate that the shape of a

plate may be described as two parallel sides growing out

from the allotriomorph to some distance and then two

planar sides that meet in an atomistically sharp tip. Such

a shape seems to be in better agreement with metallo-

graphic observations than the parabolic shape withits well dened tip radius predicted by the Ivantsov

4062 I. Loginova et al. / Acta Materialia 52 (2004) 40554063Fig. 11. Partitionless growth. Concentration eld obtained for d0 2nm, c 50 in a box 0.32 0.8 lm. Initial conditions are T 993 Kand uc1C 0:002.

Fig. 12. Diusion controlled growth. Concentration eld obtained for

d0 5 nm, c 19 in a box 0.32 0.32 lm. Initial conditions are

T 1050 K and uc1C 0:01.Our simulations reveal that the anisotropy in the

surface energy and interface thickness plays the key

role in determining whether Widmanstatten growth isThe interface velocity is essentially proportional to

1=p t except for the later stages when impingement

sets in and the system nally approaches the state of

equilibrium.

Fig. 13. Experimentally observed Widmanstatten ferrite plates thathave developed from prior austenite grain boundaries in a low-alloy

steel, white areas.solution.

LCva 190T ; 24

C C

[10] Agren J. J Scr Met 1986;20:1507.

[11] Hillert M. Metall Trans A 1975;6:5.

I. Loginova et al. / Acta Materialia 52 (2004) 40554063 4063ature.

0GcFe 237:57 132:416T 24:6643T ln T 0:00375752T 2 5:89269 108T 3 77358:5T1; 27

0GcFeC 0GcFe 77207:0 15:877T ; 28LcCva 34671: 29Gmom 9180:5 9:723T

9309:8 s4

6

s

10

135 s

16

600

if s < 1; 26

where s T=T and T 1043 K is the Curie temper-Gmom 6507:7s4

10

s

14

315 s

24

1500

if s > 1; 25We also conclude that the present two-dimensional

model predicts a transition to a massive transforma-

tion, in agreement with our previous study, if the

supersaturation is large enough. We nd it very en-

couraging that a single phase-eld formulation is ca-pable of predicting three dierent growth

morphologies of ferrite, the allotriomorphic, Wid-

manstatten and massive growth.

Acknowledgements

This work was supported by the Swedish ResearchCouncil (VR).

Appendix A. Thermodynamic description of FeC system

[8]

Gam 0GaFe uC3

0GaFeC 0GaFe

3RT uC3ln

uC3

1 uC

3

ln 1

uC3

uC3

1

uC3

LaCva Gmom ; 20

Gcm 0GcFe uC 0GcFeC 0GcFe RT uC lnuCf

1 uC ln1 uCg uC1 uCLcCva: 21The quantities introduced in the expressions above are

given functions of the temperature

0GaFe 1224:83 124:134T 23:5143T ln T 0:00439752T 2 5:89269 108T 3 77358:5T1; 22

0GaFeC 0GaFe 322050 75:667T ; 23a[12] Karma A, Rappel WJ. Phys Rev E 1998;53:432349.

[13] Loginova I, Amberg G, Agren J. Acta Mater 2001;49:57381.

[14] Odqvist J, Sundman M, Agren J. Acta Mater 2003;51:103543.[15] Almgren RF. SIAM J Appl Math 1999;59:2086107.

[16] Karma A. Phys Rev Lett 2001;87:115701.

[17] Echebarria B, Folch R, Karma A, Plapp M. arXiv:cond-mat/

0404164v1.

[18] Lan CW, Shih CJ. Phys Rev E 2004;69:031601.

[19] Debierre JM, Karma A, Celestini F, Guerin R. Phys Rev E

2003;68:041604.

[20] Uehara T, Sekerka R. J Cryst Growth 2003;254:25161.

[21] McFadden GB, Wheeler AA, Braun RJ, Coriell SR, Sekerka R.

Phys Rev E 1993;48:201624.

[22] Karma A, Rappel WJ. Phys Rev E 1999;60:361425.

[23] Amberg G. Available from: http://www.mech.kth.se/gustava/fem-

Lego.

[24] Do-Quan M, Loginova I, Amberg G. Application of parallel

adaptivity to simulation of materials processes [in preparation].

[25] Artemev A, Jin Y, Khachaturyan AG. Acta Mater 2001;49:1165.

[26] Kobayahsi R, Warren JA, Carter WC. Physica D 2000;150:141

50.

[27] Wu K, Morral JE, WangY. Acta Mater 2001;49:340108.

[28] Enomoto M. Metall Mater Trans A 1994;25:194755.

[29] Hillert M. Metall Mater Trans A 1994;25:195766.Appendix B. Kinetic parameters for FeC

Diusional mobility in a [9]

RTMaC 2 106e10115=T exp 0:5898

1

2parctan 14:985

15309

T

m2=s:

30Diusional mobility in c [10]

RTM cC 4:529 107 exp 1

T

2:221 104

17767 uC26436m2=s: 31

Mobility of a=c interface [11]

M 0:035 exp17700=T m4=J=s: 32

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On the formation of Widmanstatten ferrite in binary Fe-C - phase-field approachIntroductionPhase-field modelAnisotropy of the interfacial energyNumerical issuesFormation of Widmanstatten platesInitiationCharacteristics of growthGrowth of colonies

Transition between diffusion controlled and massive transformationConclusionsAcknowledgementsThermodynamic description of Fe-C system [8]Kinetic parameters for Fe-CReferences