SOLIDIFICATION MICROSTRUCTURES

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SOLIDIFICATION MICROSTRUCTURES: RECENT

DEVELOPMENTS, FUTURE DIRECTIONSp

W. J. BOETTINGER1, S. R. CORIELL 1, A. L. GREER 2, A. KARMA 3, W. KURZ 4{,

M. RAPPAZ 4 and R. TRIVEDI 5

1NIST, Gaithersburg, MD 20899, USA, 2Department of Materials Science & Metallurgy, University ofCambridge, Cambridge CB2 3QZ, UK, 3Department of Physics, Northeastern University, Boston, MA

02115, USA, 4Department of Materials, Swiss Federal Institute of Technology Lausanne, 1015Lausanne EPFL, Switzerland and 5Iowa State University & Ames Lab. USDOE, Ames, IA 50011,

USA

(Received 1 June 1999; accepted 15 July 1999)

AbstractThe status of solidication science is critically evaluated and future directions of research in thistechnologically important area are proposed. The most important advances in solidication science andtechnology of the last decade are discussed: interface dynamics, phase selection, microstructure selection,

peritectic growth, convection eects, multicomponent alloys, and numerical techniques. It is shown howthe advent of new mathematical techniques (especially phase-eld and cellular automata models) coupledwith powerful computers now allows the following: modeling of complicated interface morphologies, takinginto account not only steady state but also non-steady state phenomena; considering real alloys consistingof many elements through on-line use of large thermodynamic data banks; and taking into account naturaland forced convection eects. A series of open questions and future prospects are also given. It is hopedthat the reader is encouraged to explore this important and highly interesting eld and to add her/his con-tributions to an ever better understanding and modeling of microstructure development. # 2000 ActaMetallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Solidication; Microstructure; Theory and modeling (kinetics, transport, diusion); Casting

1. INTRODUCTION

Microstructures are at the center of materials

science and engineering. They are the strategic link

between materials processing and materials beha-

vior. Microstructure control is therefore essential

for any processing activity. One of the most import-

ant processing routes for many materials, especially

metals and alloys, is solidication. Over the last

decade, important advances have been made in our

fundamental understanding of solidication micro-

structures. Three main ingredients have contributed

to this progress: (i) the development of rigorous

analytical models that have focused on both steady-

state and non-steady-state microstructure evolution

with the inclusion of nucleation for the selection of

phases; (ii) the emergence of accurate simulation

methods, and in particular phase-eld and cellular

automata approaches, which have permitted a vali-

dation of analytical theories as well as enabling pre-

dictions on grain structure and morphological

evolution; and (iii) the development of more rened

experimental techniques that have led to a better

visualization and characterization of microstructural

development. The combination of these advances

now makes it feasible to address long standing

microstructure formation questions with a higher

level of scrutiny and rigor, and thus to end this mil-

lennium in a renaissance period where solidication

``science'' is ourishing and solidication technology

is leading to a better control of materials proces-

sing. We highlight in this paper the theoretical andexperimental progress made in understanding basic

aspects of microstructure formation, emphasizing

especially the critical questions that remain to be

examined in this scientically highly interesting and

technologically important area.

A decade ago, an extensive overview was given

on the topic which was based on presentations and

discussions of the rst 1988 Zermatt Workshop

Acta mater. 48 (2000) 4370

1359-6454/00/$20.00 # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 2 8 7 - 6

www.elsevier.com/locate/actamat

p

The Millennium Special Issue A Selection of Major

Topics in Materials Science and Engineering: Current

status and future directions, edited by S. Suresh.

{ To whom all correspondence should be addressed.


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dedicated to solidication microstructures [1]. In the

present overview the most important ndings of the

second 1998 Zermatt Workshop on Solidication

Microstructures are presented by the seven keynote

speakers. (Contributions to this workshop have

been published in the form of a CD [2].)

The paper is organized as follows: Section 2

describes interface pattern formation models;

Section 3 considers nucleation and growth of a new

phase during the growth of an existing phase;

Section 4 emphasizes the action of uid ow on

microstructures; Section 5 addresses the appli-

cations of the models to industrially interesting

alloys containing several solutes; Section 6 includes

dierent numerical techniques and their potential

for solving complex problems in which several

phenomena must be considered simultaneously to

predict the microstructure.

2. INTERFACE DYNAMICS

Microstructures are formed at moving solid

liquid interfaces. In this section, the evolution of

interface morphologies of a single phase solid grow-

ing into a liquid is presented. The growth in an

undercooled melt of equiaxed dendrites is rst

described. Directional solidication with planar, cel-

lular and dendritic interfaces is then considered.

Some reference is also given to recent work on two-phase growth, such as eutectic and peritectic

growth.

2.1. Equiaxed dendritic growth

During the 1980s, the study of simplied models

that incorporate surface tension in a consistent way

led to the novel insight that dendritic growth is con-

trolled not only by the balance between diusion

and capillarity, but also in a subtle way by crystal-

line anisotropy [3, 4]. This insight led to the advent

of microscopic solvability theory to predict the

selected dendrite tip velocity and tip radius [5, 6].

Over the last decade, this theory has been extended

to three dimensions [7] and it has even been vali-

dated quantitatively by fully time-dependent simu-

lations of dendritic growth in both two dimensions

[810] and three dimensions [11] (Fig. 1), with the

added insight that in three dimensions the non-axi-

symmetric tip morphology inuences the selection

for large enough anisotropy.

Beyond the understanding of steady-state growth

of the tip, the main new concept that has emerged

over the last few years, is that complex pattern for-

mation processes occurring on the much larger

scale of an entire dendrite grain structure can be

described by remarkably simple ``scaling laws''.These processes include growth transients [13, 14]

that lead to steady-state growth and the highly non-

linear competition of secondary branches behind

the tip [1517]. In addition, a deeper understanding

of the role of anisotropy has come from the discov-

ery of new steady-state growth structures (doublons

[8] and triplons [18]). Following the morphological

instability of a small spherical grain, the primary

branches of an equiaxed grain emerge along h100i

directions in cubic crystals but do not immediately

reach a steady state. These branches are much thin-

Fig. 1. Three-dimensional equiaxed dendrites calculatedwith the phase-eld method: (a) thermal dendrite withh100i growth directions [9]; (b) solutal NiCu dendrite

when the preferred growth directions are h110i instead ofh100i [12].

44 BOETTINGER et al.: SOLIDIFICATION MICROSTRUCTURES


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ner and thus grow much faster initially than in

steady state, such that the instantaneous tip velocity

V(t ) [tip radius r(t )] is a monotonically decreasing

(increasing) function of time during a transient of

duration HDaV2ss

, where Vss is the nal steady-state

growth velocity{. An analytical treatment of this

transient has been possible in two dimensions (plate

dendrites) in the limit of vanishingly small under-

cooling where the problem is analogous to anisotro-

pic HeleShaw ow and can be treated rigorously

by the conformal mapping technique [13]. The main

result is that the length and width of primary

branches, and the total area of the plate, obey

simple power laws given, respectively, by LtHt3a5,

WtHt2a5 and AtHt for t ` DaV2ssX Moreover, the

transient interface shape is described by a unique

scaling shape. Another important feature of this

transient is that although V(t ) and r(t ) can vary in

time by one order of magnitude or more, the tip

selection parameter, s 2Dd0art2Vt, remains

constant in time and xed at its value determined

by solvability theory. Physically, this follows fromthe fact that s is determined by the diusion eld

in the tip region. Thus, at low undercooling, its

value is established quasi-instantaneously on the

time scale where the interface moves one tip radius

since rt2aDrtaVtX Phase-eld simulations in

two dimensions show a good quantitative agree-

ment with these predictions at very low undercool-

ing [14]. In three dimensions, no analytical theory is

yet available to describe this transient but simu-

lations reveal the existence of some approximate

scaling behavior at short time with dierent power

laws than in two dimensions [14].

The results of two-dimensional growth transientshave immediate implications for understanding the

large-scale structure of three-dimensional dendrites,

since the mean cross-sectional shape of a three-

dimensional steady-state dendrite (perpendicular to

the growth axis) can be assumed to evolve with dis-

tance z behind the tip as a two-dimensional branch-

less plate dendrite evolves in time with t zaVss[15]. This assumption becomes exact far enough

behind the tip since the heat (or solute) ux along z

becomes negligibly small, and it yields the mean

shape xHz3a5 for z large compared with the tip

radius rss but small compared with the diusion

length D/Vss

, and xH

z for distances larger than thislength as conrmed by three-dimensional phase-

eld simulations [11]. Translated in terms of the

projection area fraction f

xz dz, the above

result gives fHz1X6 for z ` DaVss, which is in

reasonably good agreement with the scaling law

fHz1X7 obtained by detailed measurements [17] of

the morphology of pure SCN dendrites grown in a

diusive regime in space [19]. Actually, on theoreti-

cal grounds one would expect a time-varying expo-

nent slightly larger than 1.6, which is a strict lower

bound valid in the limit of vanishing undercooling.

A scaling law has also been derived that describes

how the length of ``active'' sidebranches that sur-

vive the growth competition behind the tip and the

spacing l between them depend on the distance z

behind the tip [16]. The main prediction is thatboth l and increase linearly in z. The morphology

measurements on SCN crystals yield a good quanti-

tative agreement with this linear law but only if it is

interpreted in parabolic coordinates [17], i.e. with

the length of active sidebranches measured from a

parabola tted to the tip and plotted vs distance

along this parabola. This change of coordinate in-

corporates the fact that sidebranches tend to grow

perpendicularly to the isotherms [17] and thus eec-

tively incorporates the eect of the heat ux along z

that is neglected in the analysis of Ref. [16].

Finally, the basic concept that sidebranches are

driven by small perturbations of the tip region [20],

which originated from the work of Zel'dovich et al.

on ame fronts [21], has been further developed

theoretically [16, 22] and validated by phase-eld

simulations that consistently yield branchless den-

drites (needle crystals) if numerical noise is kept

small by using ne meshes [911, 2325].

Furthermore, when noise is purposely added in a

quantitatively controlled way, phase-eld simu-

lations yield sidebranching characteristics (initial

amplitude and spacing behind the tip) that are in

good overall agreement with the predictions of the

analytical theory of noise amplication in two

dimensions [26]. These simulations presently need to

be extended to three dimensions in order to test theprediction [16] that thermal noise is responsible for

the experimentally observed sidebranching activity.

The new steady-state growth structures that have

been identied are the so-called ``doublons'' in two

dimensions [8], rst observed in the form of a doub-

let cellular structure in directional solidication [27],

and the ``triplon'' in three dimensions [18]. Both

structures have been shown [8, 18] to exist without

crystalline anisotropy unlike conventional dendrites.

The doublon has the form of a dendrite split in two

parts about its central axis with a narrow liquid

groove in between these two parts, and triplons in

three dimensions are split in three parts. For a niteanisotropy, however, these structures only exist

above a critical undercooling [28] (or supersatura-

tion for the isothermal solidication of an alloy),

such that standard dendrites growing along h100i

directions are indeed the selected structures in

weakly anisotropic materials at low undercoolings,

in agreement with most experimental observations

in organic and metallic systems. From a broad per-

spective, the existence of doublons and triplons is

of fundamental importance since it has provided a

basis to classify the wide range of possible growth{ See list of symbols in the Appendix.

BOETTINGER et al.: SOLIDIFICATION MICROSTRUCTURES 45


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morphologies that can form as a function of under-

cooling and anisotropy [29].

2.2. Directional solidication

Signicant progress has been made over the last

decade in understanding fundamental aspects of

interface dynamics in directional solidication of

alloys. The onset of morphological instability in

directional solidication has been modeled by the

classic MullinsSekerka instability [30], which pre-

dicts the instability wavelength of a steady-state pla-

nar interface. In a typical directional Bridgman set-

up, however, the planar interface does not become

unstable in steady state, but during the transient

build-up of the solute boundary layer after solidi-

cation is started. By analyzing the morphological

stability of the planar interface during this transi-

ent, and by taking into account that the instability

takes time to grow from natural modulations until

it becomes observable, it has been possible to

obtain for the rst time an accurate prediction of

the instability wavelength [31]. This prediction

agrees well quantitatively with experiments on the

onset wavelength and diers signicantly from the

wavelength predicted assuming steady-state growth

[3234] (Fig. 2).The critical role of crystalline anisotropy in inter-

face dynamics has been demonstrated experimen-

tally in directional growth [35]. This study exploited

the ability to control the orientation of the crystal

grown in a thin sample. With the h100i direction

oriented (nearly) parallel to the axis of the thermal

gradient, the typically observed stable cellular/den-

dritic array structures are obtained [Fig. 3(a)]. In

contrast, with the h111i direction oriented normal

to the glass plates, there is no second-order aniso-

tropy in the plane of the sample [i.e. d 2gyady2 0

where g(y ) is the surface energy and y is the polar

angle in this plane]. Thus growth in this plane is

rendered ``eectively isotropic'' by this judicious

choice of grain orientation. In this case, a ``sea-

weed'' structure [35] [Fig. 3(b)] whose underlying

building block is the theoretically expected doublon

Fig. 2. Wavelength of morphological instability of plane front during initial transient [31].

Fig. 3. Role of crystalline anisotropy on interface shape indirectional growth of thin transparent samples [35].



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[8] is formed in agreement with numerical simu-

lations also presented in Ref. [35]. This nding is

also consistent with the numerical nding that stan-

dard cellular structures are linearly unstable in the

absence of crystalline anisotropy, except in a very

narrow range of velocity near onset of instability

[36].

The formation of doublons and triplons has also

been suggested to play an important role in the for-

mation of ``feathery'' grains in aluminum alloys.

This peculiar morphology, known since the 1940s,

which appears as a succession of lamellae separated

by straight and wavy boundaries, was associated

with the formation of twins parallel to the lamellae

but the appearance mechanisms remained unclear.

Recent electron back-scattered diraction (EBSD)

observations combined with detailed optical and

scanning electron microscope (SEM) observations

[2, 37] have clearly shown that feathery grains are

made of h110i columnar dendrites [Fig. 1(b)], whose

primary trunks are aligned along and split in their

center by a (111) coherent twin plane. The impinge-ment of secondary h110i side arms gives rise to

incoherent wavy twin boundaries. The switch from

h100i to h110i growth morphologies was attributed

to the small anisotropy of the solidliquid inter-

facial energy of aluminum which can be changed by

the addition of solute elements such as Zn, Mg or

Ti and possible attachment kinetics eects. More

details of this kind of growth may be found in Ref.

[12].

A new experimental technique has been devel-

oped in which a brief spatially periodic u.v. laser

pulse is applied to the solidliquid interface in a

transparent organic system (succinonitrilecou-

marin), to force a desired wavelength of the mor-

phological instability [38]. These experiments have

made it possible to investigate systematically the

dynamical selection and stability of cellular struc-

tures by varying the instability wavelength, and

thus accessing cell spacings that are not normally

accessible from a planar interface.

A stability analysis of dendritic arrays [39] has

been carried out in the limit where the primary spa-

cing is larger than the diusion length. The basic

instability found to limit the array stability at small

spacing corresponds to a mode where one out of

every two dendrites in the array is eliminated in

agreement with experiments [40, 41]. The samemode is found numerically to limit the array stab-

ility of cells [36] such that its existence appears

rather universal. A range of interdendritic spacings

is therefore stable, in agreement with experimental

observations [42, 43], but experiments with the

same ``history'' lead to a reproducible spacing [44,

45]. As an elaboration of this work, a model of

``history-dependent'' selection of the primary spa-

cing has been developed. This model is based on

the picture that dendrites are eliminated continu-

ously (subject to this instability) during the long

transient that follows the initial morphological

instability of the planar interface and leads to

steady-state growth of the array with a nal selected

spacing [31]. The predictions of this model agree

reasonably well with one set of experiments [44].

Moreover, at a more qualitative level, it has been

demonstrated experimentally that the initial

instability wavelength does indeed inuence the

steady-state interdendritic spacing [46].

Cellular/dendritic arrays have also been modeled

numerically based on the traditional view that the

structure with the lowest undercooling is selected

within some stability band of spacings [47]. This

model has had some success in explaining exper-

imental data. It does not, however, control the

strength of crystalline anisotropy, which is now

understood to crucially inuence the cellular array

stability both numerically [36] and experimentally

[35]. Therefore, its validity remains to be further

investigated. An analytical approach to the primary

spacing problem by summation of the Ivantsov

elds and application of the minimum undercoolingcriterion has also been developed recently [48].

A detailed experimental study has brought new

insights into the onset of sidebranching in direc-

tional solidication [49]. In these experiments, the

cell spacing was made uniform along the array and

varied by exploiting the history dependence of

wavelength selection in this system. This technique

was used to characterize the onset of sidebranching

systematically and shows that branched and non-

branched cells in these experiments belong to the

same branch of steady-state growth solutions.

Furthermore, it has revealed that the thermal gradi-

ent plays a destabilizing role (i.e. increasing G

causes non-branched cells to branch). Theoretical

models remain to be developed to explain this role

as well as to characterize the onset of sidebranch-

ing.

2.3. High velocity microstructures

Signicant experimental studies on microstructure

formation under rapid solidication conditions have

been carried out in the last decade using the laser

scanning technique (a type of directional growth

process) and levitational techniques (undercooled

solidication).At high rates oscillatory behavior of the solid

liquid interface (banding) has been analyzed by sev-

eral authors [5052]. Band formation in the velocity

regime of strong variation of the distribution coe-

cient, k(V) [53], was shown to depend strongly on

the coupling of non-steady-state heat and solute

transport phenomena [52]. Experiments on the ab-

solute stability of SCN have been undertaken [54]

and it was shown that close to this limit, cells fol-

low a l1X5V constant relationship [55].

In highly undercooled levitated melts systematic



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measurements have been undertaken under others

by the group of Herlach [56]. These authors have

shown that over a substantial range of undercool-

ings good agreement may be obtained between the

measurements in a large number of metals and

alloys and the analytical model using the transport

solution of Ivantsov together with Marginal

Stability arguments including solute trapping eects

(a theory which we call the IMS model) [5759],

with the stability parameter s 4p21X Further it

has been recently shown that excellent agreement

with no adjustable parameters can be obtained with

this theory [60]. Note that this agreement should

only be interpreted to mean that marginal stability

arguments, although not fundamentally correct, are

still useful to make quantitative predictions of den-

drite growth rates in rapidly solidied binary alloys.

A detailed comparison of solvability and marginal

stability theory for rapid dendrite growth has

recently been carried out [61].

One of the important observations at high under-

cooling is the formation of very ne-grained struc-

tures over a range of large undercoolings. This ne-

grained structure has been explained by dendrite

fragmentation. At very high undercooling, as the

dendrite trunk diameter becomes very ne, the ten-

dency to undergo Rayleigh instability increases. A

theory has been developed which shows that frag-

mentation can occur when the characteristic time

for dendrite break-up is shorter than the post-reca-

lescence or plateau time in overall agreement withexperiments [62, 63].

2.4. Coupled and simultaneous growth

Even if most of the recent modeling was con-

cerned with single-phase growth phenomena, there

has also been some work on coupled or simul-

taneous growth of two phases. A detailed numerical

survey of the morphological instabilities of lamellar

eutectics has been carried out in two dimensions by

the boundary integral method for the transparent

organic system CBr4C2Cl6 [64] (Fig. 4). In parallel,

a detailed experimental survey of these instabilities

has been carried out in the same system [65]. There

Fig. 4. Calculated stability diagram for two-dimensionalcoupled eutectic growth in CBr4C2Cl6 [64] in excellentquantitative agreement with the experimentally measureddiagram in the same alloy [65]. Z: reduced concentrationwith the eutectic point at Z 0X3; L: lamellar spacing nor-malized by the spacing corresponding to minimum under-cooling. The basic axisymmetric state is stable within thecenter region. Other states include steady-state tilted pat-

terns (T), 2lO (spatial period doubling oscillations), 1lO(spatial period preserving oscillations), where both 1lOand 2lO oscillations can be either axisymmetric or tilted.Blank regions of the diagram are those in which the

dynamics is not yet fully understood.

Fig. 5. Quenched liquidsolid interface of simultaneous two-phase growth in peritectic FeNi alloy [66].



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is a remarkably good quantitative agreement

between simulations and experiments concerning

the regions of stability of both non-tilted and tilted

steady states in the plane of composition and eutec-

tic spacing, and the oscillatory instabilities that

limit these regions. This understanding of eutectic

stability, however, is restricted to two dimensions

and presently needs to be extended to three dimen-

sions. Eutectic cells and dendrites forming in multi-

component alloys have also been studied

theoretically and experimentally and are presented

in Section 5.

Simultaneous growth of two phases in the form

of oriented bers and lamellae has been observed in

some peritectic alloys. Figure 5 shows an example

from a FeNi alloy. For this to happen, the compo-

sition has to be between the two solid phases and

the G/V ratio close to the limit of constitutional

undercooling for the stable phase with the smaller

distribution coecient [6668]. This interesting in

situ growth phenomenon still waits for a theoretical

interpretation, although recent phase-eld calcu-lations have shown the formation of such a struc-

ture [69].

3. PHASE AND MICROSTRUCTURE SELECTION

A microstructure is dened by the morphology,

size, distribution, crystal orientation, and corre-

lation (texture), and number of phases. Phase and

microstructure selection describes the variety of

phases and microstructures that develop under

given growth conditions and growth geometries.

This section treats mainly transformations of phases

and microstructures from one structure or mor-

phology into another. It is not so much the for-

mation of a single growth form itself which is of

interest in this part of the paper (this has been

treated in Section 2) but the mechanisms of change

from one phase and morphology into another. A

detailed theory of the mechanisms responsible for

this selection is only at its very beginning. A well-

known empirical approach that is consistent with

many experimental results uses extremum criteria,

such as the highest growth temperature in direc-

tional growth. In undercooled solidication proces-

sing the highest nucleation temperature and the

highest growth rate control the nal appearance ofmicrostructures and phases.

In many materials, additional phase transform-

ations take place in the solid state which lead to the

nal microstructure. In this review only solidica-

tion will be discussed. In general all solidication

processes start with nucleation and continue with

growth. The nal phases may be controlled by

nucleation, by growth, or by a combination of

both. In all three cases that will be treated separ-

ately in the following, much progress has been

made in recent years.

3.1. Nucleation control

If suciently large undercoolings can be attained

through hindrance of heterogeneous nucleation,

then there may be access to a variety of metastable

phases, such phases having lower melting points

and liquidus temperatures. The importance of

nucleation is seen when dealing with phase selec-

tion. A typical process where nucleation plays a

dominant role is solidication processing of under-

cooled melts such as is observed in droplets [56].

There may be a spread in nucleation temperatures

even under nominally identical conditions, and con-

sequently the results are best displayed on ``micro-structure-predominance maps''. Such maps have

been constructed for binary alloys, with alloy com-

position and droplet diameter as coordinates [70

72]. It is found that: (i) microstructure correlates

very strongly with droplet diameter (which deter-

mines the availability of nucleant sites and the cool-

ing rate); (ii) the eects of processing conditions

(e.g. gas purity in atomization) can be taken into

account; and (iii) correlation with undercooling can

be found through comparison with controlled

undercooling experiments and growth modeling [71,

73] (Fig. 6). It is clear, though, that we are very far

from being able to predict nucleation undercoolings,the diversity of potential heterogeneous nucleants

being a key impediment to quantitative modeling of

most real situations.

Under given conditions it is usual for one phase

to dominate, but the primary phases can also be

mixed. A well-analyzed example is the duplex parti-

tionless solidication of b.c.c. and f.c.c. phases in

the NiV system [74]. There are many examples in

which the phase competition is between b.c.c. and

f.c.c. phases, and this has been most closely exam-

ined for FeNi. In undercooled levitated droplets,

Fig. 6. Phase predominance map (drop diameter vs com-position) for undercooled growth in FeNi alloys [73].



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the prevalence of one phase or the other can be

fairly well predicted by which phase has the lower

work of nucleation (Fig. 6) (e.g. Ref. [75]). The

b.c.c. phase is easier to nucleate than would be

expected from its relative thermodynamic stability;

it has a lower solidliquid interfacial energy. In the

FeNiCr system, it has been shown that the f.c.c.

or b.c.c. phase can be selected through the use of

an appropriate substrate put in contact with the

droplet to trigger solidication [76].

The existence of the true primary phase is some-

times revealed by a double recalescence phenom-

enon [75]. In this way, for example, the transient

existence of a hitherto unknown metastable f.c.c.

phase of rhenium has been inferred [77].

Observations of this kind, found also in some alloy

systems [78], may be crucial in analyzing nucleation

kinetics.

Particular interest has centered on analyzing het-

erogeneous nucleation kinetics. Basic treatments of

heterogeneous nucleation (taking into account vari-

ations of potency and population) have had successin predicting primary phase selection [79]. Recent

advances have been made by studying liquid dro-

plets entrained in solid matrices; when large under-

coolings b 50 K are required for heterogeneous

nucleation, the classical spherical cap model seems

to work well, but for more potent nuclei it breaks

down [80]. In that case, some success has been

achieved with a model that considers the thermo-

dynamics, if not yet the kinetics of adsorption [81].

Further heterogeneous nucleation studies have been

undertaken on liquid droplets in an emulsifying or-

ganic liquid; taking a more classical approach, roles

have been identied for dierent types of stationary

and moving surface steps [82]. Yet another advance

has been transmission electron microscopy of het-

erogeneous nuclei formed in a glassy matrix [83];

this study, of relevance for commercial grain rene-

ment of Al, shows the importance of crystallogra-

phy and chemistry in nucleation, even though

quantication of these roles remains elusive [84].

So far, it has been natural to concentrate on the

primary stage of solidication, yet there are cases

where the interest is in the formation of secondary

phases in the nal stages of solidication. An

example of considerable industrial interest is the

DC casting of very dilute Al alloys. In cases of

practical importance there is a range of intermetal-lics which can nucleate, the phase selection being

sensitive to many parameters including solidication

velocity. Directional solidication can reveal

changes in intermetallic selection and be a basis for

understanding fundamental mechanisms [85, 86].

Attempts to analyze phase selection have focused

on comparative eutectic growth kinetics [87], but

solid-state changes and nucleation eects have also

been considered. It appears that a quantitative

analysis of the phase selection may depend on the

geometry of the liquid in which the rival intermetal-

lics nucleate and grow; such an analysis has yet to

be attempted. Considerations so far suggest that the

wide range of conditions over which mixtures of

phases are obtained is indicative of a growth com-

petition [88].

3.2. Growth control

Growth-controlled phase and microstructure

selection has been successfully treated by comparing

the steady-state interface response of competing

phases. Calculating the interface response, i.e. the

growth behavior of plane front, cells and dendrites,

for all possible phases one can determine the

growth form which develops the highest interface

temperature for a given growth velocity and tem-

perature gradient [89] or the highest growth velocity

for a given undercooling{. Growth of eutectic struc-tures can also be included in this treatment. The

extremum criterion is a strong indication of the

structure to be formed. It assumes that: (i) the

microstructure selection is not nucleation controlled

(i.e. nucleation undercooling is suciently small);

(ii) interaction between competing growth forms is

negligible; and (iii) steady-state theory can be

applied. Despite its simplicity this approach is of

great help in determining microstructure maps for a

more rational alloy development. Several cases of

recent modeling of microstructure selection will be

presented in the following.

3.2.1. Stable to metastable phase transition. Using

the above-mentioned maximum temperature cri-

terion, the stable to metastable transition for direc-

tional dendrite growth has been analyzed for

peritectic systems [9092]. This allows us to ration-

alize why at high velocities a transition from a

stable to a metastable peritectic phase is often

obtained. For example in FeNi or FeNiCr

steels, high weld speeds lead to the formation of

austenite dendrites [9395], even if at low velocities

ferrite is the primary phase, with important conse-

quences for the integrity (solidication cracking) of

the weld. No nucleation is needed for this transition

to occur as the metastable phase (austenite) growsinbetween the stable phase due to microsegregation

and its growth is accelerated with velocity until the

metastable phase becomes the leading one. For the

reverse case (metastable to stable phase) this is not

true and nucleation is a necessary requirement for

the transition to happen (see under mixed control).

Similar stable to metastable phase transitions have

been analyzed in detail for FeC alloys [96].

Under conditions of rapid solidication, solute

and disorder trapping become signicant in the kin-

etics [97]. By including such eects it has been poss-{ In this approach cells and dendrites are treated as one

entity with one growth equation.



9/28

ible quantitatively to model the growth kinetics in

undercooled NiAl alloys [98, 99]. Growth kinetics

alone can be used to follow the competition

between the ordered and disordered version of one

phase. However, prediction of which basic structure

(b.c.c. vs f.c.c. again in this case [98]) will beobserved requires a modeling of the nucleation

which does not yet exist.

3.2.2. Dendrite to eutectic transition (CZ). The

limits of the so-called ``Coupled Zone'' represent

the transition between fully eutectic structure and

primary dendrites or cells with interdendritic eutec-

tic. This transition which is also hysteretic in nature

is inherently dicult to model. Karma has made

progress in this matter and his results on the stab-

ility of eutectic growth are discussed in Section 2.

Using steady-state growth theory and the extre-

mum criterion for the interface temperature, this

transition can be calculated for directional growth

(Bridgman, laser treatment, etc.) and is found to be

in good agreement with experimental evidence

[100]. In this way, following the early work of

Boettinger et al . [101], a series of solidication

microstructure selection maps has been obtained in

recent years which allows a more rational approach

to the solidication processing of technically im-

portant alloys: AlCu [102, 103], AlFe [104], AlSi

[105], AlCuSi [106], NiAl [99], and even cer-

amics such as Al2O3ZrO2 [107]. These maps have

been used as a tool for analyzing and predicting the

microstructures of laser surface treated materials. A

similar approach, but for undercooled melts with a

corresponding maximum velocity criterion, has also

been developed [108].

Another way of this type of microstructure mod-

eling is the ``inverse modeling'' which starts with in-

formation about the microstructure and optimizes

the input parameters such as the phase diagram[103, 109111] (Fig. 7). This new approach to deter-

mine stable and especially metastable phase equili-

bria is useful in cases where conventional

techniques do not work.

3.3. Mixed control

Mixed control is always found when both nuclea-

tion and growth play a controlling role in the

microstructure selection, such as in the columnar to

equiaxed transition of dendritic or eutectic struc-

tures or in low velocity microstructures in the two-

phase region of peritectic systems.

3.3.1. Columnar to equiaxed transition (CET).

Hunt's classic approach to model the CET [112] has

been applied to welding [113] and has been

extended by using more recent dendrite models

[114, 115]. In this way critical growth conditions for

the single crystalline welding of single crystal gas

turbine blades could be established and a poten-

tially interesting process for lifetime extension of

these expensive components developed (Fig. 8)

[116].

The transition from the outer equiaxed zone to

the columnar region (ECT), often observed in cast-ings can be understood in terms of the same CET

criterion. Such considerations were made for the

shape of grains continuously nucleating and grow-

ing in a thermal gradient [117]. When the ratio G/V

increases up to a critical value, the shape factor of

the grains becomes innite, meaning that the

equiaxed grains become columnar. In order to

explain the ECT, it is necessary to consider the heat

transfer coecient between the casting and the

mould (which changes strongly when solidication

starts) and the superheat of the melt. Such model-

Fig. 7. Calculated microstructure selection map (VCodiagram) for NiAl alloys (b), and optimized phase dia-gram (a) [109]. (The dierence of the eutectic temperatureofbg ' and bg eutectic is less than the width of the line.)



10/28

Fig.

8.

Epitaxiallasermeta

lformingofasinglecrystalsuperalloy(CM

SX4)showingthesinglecrystallinenatureofthelasercladdepositedontothecasts

inglecrystalsub-

strate[116].



11/28


12/28

two phases, and by growth competition between the

nucleated grains and the pre-existing phase under

non-steady-state conditions. In this case the simple

extremum growth criterion does not lead to the

right answer and nucleation in the constitutionally

undercooled zone ahead of the growth front has to

be taken into account in order to determine the

microstructure selection [119, 120] (Fig. 9).

A clear understanding of complex microstructure

formation has come from directional solidication

studies of binary alloys with compositions in the

two-phase region of the peritectic phase diagram at

large G/V ratio to suppress the morphological

instability of both the parent (a ) and the peritectic

(b ) phases, i.e. each phase alone would grow as a

planar front. Even in this simplied case, a rich var-

iety of microstructures has been identied that

depend sensitively upon the relative importance of

nucleation, diusion and convection [121125] as

shown in Fig. 10. These microstructures can be

broadly classied into the following groups based

on geometrical patterns and the underlying trans-port mechanisms: (a) discrete bands of the two

phases; (b) partial bands or particulates (or islands)

of one phase in the matrix of the other phase; (c)

single primary to peritectic phase transition; (d)

simultaneous growth of the two phases with a pla-

nar solidliquid interface; (e) dispersed phases due

to nucleation ahead of the interface; and (f) oscillat-

ing continuous tree-like structures of the primary

phase that are surrounded by the peritectic phase

[122]. Theoretical models and experimental studies

in very thin samples have shown that structures

(a)(e), can form under diusive regimes, whereas

microstructure (f) is a novel microstructure whose

formation requires the presence of oscillatory con-

vection in the melt.

In order to understand the formation of some of

the complex microstructures in the two-phase

region of peritectic systems, an analytical model of

banding in peritectic systems was rst proposed for

diusive growth in which the change in phases

occurred when the appropriate nucleation under-

coolings were reached [126]. According to this

model, a banding cycle of alternate nucleation and

growth of primary, a, and peritectic, b, phases may

continue, leading to an oscillatory behavior of the

interface and to alternate bands of a and b. The

major predictions of this diusive banding model

are: (i) the banding cycle will operate below and

above the peritectic temperature; and (ii) the band-

ing window exists only for a narrow range of initial

alloy composition in the hypoperitectic range.

In the above one-dimensional model of discrete

band formation, it was assumed that the nuclei ofthe new phase spread rapidly in the lateral direc-

tion, so that no appreciable lateral gradients exist.

However, this is generally not valid and one must

consider the relative rates of spreading of the new

phase and the continuing growth of the parent

phase. The microstructure for this complicated case

was investigated experimentally as well as by nu-

merical simulation of a two-dimensional transient

phase-eld model for a generic peritectic phase dia-

gram [69]. Several new morphologies were observed

and predicted depending on the nucleation rate.

Fig. 10. Fluid ow controlled microstructures in peritectic alloys: (a) discrete bands of the two phases;(b) partial bands or islands of one phase in the matrix of the other phase; (c) single primary to peritec-tic phase transition; (d) simultaneous growth of the two phases with a planar solidliquid interface; (e)dispersed phases due to nucleation ahead of the interface; (f) oscillating continuous tree-like structures

of the primary phase that are surrounded by the peritectic phase [122].



13/28

The results of the phase-eld model indicate that

when only a single nucleus is allowed to form on

the wall of the sample, discrete band formation in a

diusive regime is only possible for a nite range of

system sizes, Lmin ` L ` Lmax, where L is the size of

the sample or the distance between the nuclei.

Moreover, this range depends on both the compo-

sition inside the hypoperitectic region, and the

nucleation undercoolings of the two phases. For

internuclei distance L ` Lmin, discrete particles of

the b phase form inside the a matrix, and for

L b Lmax, discrete particles of a phase form in the b

matrix form. In addition to the microstructure of

discrete particles of one of the two phases

embedded inside the continuous matrix of the other

phase, more complex microstructures, including two

simultaneously growing phases form [Fig. 10(d)].

Simultaneous two-phase growth has been observed

in several peritectic systems, including SnCd [122],

AlNi [67], and FeNi [66] (see also Section 2.4).

The basic model of nucleation and growth con-

trolled structures shows that dierent microstruc-tures can form only within a narrow band of

composition in the hypoperitectic region. However,

several experimental observations of banded struc-

tures have been made for compositions outside this

banding composition window, and banding struc-

tures were reported even for hyperperitectic compo-

sitions. These observations clearly indicate that the

observed structures are not controlled by diusion

but by convection eects, as shall be discussed in

the following section.

4. CONVECTION EFFECTS

Convection eects are of utmost importance in

the development of solidication microstructures.

Despite this fact, most microstructure models are

based on purely diusive transport mechanisms.

Only recently, modeling of growth in the presence

of convection has been successfully undertaken. The

rst step in such an undertaking is modeling of con-

vection and its instabilities before coupling of con-

vection and microstructure formation is done.

4.1. Convection instabilities

For the simple problem of an innite layer withvertical temperature and solutal gradients, it is well

known that convective instabilities can occur even if

the net density of the liquid decreases with height

[127]. Similar behavior also occurs in a porous med-

ium [128]. However, usually the temperature and

solute concentration in the mushy zone of a binary

alloy are coupled by the phase diagram and this

prohibits double-diusive behavior, i.e. a density

inversion is necessary for the onset of convective

instability. Worster [129] has reviewed recent work

on convection in mushy zones. The critical

Rayleigh numbers for the onset of convection in a

binary alloy have been calculated for three dierent

models of the mushy zone during directional solidi-

cation [130]. In general, there are two modes of

instability: a mode in the mushy layer and a bound-

ary-layer mode in the melt; the wavelength of the

mushy-layer mode is small compared with the

wavelength of the boundary-layer mode. In addition

to these non-oscillatory modes, there are modes

that are oscillatory in time.

The radial segregation due to solutal convection

during the directional solidication of leadthallium

alloys with a planar crystalmelt interface has been

calculated using pseudo-spectral methods [131].

Solutal Rayleigh numbers for the calculations ran-

ged from very near the onset of convective instabil-

ity to a factor of ten above the instability onset. In

general, the ows and segregation are asymmetric,

although for special conditions axisymmetric ows

can occur.

A sudden change in ow conditions is correlated

with the interface concentration during directionalsolidication of a tinbismuth alloy [132]. The

interface concentration was monitored by Seebeck

measurements using the MEPHISTO furnace

during the USMP-3 space ight. Numerical calcu-

lations of the uid ow and solute redistribution

due to sudden gravitational accelerations caused by

thruster activation were in good agreement with the

observed Seebeck signals.

4.2. Field eects

It is well known that a uniform magnetic eld

can damp convective motions in an electrically con-

ducting uid. However, when a gradient in the

Seebeck voltage exists in the presence of a magnetic

eld and temperature gradients, there can be a

resulting ow [133, 134]. This can occur at a crys-

talmelt interface when there is a temperature gra-

dient along the interface; for example, in a binary

alloy with a non-uniform concentration. Since this

thermoelectric magnetohydrodynamic ow occurs

in the vicinity of the interface, it can play a signi-

cant role in solute redistribution. It can also be

used to counteract buoyancy-driven ow in the

mushy zone during horizontal directional solidica-

tion [135]. Freckle formation in coppersilver andaluminumcopper alloys have been examined under

dierent magnetic elds. The observed larger den-

drite spacings agree with the observation in space

experiments where the ow is signicantly reduced

[136].

There have been a number of studies of crystal

growth in very high gravitational elds using a cen-

trifuge [137139]. For germaniumgallium alloys,

solute segregation exhibits a minimum as a function

of rotation rate. This behavior can be understood

by considering non-axial temperature gradients; at



14/28

low rotation rates, the uid velocity is decreased by

the action of Coriolis forces, while at large rotation

rates the buoyancy force due to the centrifugal

acceleration, increasing as the square of the rotation

rate, becomes dominant [139]. Thus, the ow and

segregation are reduced at intermediate rotation

rates.

4.3. Eect of ow on interface morphology

The eect of simple ows on the shape of par-

ticles growing from a supersaturated solution has

been calculated [140]. The concentration and uid

ow elds are solved numerically by a mapping

technique in the Stokes ow approximation. Simple

base ows such as a uniform streaming ow or a

biaxial straining ow lead to non-spherical shapes.

The particle shape, as function of the ow magni-

tude and the anisotropy of the crystaluid surface

tension, has been studied.

The inuence of convection on morphologicalinstability and interface structure during directional

solidication was examined theoretically [141].

There have also been observations on massive

transparent specimens [142] which have revealed

that convection results in a gradient of microstruc-

ture along the interface, from a smooth interface to

dendrites. Fluid ow eects at the very scale of the

microstructure have been seen during solidication

of faceting transparent systems (e.g. salol-based

alloys) where saw-tooth patterns of millimeter size

form [143]. Surface tension-driven convection due

to the presence of uiduid interfaces and its inu-

ence on the morphology of the growth front

deserves thorough investigation, e.g. coupled

growth of bubbles of dissolved gas and monotectic

alloys in which a second liquid phase forms either

as rods in a solid matrix or droplets in the melt

[144].

Anisotropic interface kinetics stabilizes an inter-

face with respect to the onset of morphological

instability [145, 146]. Such anisotropic kinetics

arises naturally when growth is by step motion and

the crystalmelt interface is near a singular crystal-

lographic orientation. When a planar interface is

perturbed with a sinusoidal perturbation, anisotro-

pic kinetics causes a lateral translation of the sinus-

oid (traveling wave). In turn, this lateral motioncan strongly interact with shear ows along the

interface. Flow in the direction of step motion is

destabilizing while ow opposite to the step motion

is stabilizing.

Experiments on the dendritic growth of succino-

nitrile and pivalic acid from supercooled melts on

earth and in microgravity show small discrepancies

from the classic Ivantsov relation between Peclet

number and dimensionless supercooling (Fig. 11)

[147]. Under terrestrial processing conditions, con-

vection in the melt has a major impact on metallic

solidication, especially at small crystal growth vel-

ocities [56]. Previous studies of dendrite growth in

undercooled Ni melts on earth show systematic de-

viations of experimental data and dendrite growth

theory at small undercoolings. The discrepancy is

partly reduced if convection is taken into consider-

ation. Measurements of the dendrite growth vel-

ocity as a function of undercooling on pure Ni and

dilute Ni0.6 at.% C alloys under microgravity con-

ditions provide a test of dendrite growth models

[148]. The experiments were performed using the

electromagnetic levitation facility TEMPUS.

Excellent growth velocity data were obtained during

the mission in an undercooling range between 50

and 310 K. However, no dierences between micro-

g and 1 g data were detected in this temperature

range since ow due to electromagnetic forces may

be signicant.

The selection of twinned dendrites in the presence

of uid ow may be explained by a higher growth

temperature with respect to normal dendrites, in

particular as a result of doublon formation (see

Section 2.2). The eect of convection on the alter-

nating sequence of straight/coherent and wavy/inco-

herent twin is shown in Fig. 12 [37]. The alloy hasbeen produced by direct chill (DC) continuous cast-

ing and exhibits, in some regions, ``feathery grains''.

In Figure 12(a), three feathery grains labeled 13

are clearly visible: each one is made of parallel

lamellae showing an alternating sequence of colors

(green/red, light blue/purple, and yellow/violet

for grains 1, 2 and 3, respectively) separated by an

alternating sequence of straightwavy boundaries.

This corresponds to twinneduntwinned regions

separated by coherentincoherent twin boundaries

across rows of primary h110i dendrite trunks.

Fig. 11. Tip Peclet number vs supercooling for free den-dritic growth of organics under terrestrial and under

microgravity conditions [147].



15/28

Transverse melt ow was invoked to explain the

systematic alternating sequence of lamellae and

boundaries through branching mechanisms [see Fig.

12(b), in which the twinned dendrites are seen along

their trunk axis].

Numerical simulation of microscopic ow in the

melt during solidication was introduced through aphase-eld model [149151]. The eects of ow on

free dendritic growth (tip velocities, radii, and tip

selection as a function of the orientation of the ow

with respect to the crystal) were investigated.

Convection during coarsening of an isothermal

binary liquidsolid mixture has been studied, i.e.

the eects of convection on coarsening and of coar-

sening on the permeability were examined. The

eect of convection on equiaxed dendrite growth

and associated macrosegregation has also been stu-

died. New results have been obtained on the size

evolution and settling velocity of NH4Cl equiaxed

crystals growing from supercooled NH4ClH2O sol-

ution [152]. The results have been analyzed with the

theory for an isolated dendrite growing in an axi-

symmetric melt ow [153]. In the range of the ex-

perimental settling velocities (711 mm/s), the best

t for the stability constant was found to be 3.12

times greater than the value measured for the

purely diusive case [154].

Extension of a Cellular Automaton (CA) tech-

nique coupled with a nite element (FE) method

(CAFE model [155]), improves the modeling of den-

dritic grain structures in the presence of convection.

The movement of equiaxed crystals in the liquid to

form a sedimentation cone, as well as the modi-

cation of the columnar-to-equiaxed transition in the

presence of convection, are well described qualitat-

ively by the CAFE model.

There has been interesting experimental evidence

on the mushy zone interactions with melt ows in

transparent organics. Quantitative measurements of

the ow eld during solidication could be made[156].

When the JacksonHunt model of eutectic

growth [157] is applied to the growth of monotectic

composites, the predicted value ofl 2V is more than

an order of magnitude smaller than the experimen-

tal value for aluminumindium monotectic alloys

which grow with rods of indium-rich liquid in an

aluminum-rich solid matrix; here, V is the growth

velocity and l is the interrod spacing. Allowing for

diusion in the rod phase does not improve the

agreement between experiment and theory [158].

While the discrepancy could be due to inaccurate

values of the thermophysical properties, another

transport mechanism such as convection could

account for the discrepancy. Such uid ow could

arise from surface tension variations along the

uiduid interface; a pressure-driven ow could

also occur at the uiduid interface due to the

requirement of satisfying both the GibbsThomson

and YoungLaplace equation at this interface [159].

Convection eects have also been found to give

rise to new microstructures that are not observed in

the diusive growth regime. For example, Fig. 10(f)

shows a novel microstructure whose formation

requires the presence of oscillatory convection in

the melt of peritectic systems. A detailed study of

the three-dimensional shape of the microstructurerevealed that the bands were not discrete, but both

the a and the b phases were continuous [160]. It

was shown that the microstructure, which appears

like discrete bands on a section close to the surface

of the sample, is in fact a more complex structure

made up of two continuous interconnected phases

in three dimensions. In particular, the microstruc-

ture consists of a large tree-like domain of primary

a phase that is embedded inside the peritectic b

phase. The formation of this structure is governed

by oscillating convection present in a large diameter

Fig. 12. Twinned dendrites in AlCu alloy. (a) EBSDreconstruction of the microstructure in a transverse sec-tion to the thermal gradient containing three grains. (b)Schematic view of the eect of convection on twinned

dendrite formation [37].



16/28

(6.0 mm) sample [124]. Besides the tree-like struc-

ture, several other new oscillating microstructures

were observed experimentally, and predicted nu-

merically, depending upon the intensity and modes

of convection [160].

5. MULTICOMPONENT SYSTEMS

The application of solidication modeling to

practical technology is closely linked to our ability

to model microstructural development in multicom-

ponent alloys (three or more components). Over the

past ten years signicant progress has been made in

this area.

5.1. Thermodynamics

Solidication models, which use local interfacial

equilibrium, have been successfully coupled to

phase diagram information obtained via the

Calphad method [161]. Examples include analyses

of Scheil solidication path, dendrite tip kinetics,solid (back) diusion and macrosegregation.

Commercial databases are available for Al, Fe, Ni

and Ti base alloys (ThermoTech, Ltd{) and others

are distributed with the various thermodynamic

computational codes available: ThermoCalc (KTH,

Stockholm), MTData (NPL), Chemsage (RWTH,

Aachen). All of these computational codes can be

interfaced with solidication models. As an example

[162] a set of subroutines, LEVER, SLOPE and

HEAT, have been built on top of a modied ver-

sion of the Lukas code. LEVER gives the phase

fractions and phase compositions at equilibrium for

a specied temperature T and overall composition.

SLOPE gives the liquidus temperature, the solid

phase concentrations, and the liquidus slopes for a

specied liquid composition and solid phase. HEAT

gives the enthalpy per unit mass for a specied

phase for a given temperature and phase compo-

sition.

This thermodynamic approach naturally enables

a Scheil analysis of the solidication path; i.e. the

evolution of the liquid and solid concentrations and

the phase fractions during cooling under the

assumption of complete liquid diusion and no

solid diusion. This approach easily treats the

appearance of new phases at eutectic reactions, or

the disappearance and appearance of phases at peri-tectic reactions. A Scheil analysis provides the basis

for a good estimate of very practical information

for castings: (a) how the heat of fusion evolves

during cooling (for coupling to macroscopic heat

ow analysis); and (b) how the density of the

mushy zone changes (for coupling to uid ow

modeling for macrosegregation, porosity and hot

tearing analysis).

Under rapid solidication conditions, when local

interface equilibrium is invalid, thermodynamic cal-

culations for multicomponent alloys can be used to

compute the thermodynamic driving ``force'' and

the energy dissipated due to solute drag (if a model

of diusion through the interface is prescribed).

The thermodynamic driving ``force'' is required for

analysis of the interface response functions for

rapid solidication. In this area, the AzizKaplan

model of solute trapping [163] has been extended to

multicomponent systems for the case when the dif-

fusive speed for all of the solutes is identical [164].

An open question remains regarding the impact of

dierent diusive speeds for dierent solutes in

multicomponent solute trapping models of rapid

solidication. Experimental work in this area would

be useful.

5.2. O-diagonal diusion terms

O-diagonal diusion terms are usually neglected

for multicomponent liquids, yet there is little justi-

cation. Moreover the diagonal terms are usually

assumed to be identical. When diusion uxes are

related to chemical potential gradients through

appropriate mobilities, the absence of o-diagonal

mobility terms does not imply the absence of o-di-

agonal diusion terms. O-diagonal terms tend to

be strongly concentration dependent. One set of ex-

periments [165] measured the o-diagonal diusion

terms in liquid ternary Al alloys. Diusion couples

with a step change in one component but a con-

stant value of the second were analyzed. In thealloys tested, there was no detectable change in con-

centration of the second component; i.e. negligible

o-diagonal terms.

It has been shown [166] that analytical models

for plane front and dendritic growth developed for

binary alloys can be extended to multicomponent

alloys by taking into account the o-diagonal terms

of the diusion matrix. The diusion elds for the

n solutes are given by linear combinations of the n

binary solutions using the eigenvalues and eigenvec-

tors of the diusion matrix instead of the diusion

coecients (Fig. 13). For the time being however,

use of these solutions is limited by the lack of

measured diusion coecients and methods todetermine the o-diagonal terms. A theoretical

approach to this problem is needed.

5.3. Fundamental morphological stability issues

A complete linear stability analysis of planar

growth under the assumption of local equilibrium

for a ternary alloy with no o-diagonal diusion

terms has been performed [167]. When the pertur-

bation wavelength is not assumed to be small com-

{ Trade names are used for completeness only and do

not constitute an endorsement of NIST or any other or-

ganization.



17/28

pared with the liquid diusion lengths for all of the

solutes Di/V, the perturbation growth rate, e, not

only depends on the partition coecients, ki, and

liquidus slopes, mi, but also on the derivatives of

liquid concentrations with respect to solid concen-

trations evaluated on the liquidus surface, quantities

that can be computed from the thermodynamic

approach. When the perturbation wavelength is

small, these factors disappear. In this case, the ex-

pression for e contains a denominator, which can

vanish due to the fact that miki 1 can be nega-

tive for one or more of the solutes in multicompo-

nent systems. This has the potential to lead to

oscillatory instabilities. Whether this can occur in

an experimental system is not known.

Other stability issues, such as the cell to dendrite

transition, have not been adequately resolved, even

for binary alloys. One situation peculiar to ternary

systems is the formation of eutectic cells by the pre-

sence of a dilute ternary solute. The full stability

spectrum of a steady-state lamellar interface in the

presence of a ternary impurity has been calculated

and an analytical form of this spectrum has been

derived in the limit where the wavelength of the

perturbation is large compared with the lamellar

spacing [168]. Also preliminary phase-eld calcu-

lations of the growth of eutectic cells to treat large

amplitude perturbations have been performed as

shown in Fig. 14 [169]. (Such calculations are de-

nitely at the limit of what is actually possible; this

simulation took approximately 60 CPU hours on 32

processors of a CRAY T3E.)

5.4. Microstructure prediction

5.4.1. Dendritic growth and solid diusion. Even

though many issues remain regarding the funda-

mental role of anisotropy on dendrite tip radius

selection (even for pure materials), models used for

practical materials typically use the Ivantsov/

Marginal Stability (IMS) approach [5759]. This

model for binary alloys has been extended to multi-

component alloys [170, 171]. The equiaxed growth

model [172] has been generalized to multicompo-

nent alloys [173] as well as the standard model of

secondary spacing [173, 174]. The FloodHunt

method coupling dendrite tip models to the Scheil

analysis [175] has been modied to treat multicom-

ponent alloys. The modication also conserves

solute, but only for the case of diagonal and equal

liquid diusion coecients [176].

Modications have been made to the Scheil

approach to deal with solid diusion for multicom-

ponent alloys. An approximate treatment of solid

diusion [177] has been extended to multicompo-nent alloys and coupled to phase diagram calcu-

lations [162]. This method is convenient for node by

node coupling to macroscopic heat ow calculations

because it reduces computation time compared to a

full solution of the diusion equations. Solution of

the full diusion equations has been performed

using DICTRA [178], a diusion analysis code built

on top of ThermoCalc. An approach to model solid

diusion during monovariant eutectic solidication

in addition to primary solidication has also been

performed [179].

Fig. 13. Plane front concentration elds for a three-component system with the liquid solute diusioncoecients; D11 6 10

9, D22 2 109, D12 D21 3 10

9 [166].



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5.4.2. Eutectic coupled zone and associated micro-

structure maps. By computing the competition

between dendritic and eutectic growth for a speci-

ed alloy composition, microstructure maps that

dene the range of solidication speed and tempera-

ture gradient required to form a specied growth

form (hence microstructure) have been developedfor ternary alloys [106]. Similarly the code PHASE

[180], which has been extended to multicomponent

multiphase alloys [181], computes the dominant

growth microstructure and the resultant microsegre-

gation during cooling or during steady-state direc-

tional growth through a numerical analysis of

competitive growth and solid diusion. Phase dia-

gram information is obtained using graphs as input.

The analysis of the microstructure selection has

been extended to a ten-component superalloy

(CMSX4) [116, 182]. In this way processing win-

dows for laser metal forming with application to

repair of single crystal superalloy turbine com-

ponents could be calculated. The processing con-

ditions required to avoid stray grain formation were

evaluated using Hunt's columnar-equiaxed tran-

sition model [112] but with numerical evaluation of

dendrite tip kinetics using IMS with solute trapping

modications [115] and with phase diagram infor-

mation delivered via coupling to ThermoCalc.

Computation of macrosegregation using the sub-

routines described in Section 5.1 has also been per-

formed [183]. Other practical solidication

problems in multicomponent alloys are being ana-

lyzed. For example, the relative importance of

nucleation vs growth competition in understanding

the identity of the primary phase (f.c.c. or b.c.c.) in

FeCrNi alloys near the monovariant eutectic line

of the ternary liquidus surface of the phase diagram

has been analyzed [184]. The mechanism for the

formation of austenite dendrites in the so-called

eutectic region of the microstructure of FeCSi

spheroidal cast irons [185] is another example.

6. SIMULATION METHODS

With the advent of very powerful computers,

advanced numerical methods and better under-

standing of the physical phenomena involved in

solidication, it is not surprising that computer

simulations are becoming increasingly used for the

modeling of microstructure formation and associ-

ated characteristics or defects (e.g. microsegregation

pattern, porosity formation, etc.). Over recent

years, three major contributions have emerged: (1)modeling of microstructure formation using phase-

eld or front-tracking-type methods; (2) modeling

of solidication processes and microstructural fea-

tures using averaging methods; (3) modeling of

grain structure formation using physically based

Cellular Automata or ``Granular Dynamics''

methods. All three are important since the macro-

scopic scale of a solidication process (typically

cmm), the grain size (typically mmcm) and the

characteristic length of the microstructure (mm)

encompass six orders of magnitude and cannot be

Fig. 14. Colony structure simulated using a phase-eld model for the directional solidication of aeutectic alloy with a dilute ternary impurity [169].



19/28

taken into account simultaneously. Their main

characteristics are briey discussed hereafter. It

should be emphasized that the smallest size of the

microstructure (mm) is still three to four orders of

magnitude larger than the size of the atoms or mol-

ecules or the thickness of the solidliquid interface.

This much ner scale still sets another limit to be

accounted for in molecular modeling (which will

not be treated here) or in any realistic phase-eld

simulation, which precisely intends to model the

gradual transition from liquid to solid.

6.1. Modeling of microstructure

In most metallic alloys solidied under normal

conditions, microstructure formation is controlled

by solute diusion and curvature, heat diusion

occurring over much longer distances (i.e. Lewis

number of the order of 104 for most metals).

Simulation at this level normally requires following

the interface separating the solid and liquid phases(front tracking). This has been achieved successfully

in simple two-dimensional geometry using either the

boundary element method (BEM) [186] or the nite

element method (FEM) [187]. In the rst technique,

only the interface is enmeshed and the Greens func-

tions are used to solve the diusion problem. In the

second method, dynamic remeshing of the domain

is necessary. These methods are accurate but di-

cult to implement even in two dimensions.

Furthermore, topological changes such as coalesc-

ence (merging of two dendrite arms) cannot be

handled. They have been of great use to calculate

the transient from a planar front to cells and the

growth kinetics of the dendrite tip [186].

In pseudo-front-tracking techniques [188190],

the solidliquid interface is spread over only one

mesh of the nite dierence (FDM) or nite volume

(FVM) enmeshment and the concept of the volume

fraction of solid, f (or liquid), is introduced: it is

equal to unity in the solid, zero in the liquid and in-

termediate for the ``interface meshes''. Among the

advantages of such methods, fairly easy implemen-

tation and computation speed can be mentioned.

However, the error associated with the estimation

of curvature from the divergence of the normalized

gradient of f is large (H1030%). Since preferred

growth directions and dendrite tip kinetics are gov-erned by the small anisotropy of the interfacial

energy (H110% in metallic alloys), such methods

can only give qualitative results.

In the phase-eld method, the diuse nature of

the solidliquid interface of metallic alloys is con-

sidered and f varies continuously from 0 to 1

over a certain thickness, d. Using a free energy or

entropy formulation, two equations governing the

evolution of the phase eld and the evolution of

either heat or solute can be derived and solved

using an explicit FDM. No front tracking being

required, the technique is ecient and capable of

reproducing most of the phenomena associated

with microstructure formation (dendrite tip kin-

etics, preferred growth direction, coarsening, co-

alescence, etc.). Initially developed for thermal

dendrites in two dimensions, it has been extended

to solutal dendrites [24] and three dimensions [9,

12]. As an example, Fig. 1 shows a thermal den-drite growing along h100i directions [9] and a solu-

tal NiCu dendrite growing along h110i directions

[12]. However, the technique also has some disad-

vantages. The rst one is related to the eective

thickness of the diuse interface, d, of alloys (H1

5 nm) which is three to four orders of magnitude

smaller than the typical length scale of the micro-

structure. Since it must spread over several points

of the mesh, this considerably limits the size of the

simulation domain, even if d is multiplied by some

arbitrary factor (10100). It is to be noted that

this upscaling of d biases curvature eects by

introducing some ``numerical curvature'' and also

induces coalescence of dendrite arms at a much

earlier stage of growth. The second problem arises

from the attachment kinetics term that plays a sig-

nicant role in the phase-eld equation, unlike

microstructure formation of metallic alloys at low

undercooling. These two factors have so far lim-

ited phase-eld simulations of alloy solidication

with realistic solid-sate diusivities to relatively

large supersaturations. Recent mathematical and

computational advances, however, are rapidly

changing this picture. Some of the recent advances

include: (1) a reformulated asymptotic analysis ofthe phase-eld model for pure melts [9, 11] that

has (i) lowered the range of accessible undercool-

ing by permitting more ecient computations with

a larger width of the diuse interface region (com-

pared with the capillary length), and (ii) made it

possible to choose the model parameters so as to

make the interface kinetics vanish; (2) a method to

compensate for the FDM grid anisotropy [11]; (3)

an adaptive FEM formulation that renes the

zone near the diuse interface and that has been

used in conjunction with the reformulated asymp-

totics to simulate two-dimensional dendritic growth

at low undercooling in two dimensions [10]; (4) astochastic Monte Carlo treatment of the large-

scale diusion eld that provides an alternative to

adaptive mesh renement that has been im-

plemented in both two dimensions and three

dimensions (Fig. 1) at low undercooling [14]; (5)

the implementation in the method of uid ow

eects [191193]; and (6) the extension of the tech-

nique to other solidication phenomena including

eutectic [194196] and peritectic reactions [69], and

the interaction of dendrites with surfaces [197].



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6.2. Modeling of processes and average

microstructural features

Modeling of solidication processes and micro-

structural features has beneted from two main

contributions: (i) the introduction of averaged con-

servation equations previously developed for dipha-

sic media [198200]; and (ii) the coupling of these

equations with microscopic models of solidicationdescribing grain structure formation and other

microstructural features (e.g. secondary arm spa-

cing, microsegregation model, etc.) [200204].

When conservation equations are averaged over

the liquid and solid phases, the interfacial continu-

ity condition automatically vanishes and average

entities (e.g. mean specic mass, enthalpy or solute

concentration) appear. Because the constitutive

equations of the solid and liquid are widely dierent

(typically the viscosity increases by 20 orders of

magnitude during solidication), the momentum

conservation equation is averaged only over the

liquid phase. This introduces the interfacial bound-

ary condition in the form of a drag term when the

solid is supposed to be xed (packed bed).

Beckermann and co-workers have extended this

averaged equations formalism in order to encom-

pass the situation of free moving equiaxed grains

and make a smooth transition with the packed bed

limit [205].

The same authors have also coupled these aver-

aged conservation equations with microscopic

models of grain structure formation in a way simi-

lar to the micromacro approach reviewed in Ref.

[206] but including the eect of convection [207

209]. An example of this is shown in Fig. 15. At

early times (rst panel), the grains nucleate at the

left wall and are swept by the convection currents

around the cavity. At later times, when the grains

have grown, they settle and form a bed of increas-

ing height (next panels). The nal structure is highlynon-uniform (right panel), with a lower grain den-

sity (i.e. larger grains) observed near the top due to

the settling eect (after Ref. [208]). Convection

modies the growth kinetics of the dendrite tips

and thus of the grain envelope [210], but also

entrains free grains. Experimentally based laws

similar to Stokes drag are now available for high

Reynolds number, non-spherical and porous grains

[211]. The diculty in such an approach is to

include the fragmentation of dendrites induced by

convection [198, 212], although a preliminary

attempt was presented in Ref. [212].

Averaged conservation equations, coupled or not

with detailed microscopic models of solidication,

have been applied primarily to the problem of

macrosegregation induced by thermal or solutal

convection. Most of the developments in this area

are based on structured FVM meshes and for two-

dimensional geometries. These methods have been

extended recently to small three-dimensional

domains and FEM [213] and a benchmark compari-

son between FVM and FEM has shown that these

methods are quite sensitive to the formulation, in

Fig. 15. Computed time evolution of the number density of equiaxed dendritic grains during solidica-tion of an Al4% Cu alloy in a cavity cooled from the left side [208].



21/28

particular regarding the formation of freckles (seg-

regated channels) [214]. Implementation of a macro-

segregation model based on averaged equations into

a full-scale, three-dimensional simulation was

recently accomplished [215]. The eect of grain

movement on macrosegregation has been addressed

in Refs [208, 209].

6.3. Modeling of grain structures using stochastic

methods

Grain structure formation can be modeled suc-

cessfully with averaging methods (see previous sec-

tion). Such methods are particularly suitable when

the grain size is small with respect to the scale of

the process (e.g. in continuous casting of inoculated

aluminum alloys) and/or when only one mor-

phology is present (e.g. columnar or equiaxed).

They are however unable to predict grain compe-

tition in the columnar zone and the associated tex-

ture evolution, and furthermore cannot provide a

representation of the microstructure. The predictionof morphology transitions (from outer equiaxed to

columnar and from columnar to equiaxed) [175,

204, 216] is also quite dicult with averaging

methods. In order to overcome these shortcomings,

stochastic methods have been developed over the

past decade [217225]. It should be pointed out that

the stochastic aspect is only related to nucleation

(random location and orientation of nuclei) whereas

growth is usually treated in a deterministic way.

Two types of models can be distinguished:

(i) Cellular Automata (CA) have been developed

for dendritic grain structures and can treat arbi-

trary shapes and grain competition [217222]. Inthis technique, the solidication domain is

mapped with a regular arrangement of cells and

each grain is described by a set of cells, those

located at the boundary (i.e. in contact with

liquid cells) being active for the calculation of

the growth process.

(ii) In ``Granular Dynamics'' (GD) techniques,

the surface of each grain is subdivided into an

ensemble of small facets [223225]. The growth

stage of each grain is then described by a set of

parameters, e.g. the position of its center, the

radial positions of its facets and their status

(contact with the liquid or with another grain),

etc. This latter technique is more appropriate for

nearly spherical morphologies (e.g. equiaxed

eutectics or globulitic grains) and can handle the

transport of equiaxed grains fairly well.

Although it has been demonstrated that CA can

also treat the movement of equiaxed grains, it is

particularly well adapted to describe grain compe-

tition and texture evolution in dendritic columnar

specimens, as

SOLIDIFICATION MICROSTRUCTURES

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