Review 2006 (Spittle)

8102019 Review 2006 (Spittle)

httpslidepdfcomreaderfullreview-2006-spittle 123

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Columnar to equiaxed grain transition inas solidified alloys J A Spittle

The generally reported observations pertinent to any proposed interpretation of the columnar toequiaxed transition (CET) in as solidified alloys are initially considered The review then proceedsto consider the proposed mechanisms of equiaxed grain formation the influence of alloy andprocessing conditions on the CET criteria for the termination of columnar growth and finallydeterministicstochastic models for predicting the CET In conclusion the present level ofunderstanding of the CET and current modelling capabilities are summarised and assessedKeywords Columnar to equiaxed grain transition As solidified alloys Modelling

IntroductionAs solidied metals (shaped castings ingots continu-ously cast alloys directionally solidied materialswelds etc) consist of grains (single phase or multi-phase) formed during solidication by one or more of the various types of phase transformation that can occuron cooling eg primary phase freezing from the melteutectic solidication and peritectic transformationSubsequent to nucleation these grains either continueto grow preferentially in a direction normal to theliquidus isotherm in the solidifying system (ie theybecome elongated in one dimension and are termedcolumnar) or they grow in suspension in supercooledliquid (termed equiaxed)

To manufacture useful products most alloys arefabricated either by solid state deformation of simplecast forms such as sheet billet or slab (wrought alloys)or by lling complex shaped mouldsdies with moltenmetal (casting alloys) The as cast microstructures of wrought alloys (which are often continuously or semi-continuously cast) consist predominantly of solid solu-tion grains of the initial primary phase However this is

not always the case eg in low carbon steels the initialprimary phase d ferrite is transformed to single-phase caustenite at a lower temperature On the other handcasting alloys in most alloy systems often deliberatelycontain large volume fractions of eutectic (eg alumi-nium and zinc-base alloys and cast irons) Therefore if the alloy compositions lie close to the eutectic value themicrostructures will be dominated by eutectic grains(cells) However other casting alloys eg hypoeutecticAlndashSi alloys may contain signicant amounts of primary solid solution phase

In as solidied structures growth of columnar grainsoften terminates with the appearance of an equiaxed

zone or possibly a band of equiaxed grains (which thenundergoes a further transition back to columnar

growth) This is known as the columnar-to-equiaxedtransition (CET) (Fig 1) In addition a band of equiaxed grains (the chill zone) may be seen at theoutside of the casting the formation of which precedesthe growth of the columnar grains

When investigating the CET experimental observa-tions and models of the transition have usually beenconned to the grain structures of primary solid solutiongrains (although it can also be observed in the eutecticgrain structures of impure binary alloys or multi-

component alloys) In alloys these solid solution grainsinvariably grow with non-faceted cellular or dendriticinterfaces Columnar grains freeze with a preferredorientation such that the long axis is parallel to a speciccrystallographic direction 1 Equiaxed dendritic grains onthe other hand are randomly oriented

The CET has been examined in wrought and castingtype alloys because of the advantages offered byequiaxed grain solidication in both cases in themajority of situations In the case of wrought alloysne-grained equiaxed structures reduce susceptibility tohot tearing and generally improve structural homoge-neity (eg prevent the growth of columnar lsquofeathercrystalsrsquo in Al alloys) Casting alloys for shapedcastings are usually far less susceptible to hot tearingand ne grain sizes enhance feeding (eg of long freezingrange aluminium alloys) improve the distribution of shrinkage porosity and increase fatigue lives

In commercial practice attempts are made to produceeither wholly columnar structures (this is rare but theclassic example is the production of directionallysolidied turbine blades) or wholly equiaxed structuresInterest in the CET therefore stems from the wish totheoretically understand the solidication conditionsthat dene the transition between these two extremes It

is anticipated that a mixed columnar-equiaxed structurewould be undesirable in any situation (if it can possiblybe avoided) The majority of the experimental studiesdeveloped models of the CET have excluded theaddition of deliberately added inoculants to promoteheterogeneous nucleation and grain renement Thisenables mixed columnar-equiaxed structures to be more

Materials Research Centre School of Engineering University of WalesSwansea Swansea SA2 8PP UK

Email jaspittleswanseaacuk

2006 Institute of Materials Minerals and Mining and ASM InternationalPublished by Maney for the Institute and ASM InternationalDOI 101179174328006X102493 International Materials Reviews 2006 VOL 51 NO 4 24 7



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easily achieved experimentally and modelled since thenumberpotency of nucleant particles will be lowerHowever commercially the use of grain reners may bethe normal practice eg in the DC casting of Al alloysand one recent study has attempted to model the CETduring directional solidication of Al alloys with grainrener addition 2

Because of the problems associated with the manip-ulation of processing parameters and structural exam-ination on a commercial scale most qualitative andquantitative studies of the CET have been performed

experimentally on relatively small volumes of lowmelting point materials (usually aluminium alloys)Generally speaking most studies have involved one of two approaches either (i) investigation of the inuenceof various parameters on the as solidied grainstructures of simple shapes eg cylinders of eitherpoured castings or alloys melted and solidied lsquo in situ rsquo

or (ii) the directional solidication of alloys either underBridgman or non-steady (solidication from cooledchills) conditions Alternatively because of the opacityof metals the freezing of transparent non-metallicanalogues of alloy systems has been studied (egNH 4ClndashH 2O cyclohexanolndashphenol red)

Although the CET has been reported and discussed inmany early studies including those of Stead 3 Howe 4

Genders 5 and Hensel 6 signicant renewed interest arosefrom the initial quantitative derivation of the conditionsnecessary for the breakdown of a planar interface as aresult of lsquoconstitutional supercoolingrsquo (CS) by Tilleret al 7 [equation (1)] CS leading to the instability of planar growth will occur when

G =R v mC 0(1 k )=kD (1)

where G is the temperature gradient in the liquid R theinterface velocity (in this review R and V are used inter-changeably for interface velocity for ease of presentingreported data) m the liquidus slope C 0 the initial alloy

composition k the equilibrium distribution coefcientand D the solute diffusion coefcient in the liquidEquiaxed grain formation and the CET have been

extensively studied over the last half century and theimportant aspects that have been examined can besummarised as follows

(i) investigation of the mechanisms of equiaxedgrain formation (how and where do theequiaxed grains originate)

(ii) qualitative and quantitative experimental eva-luation of the parameters influencing the CET

(iii) investigation of the conditions causing thetermination of columnar freezing (what physi-cally causes columnar grains to stop growing)

(iv) development of deterministic and stochasticmodels for predicting the CET

The most recent comprehensive review of the CETgiving consideration to all of these aspects is that due toFlood and Hunt 8 In the intervening years there hasbeen additional experimental research and further modeldevelopment particularly the use of stochastic modelsfor simulating the nucleation and growth of individualcolumnar and equiaxed grains

Requirements and features of CET Unless otherwise stated the remainder of this reviewrelates to the grain structures of solid solution grains inalloys (usually binary alloys) Before embarking on adetailed consideration of the CET it is useful to state theobvious requirements for a CET and to list some simpleobservations pertinent to the discussion of how it isinuenced by different solidication parameters Toobtain a structure revealing a CET (ie showingcolumnar growth arrested by equiaxed grains) thefollowing are required

(i) the presence of supercooled liquid ahead of ornear the columnar front in which either (a)

equiaxed grains nucleate and grow or (b) towhich grainsfragments of grains nucleated else-where are transported and then grow In thelatter case during the early stages of solidifica-tion conditions must exist in the melt for thenucleationfragmentation of grains elsewhere andtheir transport and survival

1 Columnar to equiaxed (CET) grain transition in an Alndash4 wt-Cu alloy 56

Spittle Columnar to equiaxed grain transition in as solidified alloys

248 International Materials Reviews 2006 VOL 51 NO 4



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(ii) equiaxed grains of sufficient size andor numberbesideattached to the columnar interface toarrest columnar growth

These statements make no comment on the origin of thesupercooling or the mechanism of columnar growtharrestment

Observations that are signicant to further under-standing of the transition are

(i) pure metals poured into cold moulds haveeither mixed chillcolumnar or wholly columnarstructures

(ii) a cast solid solution alloy may have structuresranging from columnar to mixed columnar-equiaxed to wholly equiaxed depending oncasting conditions

(iii) in binary alloys the same amount of differentsolutes (atomic or weight ) produces differentgrain structures

(iv) equiaxed zones can be observed in ingots withcellular columnar zones

(v) equiaxed grains are formed in alloys even in theabsence of deliberately added nucleants Thisimplies either the heterogeneous nucleation of equiaxed grains on unknown impurity nucleantsor alternative mechanisms of origin

(vi) solidified weld metals rarely display a CET(vii) DC cast wrought Al alloy ingots are dominated

by columnar structures in the absence of addedgrain refiners

(viii) because of fluctuations in the solidificationconditions VAR ingots of high melting pointalloys can display alternating columnar andequiaxed structures

(i)ndash(iii) emphasise the importance of solute level andalloy system on the CET and (ii) (vi)ndash(viii) theimportance of solidication conditions

Because of the difculty of tracking the columnarinterface under transient growth conditions and of accurate measurement of shallow temperature gradientsexperimental studies of the multidirectional freezing of poured castings or of in situ melted and solidied alloysare generally conned to qualitative investigations of theorigins of equiaxed grains and of the relative effects of different parameters on the CET Quantitative evalua-tion of the conditions at the columnar interfaceimmediately preceding the CET usually requires the

directional freezing of alloys However depending onthe manner in which directional freezing is experimen-tally achieved one or more of the proposed mechanismsof equiaxed grain formation may be eliminated

Mechanisms of equiaxed grain formationThere have been a number of reviews of the proposedmechanisms of equiaxed grain formation which includethose of Kisakurek 9 Flood and Hunt 8 and Hutt and StJohn 10 These reviews particularly that of Hutt and StJohn also include detailed information on experimentalstudies designed to try to validate or repudiate

individual mechanisms As will be seen below theproposed mechanisms differ according to how whereand when the equiaxed grains are thought to originateduring solidication

It is extremely confusing for a reader attempting togain an insight into the CET to be presented with thesevast amounts of detailed and often conicting data

Sufce to say that a variety of techniques designed toeither (i) remove the possibility of certain mechanismsever operating or (ii) provide sufciently strong evidenceto support the conclusion that a particular mechanismswas likely or unlikely to have operated during solidica-tion have been devised These techniques include thepouring of castings into heated moulds in situ meltingand solidication (ie no pouring) the use of magneticelds to either enhance stirring or to remove convectionat specic stages of solidication the use of mechanicalbarriers (assumed not to be thermal barriers) positionedboth horizontally and vertically in castings vibration of castings the direct observation of the solidication of metals and non-metallic analogues of metals andthermal analysis Consideration of the arguments tovalidaterepudiate individual mechanisms will thereforebe omitted but the reader can refer to one of thesuggested reviews After presenting the proposedmechanisms a summary of current thinking regardingthe mechanisms will be given

As mentioned above the use of directional solidica-tion to quantitatively examine the CET may automati-cally preclude certain mechanisms from operating Insuch studies experimentalists usually conclude whichmechanisms they believe were operative and directionalsolidication modellers may assume that a specicmechanism is operative see the section lsquoModels forpredicting the CETrsquo below

The proposed mechanisms of equiaxed grain forma-tion in the absence of added nucleants are consideredbelow It is normally assumed that the solutes haveequilibrium distribution coefcients less than unity sothat solute is rejected at the solidliquid interface onfreezing Solute accumulation at the interfacelowers the liquidus and depresses the interface freezingtemperature

Constitutional supercooling hypothesisThis hypothesis originally proposed by Northcott 11 andlater by Winegard and Chalmers 12 assumes that soluteaccumulation at the tips of the cellulardendriticcolumnar grains depresses the tip temperature leadingto constitutional supercooling ahead of the tips If acritical undercooling is exceeded heterogeneous nuclea-tion is suggested to occur on unknown nucleants in themelt (Fig 2)

The Big Bang hypothesisProposed by Chalmers 13 this theory argues that thegrains in the equiaxed zone nucleate at the time of pouring in the thermally supercooled region beside themould walls Grains not attached to the wall will beswept into the bulk liquid by convectionpouringturbulence and those that survive remelting will growin the constitutionally supercooled liquid ahead of theinterface advancing from the mould walls The crux of this theory is that all grains seen in the casting areheterogeneously nucleated at the onset of solidication

Another possible aspect of this proposed mechanismnot dwelt on by Chalmers is that the grains nucleated inthe thermally supercooled region will initially grow veryrapidly into the supercooled liquid rejecting solute attheir freezing interfaces This will lead to soluteaccumulation depression of the liquidus and lsquogrowthrestrictionrsquo of the grains This growth restriction may


International Materials Reviews 2006 VOL 51 NO 4 24 9



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result in a further increase in undercooling and anincrease in size of the supercooled region Underparticular circumstances (controlled by alloy parametersand solidication conditions) undercooling may extendthroughout the casting at the time of pouring giving awholly equiaxed structure

Dendrite arm remelting From observations on the solidication of cyclohexanol(an analogue of a pure metal) and cyclohexanolwith uorescein (an analogue of an alloy) Jacksonet al 14 concluded that the dendrites of pure materialsdiffer from those of an alloy In a pure materialthe diameters of the main stem and branches aresimilar However in the alloy case as soon as thebranch grows through the impurity layer around themain stem it broadens so that it is attached to the mainstem by a narrow neck It was also demonstrated in acontinuously solidifying analogue system that uctua-tions in growth rate could lead to branches remelting off the main stem and the formation of isolated crystalsThey therefore proposed that dendrite fragmentsresulting from local recalescence owing to uctuationsin growth rate caused by convective mixingstirring arethe nuclei of equiaxed grains The fragments are carriedinto the bulk melt by buoyancy or convection wherethey grow as new grains in the constitutionally super-cooled melt

Showering of dendrite particlesSouthin suggested from observations on the solidiedstructures of laboratory ingots of Al Alndash0 1 and Alndash 2Cu that as heat is lost from the surface of an ingot azone of coarse dendritic grains forms as an upperlayer 15 At some stage dendrites or dendrite fragmentsare dislodged from this layer by some unspeciedmechanism and sink until they meet the solid metalgrowing from the mould walls He observed that the

grains in the equiaxed zone are comet shaped with acoarse dendritic head and a tail that grows with the samestructure as the columnar zone

Separation of equiaxed crystals from mould wallThis theory proposed by Ohno et al 16 bears aresemblance to that of Chalmers in the section lsquoTheBig Bang hypothesisrsquo above in that the grains compris-ing the equiaxed zone are thought to originate in thevery early stages of solidication Ohno et al usedoptical microscopy to directly observe the start of solidication at the mould wall for the unidirectional

freezing of SnndashBi BindashSn SnndashPb and SnndashSb alloys inhorizontal Pyrex tubes They observed that in thepresence of solute growth of granular shaped equiaxedcrystals took place which were attached to the mouldwall by narrow necks Subsequently and before theformation of a complete solid shell at the mould wall itwas observed that equiaxed crystals separated from the

2 Schematic series ( a ndashe ) illustrating change of liquidus temperature and actual temperature in bulk liquid ahead ofadvancing columnar front and origin of constitutional supercooling 13





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mould wall owing to remelting of the necks caused bythermal convection The investigators considered thatthese detached equiaxed crystals act as the nuclei for theformation of the grains in the equiaxed zone

Overall consideration of proposed mechanismsof equiaxed grain formation

The various proposed mechanisms of equiaxed grainformation can be grouped into two categories(i) those mechanisms that involve direct heteroge-

neous nucleation of equiaxed grains in the bulkliquid 1213

(ii) those mechanisms that involve the detachmentof dendritesdendrite fragments from grainsnucleated and growing from the mould wallsupper liquidndashair surface of the ingot 14ndash16

In each category the differences between the mechan-isms are governed by when during solidication andwhere in the casting the mechanisms are thought tooperate Without exception experimental data exist tosupport every one of the ve mechanisms under theconditions prevailing in the individual experimentsHowever the experimental arrangements often smallcast cylinders of alloys or the freezing of non-metallicanalogues differ considerably from the shapes andsolidication conditions found in shaped castings largeingots and continuously cast alloys In reality it is likelythat more than one of the mechanisms may be operativein a particular casting situation However speciccasting conditions are likely to favour different mechan-isms In the case of shaped castings which often havethin sections signicant undercooling (both in degreeand depth) and convectionpouring turbulence arethought to favour those mechanisms that operate atthe onset of solidication 1316 In the case of large ingotsand continuously cast alloys (associated with longsolidication times and signicant interdendritic con-vective ow) or processes such as vacuum arc remeltingwhich can be subject to major uctuations in thermalsolidication conditions dendrite remelting is likely tobe enhanced In the absence of hot tops solidication onthe top surface of large ingots may also contribute to theformation of equiaxed grains On the other hand underconditions of perfect directional freezing into a positivetemperature gradient in the absence of convective owor any pouring turbulence if a CET is observed it mustoriginate from grains heterogeneously nucleated inconstitutionally supercooled liquid

All the above mechanisms have also been consideredwhen examining the CET in other as solidiedstructures eg welds In the latter case an additionalalternative mechanism termed lsquograin detachmentrsquo hasbeen proposed 17 This mechanism involves the detach-ment of small grains from the partially molten part of the heat affected zone This requires ne-grained alloyswith relatively large freezing ranges

Parameters influencing CET Many experimental investigations of the CET have beenqualitative in nature recording the inuence of differentparameters such as superheat and alloy content on therelative sizes of the columnarequiaxed zones columnarzone length grain sizes etc for a specic experimentalarrangement Although as will be described below

there have been numerous attempts to quantitativelydetermine the precise set of conditions that exist atthe location of a CET these conditions are still notsufciently well understood to facilitate the prediction oravoidance of a CET for different solidifying geometriessolidication processes solidication conditions andalloy systems Flood and Hunt 8 list the importantparameters as alloy factors superheat uid ow castingsize mechanical vibration and inoculation grain rene-ment Mechanical vibration is not considered at all inthe present review and there have only been a limitednumber of studies of the CET in the presence of addedgrain reners either experimentally or by modelling

Alloy parametersOmni-directional freezingAlthough equiaxed crystals can be found in thermallyundercooled lsquo in situ rsquo melted and solidied pure metals 18

it is accepted that in castingswelds of pure metalscolumnar grain growth predominates In castings and

welds solutes (soluble in the liquid) is thereforerequired to cause a CET This solute provides theconstitutional supercooling necessary for the survivaland growth of equiaxed grains and possibly theirnucleation A variety of alloy parameters can inuencethe transition including m k D and C 0

A number of generally accepted conclusions havebeen reached from observations on poured or lsquo in situ rsquocastings For a given alloy system and a constantsuperheat increasing C 0 tends to decrease the columnarzone length and reduce the equiaxed grain size iepromote the CET 151920 Tarshis et al 21 in a classicpaper examined the variation in grain size in binary Ni

alloys and Al alloys For each alloy system a variety of solutes were employed in order to vary the magnitudesof the different alloy parameters The grain sizes of aseries of poured binary alloy castings were compared fora xed superheat and a xed level of solute additionThree series of alloys were examined Nindash1 at- soluteNindash5 at- solute and Alndash1 at- solute For each seriesit was found that the relative grain size decreased as theparameter P increased where P is given by

P ~ mC 0(1 k )=k (2)

At low P values the structures were columnar andchanged to columnar-equiaxed and nally equiaxed as P increased (Fig 3)

In a given binary system if the solidus and liquiduslines are straight for compositions below the solubilitylimit P (since termed the constitutional supercoolingparameter) is equal to the equilibrium freezing rangeTarshis et al 21 proposed that since this parameterpredicts grain size it permits the selection of solutes asgrain rening additions (in the absence of addedinoculants) It should be borne in mind however thatthe quantitative observations relate to a particularmould material casting size and set of casting condi-tions Prediction of the CET if one or more of these is

varied is impossible The form of the variation of caststructure with P has been conrmed for various binaryAl alloy systems 22 From observations on AlndashZn alloyscontaining up to 85 wt-Zn in which P and freezingrange vary independently with composition Dohertyet al 23 found no direct correlation between the CET andfreezing range but a reasonable correlation with P In





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recent years when examining the inuence of phasediagram parameters it has become more common torelate grain sizes to the lsquogrowth-restriction parameterrsquoQ rather than P where Q is given by

Q~ kP (3)

Although as illustrated above numerous examples existof where solute redistribution on solidication promotesgrain renement and the CET in some alloy systemssolute addition can result in grain coarsening Animportant example is seen in castings of AlndashSi alloysa(Al) grain size initially decreases with Si content up to 2ndash3Si and thereafter continues to increase 2425 PbSb and Bi also cause coarsening of the Zn solid solution

grains in Zn and Zn-base alloys26

Unidirectional solidificationInstead of trying to experimentally quantify the inu-ence of different alloy parameters on the CET for agiven set of casting conditions during omni-directionalsolidication an extensively used alternative approachhas been to try to determine the conditions existing atthe location of the CET for a given alloy system Studiesof this type involve unidirectional solidication

Plaskett and Winegard 27 examined the unidirectionalsolidication of AlndashMg alloys under non-steady condi-

tions The alloys were melted in situ in a graphitecrucible held in a furnace and directionally solidiedfrom a cooled chill For each alloy the values of G andR were determined at the location of the breakdown of columnar growth Over the composition range investi-gated 0ndash10Mg an almost linear relationship wasobtained between C 0 and the value of G R12 at

breakdown They suggested that the results supportedthe Winegard and Chalmers mechanism of equiaxedgrain formation 12 Elliott used a thermal valve techni-que which permits independent control of G and R toinvestigate the directional freezing of PbndashSn alloyscontaining up to 6 wt-Sn 28 A linear relationshipbetween G R and C 0 for the CET was obtained inagreement with the analytical model of Tiller 29 for thetransition (see the section lsquoDeterministic modelsrsquo below)(Fig 4) This model again assumes that equiaxed grains

3 Variation of relative grain size of Alndash1 at- solute alloys as a function of parameter P 21

4 Experimental plot illustrating linear relationshipbetween wt-Sn and G R for columnar to equiaxedtransition in PbndashSn alloys 28





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form by heterogeneous nucleation ahead of the colum-nar front due to CS Tiller predicted that the equiaxedzone would form when the maximum undercoolingexceeded a critical value

The signicant renewed interest in the CET since the1980s has resulted from the emergence of new modellingapproaches to investigate the transition which as aconsequence have prompted further experimental stu-dies The studies have been performed on a variety of alloy systems and the data have often been compared tothe predictions of models Alternatively modellers haveused experimental data to assess the general validity of their models

Mahapatra and Weinberg 30 and Ziv and Weinberg 31

using a 1D nite difference heat transfer modelinvestigated the CET under non-steady freezing condi-tions for in situ melted and directionally solidiedalloys of Sn containing 5 10 and 15 wt-Pb and Alndash 3 wt-Cu respectively In the case of the SnndashPb alloysthey reported that the CET occurred when thetemperature gradient ahead of the advancing dendritetips for each alloy fell below a critical value The values(which were determined from a nite difference heattransfer model of the solidifying ingot) were 1 0 11 and13 K cm ndash1 respectively The CET could not be directlyrelated to dendrite tip velocity For the Alndash3 wt-Cualloy the CET occurred when the gradient fell to06 K cm ndash1 which was found to be in agreement with agradient prediction from Huntrsquos analytical model of theCET (see the subsection lsquoAnalytical modelsrsquo below) 32

From adding different amounts of nominally Al5Ti1Bgrain rener to the AlndashCu alloy it was found that adenite amount was required in order to effect the CET

They suggest this indicates that a critical high density of nuclei is required to form a ne-grained structureSuri et al 33 again using a 1D heat transfer model to

determine the values of G and V at the positions of theCET studied Alndash4 5Cu alloys directionally solidiedunder non-steady conditions for different superheatsand conditions of heat extraction They reported thatthe transition occurred if

G v 074V 064 (4)

Kim and Grugel 34 using a Bridgman type furnaceexamined the CET of the Cu dendrites in PbndashCu alloyscontaining 4 9 and 14 wt-Cu The alloys weredirectionally solidied at velocities ranging from 1 to100 mm s ndash1 and a temperature gradient of 4 5iexcl05 K mm ndash1 In this composition range it was foundthat as growth velocity increased there was a transitionfrom fully columnar to fully equiaxed The growthvelocity to effect a fully equiaxed structure was observedto drop rapidly with decreasing Cu content The investi-gators concluded that the results were in qualitativeagreement with Huntrsquos theory of the CET 32

Ledgard and McCartney 35 using a Bridgman typearrangement examined the directional solidicationof Alndash1 8 wt-Si alloys to which an Alndash6 wt-Tindash

002 wt-C grain rener was added at levels of 0

01003 or 0 05 wt-Ti Alloys were prepared from both

superpurity (99 995 wt-) and commercial purity(997 wt-) Al Pull rates of 1 4 10 30 and60 mm min ndash1 were used and growth velocities were saidto be within 10 of these rates Temperature gradientsin the liquid were typically 5 K mm ndash1 at the highest

growth rate rising to 10 K mm ndash1 at the lowest growthrate It was found that the superpurity alloys did notproduce equiaxed structures for any grain reneraddition level or growth rate employed In the case of the commercial purity alloys equiaxed structures wereobserved for all growth rates at the 0 03 and 0 05Tilevels At the 0 01 level columnar grains wereobserved at the lowest growth rate but equiaxedstructures were obtained at all pull rates in excess of 4 mm min ndash1 The results for the critical velocity for theCET were considered using the Hunt model 32

Reasonable agreement with the model was found foran assumed nucleation undercooling of 0 75 K and anestimated density of nucleating sites of 200 mm ndash3

Pollock and Murphy 36 using Bridgman type furnacesexamined the breakdown of single crystal solidicationin high refractory nickel-base alloys Ten different alloyswere studied under conditions of directional solidica-tion All 10 alloys were studied using xed values for thegrowth rate and thermal gradient However one of the

alloys (Al60 Cr4

5 Co12

5 Hf0

16 Re6

3 Ta7

0W5 8 Nibal) was investigated for withdrawal rates

ranging from 4 26 10 ndash4 to 1 136 10 ndash2 cm s ndash1 and ther-mal gradients in the range 0 3ndash140 K cm ndash1 No graindefects were observed when gradients were greater thanapproximately 15 K cm ndash1 For a xed withdrawal ratethey observed that as the gradients decreased below thiscritical value an abrupt transition from single crystaldendritic to equiaxed solidication was not observedInstead there was a large intermediate range of conditions where grains nucleate and grow withoutcompletely interrupting single crystal columnar growthThe primary dendrite arm spacing at which isolated

grains began to nucleate and grow corresponded withthat which marked the onset of freckling The authorsconclude that thermo-solutal convection promotes thedevelopment of both types of defect and that dendritedetachmentfragmentation contribute to the develop-ment of isolated grains and ultimately to the transitionto polycrystalline equiaxed solidication This conclu-sion regarding the origin of the equiaxed grains is inmarked contrast to those in the directional studiesalready described where it is assumed that the equiaxedgrains nucleate in supercooled liquid ahead of theadvancing front

Gandin 37 has studied the directional solidication of AlndashSi alloys containing 3 7 and 11Si under non-steady freezing conditions Liquid alloy is initially heldat a uniform temperature within a mould located in afurnace The furnace is lowered away from the mouldand directional freezing initiated by raising a water-cooled copper chill to contact the bottom of the mouldIn all three alloys a CET was observed at approximatelythe same position two-thirds along the ingot lengthGandin suggested that the equiaxed grains may haveoriginated from a dendritic surface layer at the metalairinterface by heterogeneous nucleation in the bulk liquidor by dendrite arm detachmentfragmentation

Ares and Schvezov38

studied the CET in PbndashSn alloysin the range 2ndash40Sn Again directional freezingoccurred under non-steady conditions During solidi-cation temperatures were measured at ve locationsalong the ingot length at 10 s intervals From thecooling curves a variety of parameters were calculatedThese included superheat cooling rates positions and





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L t d

velocities of the liquidus and solidus fronts length of themushy zone local solidication times and temperaturegradients For an alloy containing 2Sn their resultsshowed that there was no correlation between columnarlength and superheat and that the length of thecolumnar zone increases with the average cooling rateof the melt For a given cooling rate it was also foundthat the length of the columnar zone increases with alloycomposition From the ve thermocouples values of thetemperature gradient ahead of the liquidus interface andthe velocity of the liquidus interface were calculated forall the experiments performed over the compositionrange 2ndash40Sn These were used to plot gradient versusinterface velocity indicating the type of grain structure

present for each pair of values (Fig 5) When observedthe CET was not sharp but showed a transition zonewhere some equiaxed grains coexisted with columnargrains In all the experiments the gradient for the CETwas calculated to be within ndash0 8 to 1 K cm ndash1 It was alsofound that the CET corresponded to a critical interfacevelocity of 0010 iexcl 0005 cm s ndash1 The data in Fig 5the inuence of cooling rate on columnar length and thefact that the critical values of temperature gradient andinterface velocity were independent of alloy compositionand position of the transition led the investigators tosuggest that the process is mainly controlled by heatextraction From calculations they showed that theamount of heat ow decreases with time and reaches aminimum critical value of 010iexcl 004 J cm ndash2 s ndash1 at theCET A qualitative t with Huntrsquos model 32 was obtainedby adjusting the values of DT N and N 0 to t theexperimental results However in order to t theexperimental data the patterns of the variations of these two parameters with Sn content appeared to beunrealistic

Martorano and Capocchi 39 examined four castings of Cundash8Sn which were unidirectionally solidied undernon-steady conditions by pouring the molten metal intoan insulated mould standing on a copper base In two of the experiments castings were poured at 1110 u C onto anon-cooled base with and without the addition of 008Zr as an inoculant In the other two the alloyswere poured onto a water-cooled base again with andwithout the same level of inoculant addition In theabsence of the inoculant raising the pouring tempera-ture and increasing the heat ux from the base increased

the columnar length ie delayed the CET Theinoculated castings were completely equiaxed

Siqueira et al 40 examined the CET in SnndashPb alloys(10 and 30 wt-Pb) and AlndashCu alloys (2 5 8 and10Cu) again under non-steady directional freezingconditions The alloys were preheated in a mould heldin a furnace to the desired superheat The furnace wasthen switched off and water-cooling applied to the baseof the mould The CET was analysed for differentsuperheats and heat-transfer efciencies at the base Themould consisted of a stainless steel cylinder closed at thebottom with a disc of carbon steel The inner verticalsurface was coated with a layer of insulating alumina Insome experiments the heat-extracting surface of the

mould was coated with an alumina-based wash and inothers it was polished Temperatures along the length of the casting were monitored from a series of thermo-couples These measured temperatures were used in a 1Dnite difference heat transfer programme to determinethe transient heat transfer coefcient representing theglobal coolantndashcasting heat exchange Experimentalresults of the position of liquidus isotherms as a functionof time gave good agreement with those numericallypredicted using the corresponding transient heat transfercoefcient The numerical model was therefore used tocalculate certain solidication parameters associatedwith the CET transition These were tip growth ratetemperature gradient in the liquid and tip cooling rateFor all test conditions examined fully columnarstructures were always observed in the SnndashPb alloysFrom observations on the AlndashCu alloys the CET wasobserved to occur rapidly on a near horizontal plane andfurther from the chill with increasing heat transfercoefcient and increasing superheat For these AlndashCualloys it was reported that the CET occurred at tipgrowth rates ranging from 0 28 to 0 88 mm s ndash1 and fortemperature gradients in the liquid at the liquidusisotherm ranging from 0 28 to 0 75 K mm ndash1 Theinvestigators therefore concluded that a criterion forthe CET could not be based solely on tip growth rate orsolely on temperature gradient They suggested that amore realistic criterion should encompass both tipgrowth rate and temperature gradient through the tipcooling rate For the 15 tests on the AlndashCu alloys theCET occurred when the cooling rate fell below thecritical value of 02 K s ndash1 By comparing the gradients

5 Temperature gradients in liquid versus velocity of liquidus interface 38





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calculated by the authors for the CET for the veexperiments conducted with Alndash5 wt-Cu with thosepredicted by the Suri et al 33 criterion for the transitiononly two supported the latter criterion

Vandyousse and Greer 41 using a Bridgman techni-que investigated the inuence of solidication frontvelocity on the grain structures of Alndash4 15 wt-Mgalloys with and without inoculation The rener usedwas Alndash3 16 wt-Tindash0 17 wt-C and the addition levelwas varied After temperature stabilisation at 720 u C thealloy contained in an alumina tube was lowered throughthe furnace at a velocity between 0 05 and 1 0 mm s ndash1 The temperature gradient in the liquid was xed at10iexcl 2 K mm ndash1 Without inoculant addition the struc-tures were always columnar With low to medium levelsof addition (2 to 10 parts per thousand) mixed non-equiaxed and equiaxed structures were foundSolidication of the alloys was also studied using acommercial CA-FE (cellular automaton-nite element)package CalcoMOSTM 42 see the section lsquoStochastic

modelsrsquo below43

Both experiment and CA-FE model-ling predict that the CET is gradual with intermediatestages of short columnar or elongated grains Micro-structural studies of quenched interfaces appeared tosupport the contention of Hunt 32 that the CET is theresult of the competition between continued growth of existing grains and the appearance of new grains in theconstitutionally undercooled region ahead of the maingrowth front Experimental results were plotted on aninterface velocityndashtemperature gradient map showingthe CET and compared with simulation from Huntrsquosanalytical model Again there was reasonable qualita-tive agreement

Two very recent investigations on aluminium alloyshave also been carried out both with and without thedeliberate addition of grain reners using Bridgmantype furnaces As part of a European Space Agencyprogramme on the columnar to equiaxed transition insolidication processing Sturz et al 44 have studied thedirectional freezing of Alndash7 wt-Si alloy rods 10 mm indiameter and 200 mm in length The grain rened alloyscontained 215 mg g of titanium and 15 mg g of boronDuring solidication the temperature gradient wasdecreased and the solidication rate increased simulta-neously to initiate the CET in a transient experimentExperiments were performed with different coolingrates From thermocouples located along the samplesthe values for G and V at the CET were determined Alinear decrease in columnar grain length with increasingcooling rate was found for non-rened alloys Thecritical experimental values of G and V at the CET werecompared with the models of Hunt 32 and Martoranoet al 45 (see section lsquoDeterministic modelsrsquo below)calculated for Alndash7 wt-Si for three different valuesof the critical undercooling 0 3 and 5 K The criticalexperimental values were found to be in good agreementwith the model of Martorano et al for a criticalundercooling of about 5 K Grain renement resultedin a lower critical undercooling a higher critical tem-perature gradient and higher grain densities in thecolumnar and equiaxed regions The CET was smootherwith rened alloys

Reinhart et al 46 have used a novel approach to makedirect observations of the solidifying interface inparticular at the CET Synchrotron X-radiography has

been used to examine vertically solidied rened andnon-rened Alndash3 5 wt-Ni alloys in a Bridgman fur-nace The samples were about 40 mm in length by 6 mm

in width by 150ndash200 mm in thickness In any experimentsolidication was started with a low pulling velocity toproduce a columnar dendritic structure The CET wasthen initiated by suddenly applying a sharp increase inpull rate keeping the gradient constant Figure 6 a to d shows a sequence of images following an increase in pullrate from 1 5 to 15 mm s ndash1 for a gradient of 2 K mm ndash1 for a grain-rened alloy with 0 5 wt- of an Al5Ti1Bgrain rener added Direct observation of the CETreveals some interesting features A short time after thevelocity jump a band of equiaxed crystals appears in thesupercooled liquid beside the columnar dendritic frontSome are nucleating around the columnar dendrites butsome are nucleating in an almost horizontal band Whenenough grains have nucleated and grow they may blockthe columnar growth leading to the CET The fact thatboth columnar and equiaxed dendrites are blockedbefore the grains are touching each other (Fig 6 c and d )led the investigators to suggest that the blocking ismostly solutal as proposed by Martorano et al 45 seesections lsquoProposed mechanismscriteria for the termina-tion of columnar growthrsquo and lsquoModels for predicting theCETrsquo below Another interesting feature was that someof the newly nucleated grains fall down either on thecolumnar dendrite or towards the eutectic front ie theliquid area on both sides of the columnar dendrite canbe gradually lled in with equiaxed dendrites Thereforea post-mortem analysis of the sample could lead to thefalse conclusion that the structure is mixed columnar-equiaxed whereas it is only caused by sedimentation

Direct comparison of the data sets from the variousdirectional solidication studies is virtually impossible

6 Synchroton X-ray images recorded a 42 s b 63 s c

87 s and d 111 s after a sharp increase in pull ratedashed line marks eutectic front position 46





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First the studies cover a wide range of alloy systemsincluding AlndashCu AlndashMg AlndashNi AlndashSi PbndashSn SnndashPbPbndashCu CundashSn and Ni-base alloys Second even wherethe studies have been made on the same systemsometimes a single composition has been examinedwhereas in other studies a range of compositions wereinvestigated Third some investigations were carried outunder steady controlled freezing conditions (using aBridgman technique) whereas others were performedunder non-steady conditions Fourth in some studiesgrain rening inoculants have been deliberately addedwhereas in the others equiaxed grains have either beenheterogeneously nucleated on unknown substrates orformed by other means Depending on the alloy systemand directional freezing method employed other majordifferences can be recognised In some alloy systemssolutes may have partition coefcients k less than unityand in others values greater than unity In some of thesystems solute redistribution on freezing may result inthe interdendritic liquid becoming less dense than thebulk liquid leading to thermosolutal convection whensolidifying upwards This is the case in those k 1systems where the solutes are less dense than the solventor those k 1 systems where the solutes are more densethan the solvent Different mechanisms of equiaxedgrain formation may be operating in the differentstudies even in the absence of deliberately added grainreners Under steady controlled freezing conditionsand in the absence of thermosolutal convection hetero-geneous nucleation in the bulk liquid ahead of theadvancing front may be a plausible mechanism In thepresence of thermosolutal convection dendrite remelt-ingfragmentation may be the mechanism as evidenced

in the study by Pollock and Murphy36

Under non-steady freezing conditions loss of heat from the upperliquid surface may result in the formation of a surfacedendritic layer Fragmentation of this layer could lead toshowering of dendrite fragments which grow asequiaxed crystals From the directional solidicationstudies it is only possible to draw certain generalisedconclusions As stated by Quested and Greer 2 in manyinstances equiaxed microstructures are found to befavoured by high interface growth velocities lowtemperature gradients ahead of the advancing fronthigh solute levels and a larger number of equiaxednucleation events However the precise relationship

governing the CET between the local solidicationconditions at the advancing interface during directionalsolidication and alloy parameters is still far from clearAttempts to date to compare experimental data withHuntrsquos analytical model for the CET have only beenqualitative in nature because of the assumptionsrequired regarding the number of nucleant particlesand the undercooling for nucleation of the equiaxedgrains

SuperheatNumerous authors have reported that increasing super-

heat increases columnar grain length ie suppresses theCET (Fig 7) 131947ndash51 Although this is true for labora-tory scale experiments frequently performed on cylind-rical cast shapes of less than 500 cm 3 in volume theobservation cannot be extrapolated to larger volumes

Morando et al 50 examined the solidication of aseries of Alndash2Cu cylindrical ingots varying in volume

from 50 to 500 to 5000 cm 3 poured into graphitemoulds with a xed ratio of internal diameter to internalheight of 0 7 For each volume ingots were poured witha range of superheats from 20 u C to 150 u C Thecolumnar length from the ingot base was used as themeasure For the 50 cm 3 volume columnar lengthincreased with superheat There was also some lengthvariation in the 500 cm 3 ingots However in the5000 cm 3 ingots the columnar length was invariablewith superheat As discussed by the investigators for thesmallest volume solidication times are short andsettling of crystals is unimportant However theyobtained evidence that with increase in size Southinrsquosmechanism of equiaxed grain formation 15 comes intoplay and as solidication time increases crystal settlingbecomes more important

Because of their low thermal conductivities and longfreezing times the solidication of small castings of transparent non-metallic systems eg NH 4ClndashH 2O isoften taken as an analogue of the freezing of large ingotsSuch studies have revealed the importance of dendriteremelting 52 or alternative mechanisms of equiaxed grainformation 53 and the importance of equiaxed grainsettling

When dealing with ingot shapes the often reportedinuence of superheat on the CET is therefore only validfor small laboratory scale castings However superheatwill also be of importance in larger commercial thinwalled castings because of the higher surface to volumeratios The inuence of superheat on primary phasegrain structures in thin section castings has not beenrigorously studied primarily because a certain minimumsuperheat must be exceeded in order to provide

adequate uidityAssuming for the case of small laboratory ingots( 500 cm 3) that grains nucleated near the mould wallon pouring are the probable origin of the centralequiaxed zone 1316 increasing superheat will decreasethe degree and depth of supercooling and increase thetime required for dissipation of the superheat Fewernuclei will therefore initially form and survive remelting

Some of the non-steady unidirectional solidicationstudies described in the section lsquoUnidirectional freezingrsquoalso report the effect of superheat on the CETGenerally speaking as might be anticipated increasingsuperheat increases columnar zone length and delays the

CET

Fluid flowInvestigations of the signicance of uid ow to theCET have centred on two aspects First determinationof the inuence of natural convectionpouring turbu-lence on the transition and second application of forcedow to promote equiaxed grain formation Much of thiswork which has been qualitative in nature was carriedout by Cole and Bolling 1948495455

Considering the role of natural convection a varietyof techniques have been used to reduceremove convec-

tion in the melt1947495056

Cole and Bolling insertedgrids across the mould diameter 19 or slowly rotated themould about its vertical axis during solidication 49

Others have solidied alloys in a static magneticeld 475056 either applying the eld at all times or overselected time periods during solidication in order todifferentiate between potential mechanisms of equiaxed





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grain formation Generally speaking increasing uidow decreases columnar grain length This is due to oneor more of the following

(i) increase in the number of dendritesdendritefragments transported from the vicinity of themould wallsmelt surface into the bulk liquid

(ii) increase in the rate of loss of superheat from thebulk liquid thus increasing the chance of nuclea-tion andor survival of equiaxed grains ahead of the advancing front

(iii) increase in the number of nuclei for equiaxedgrains formed by dendrite fragmentation

The results support the contention that convectionincreases the rate of loss of superheat They alsohighlight the signicance of equiaxed grain formationin the early stages of freezing in small ingots and theprobable importance of dendrite fragmentation in largeringots

Regarding forced ow early studies were made usingLorentz interaction between a current and magneticeld 48 and oscillation of the ingot mould during

solidication55

These studies indicated that forced owcauses grain renement primarily it was thought byenhancing the fragmentation of dendrites

More recently several studies have been made of theinuence of forced ow on the CET in alloys unidir-ectionally solidifying under non-steady conditions 57ndash59

Since most of the early experiments associated with

investigating the inuence of uid ow on grainstructure were qualitative in nature Grifths andMcCartney 57 set out to try to quantify the effect of melt velocity on structural transitions A series of AlndashSialloys was directionally solidied downwards in arectangular mould 150 mm long by 65 mm wide by210 mm deep by chilling the top surface of the meltThe alloys which contained 2 5 56 and 8 5 wt-Siwere made from 99 7 wt-Al and 98 4 wt-Si Experi-ments were also performed with an Alndash2 5 wt-Si alloyprepared from higher purity materials The mould wasinitially preheated to above the alloy liquidus to preventnucleation on pouring Experiments were performedwith and without electromagnetic stirring A representa-tion of the uid ow in the mould under conditions of natural convection and stirring was obtained using thecommercial CFD software package FLUENT Tem-peratures were recorded from a series of thermocouplesaligned vertically along the central axis of the mouldIngots were sectioned longitudinally along the mid-plane The position of the CET was traced and the area

percentage of the equiaxed region calculated Underconditions of natural convection there was no apparentrelationship between Si content and extent of theequiaxed region However when electromagnetic stir-ring was applied the extent of the equiaxed zone wasfound to increase systematically with both increasing Sicontent at a xed stirring current and with increasing

a 40 u C superheat b 80 u C superheat7 Effect of superheat on grain structure of Alndash2 wt-Cu alloys 50





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stirring current at a xed Si content Electro-magnetic stirring therefore promotes the CETElectromagnetic stirring resulted in the rapid removalof the bulk liquid superheat and liquid velocities up to025 m s ndash1 were predicted The investigators concludedthat their results support the contention that forced uidow promotes the equiaxed region and the CET byfragmentation of the solidifying front The role of heterogeneous nucleation was also investigated bycomparing ingots of Alndash2 5 wt-Si made from basematerials of different purities solidied under conditionsof natural convection (no forced stirring) The equiaxedzone in the higher purity alloy (lower Ti and B levels)contained fewer and larger grains than in the commer-cial purity alloy This suggested that under conditionsof natural convection the CET is inuenced by thenumber of heterogeneous nuclei With stirring thenumber of heterogeneous nuclei becomes less importantbecause of the creation of dendrite fragments fromwhich growth of equiaxed dendrites can proceed

Willers et al 58

carried out an experimental study toinvestigate the inuence of bulk ow in the melt on theunidirectional solidication of cylindrical samples of alloys of Pb and Sn The bulk ow was generated by arotating magnetic eld The alloys studied were Snndash 15 wt-Pb Snndash38 wt-Pb and Pbndash25 wt-Sn Thesecond alloy corresponds to the eutectic compositionThese compositions were chosen to provide differentdensity ratios between interdendritic liquid bulk liquidand primary crystals and also to provide an alloy(eutectic) that is not subject to macro solute rejection atthe solidifying front or grows dendritically The alloyswere preheated in cylindrical moulds 50 mm internal

diameter by 100 mm in height to a superheat of 90 KThe alloys were then solidied upwards by placing themould on a water-cooled copper chill the wholeassembly being located inside a magnetic inductorStirring of the melt was initiated simultaneously withthe cooling Temperatures were recorded along thecentral axis of the cylinder and experiments wereconducted with variations of the magnetic eld strengthIn the absence of a magnetic eld the alloys werecompletely columnar Melt agitation promoted grainrenement The present study revealed in all casesincluding the eutectic alloy that increasing eld strengthdisplaced the CET towards the bottom of the cylinderThe CET appears for critical values of the cooling rateof about 0 4 K s ndash1 and for temperature gradientsbetween 0 6 and 1 0 K mm ndash1 These are about an orderof magnitude greater than those reported for non-stirredmelts 3040 Measurements revealed the existence of remarkable temperature uctuations in the mushy zonelending support to thermal melting of dendrites Soluteaccumulation at a CET was considered to be responsiblefor obstruction of columnar growth in accordance withRef 45 see the section lsquoProposed mechanismscriteriafor the termination of columnar growthrsquo below Theorigin and the effect of magnetic eld strength on theCET in eutectic alloys are unclear It appears thatdendrite fragmentation and solutal interaction betweencolumnar and equiaxed growth cannot be responsibleand the investigators simply suggest that stirring leads toa multiplication of nuclei

In a continuation of the research described in theprevious paragraph using the same experimental

arrangement Eckert et al 59 studied the solidicationof Snndash15 wt-Pb alloys in a rotating magnetic eld Theresults were compared with numerical simulations forthe temperature and velocity elds in the liquid phase Inone set of experiments the rotating magnetic eld wasnot switched on at the onset of cooling but after timedelays These experiments gave no indication of consi-derable thermal remelting of the solidication front Itwas therefore concluded that the activation of the CETby stirring is not necessarily based on fragmentation of the columnar dendrites

Electromagnetic stirring to promote the CET has beenapplied commercially to the continuous casting of steelThe low thermal conductivity of steel leads to deepliquid pools inside the solidifying shell within which thetemperature gradients are shallow Electromagneticstirring has also been applied to the direct chill castingof wrought aluminium alloys

Overall consideration of parameters affecting

CET The inuence of the principal phase diagram parametersm k and C 0 on the CET are now qualitatively wellunderstood although in a particular system they cannotbe varied independently The CET is encouraged as theconstitutional supercooling parameter P increasesSolute redistribution on freezing lowers the solidliquidinterface temperature promotes CS and restricts graingrowth prompting further nucleation In the case of small experimental ingots columnar length increases assuperheat increases This is primarily because of theinuence of superheat on the remelting of equiaxeddendrites nucleated in the early stages of solidicationThe effect of superheat decreases as ingot size increasesbecause the origins of the equiaxed grains changes andequiaxed crystal settling becomes more dominant indetermining the cast grain structure Natural andenhanced uid ow encourage the CET by transportingdendritesdendrite fragments enhancing dendrite frag-mentation and increasing the rate of loss of superheat

The collective interaction of all the above parametersis extremely complex which makes quantitative predic-tion of the CET very difcult

Proposed mechanismscriteria for

termination of columnar growthVarious mechanisms have been proposed to try toexplain the local interaction of columnar grain growthand equiaxed grain formation that gives rise to a CETThese consider to varying extents the solute andtemperature elds uid ow and the growth of equiaxedgrain nuclei Whatever the mechanism a transition willonly occur if the equiaxed grains are sufcient in size ornumber to arrest columnar grain growth This issupported by the observation that individual equiaxedcrystals can be found trapped within columnar zones ACET will occur if the equiaxed grains cannot be pushed

ahead of the growing columnars and the columnars areincapable of growing between the equiaxed grains Theconditions associated with the local columnarequiaxedinteraction will be determined as discussed in earliersections for the case of castings by casting size andgeometry type of casting process process operationparameters and alloy characteristics Most of these





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proposed mechanisms are only of academic interesthaving little relevance to the control of the CETHowever some criteria for the transition have beensuggested eg based on equiaxed grain size volumefraction of equiaxed grains or solutal interactionsbetween the growing equiaxed dendrites and theadvancing columnar front that can be used in CETmodels

As stated by Kisakurek 9 two possibilities for thetransition exist Either

(i) columnar growth may slow down allowing thedominance of equiaxed grains or

(ii) equiaxed grains take over and force the columnargrowth to halt

The proposed mechanisms therefore describe likelyscenarios leading to one or other of these two conditions

Winegard and Chalmers 12 suggested that the CEToccurs by the impingement of the columnar dendriticinterface upon the dendritic skeleton of oatingequiaxed crystals

Biloni and Chalmers20

proposed that the oatingequiaxed crystals probably grow initially in a lsquopreden-driticrsquo manner and that each crystal is surrounded by itsdiffusion eld of higher solute content They suggestthat columnar growth is arrested when the diffusionelds of the oating crystals ahead of the columnarshave impinged A cellular dendritic sub-structure thendevelops in the crystals

Fredriksson and Hillert 60 suggested that the centralequiaxed zone observed in experimental ingots of a Pbndash 2Sb alloy was formed by two different mechanismsgrowth of a sedimentation layer on the bottom andgrowth of crystals which have adhered to the vertical

solidication front In the latter case it was anticipatedthat as solidication progresses the vertical columnarsolidication front will become jagged and be able tocatch a oating crystal Roughness therefore increasesand the chance of catching more crystals is increasedThis process may then spread along the whole solidica-tion front The subtlety of this argument is that the CETis caused by the development of individual equiaxedgrains Southin 15 has also suggested that dendritesdendrite fragments dislodged from a surface dendriticlayer sink in the liquid until they are met by solidgrowing from the mould walls In a later paperFredriksson and Olsson 61 reiterate that the CET occurswhen the free equiaxed crystals are sufciently large ornumerous to physically block columnar growth byadhering to the solidication front From observationson a steel ingot they also proposed a possible criterionfor the CET namely that it occurs at the time when thetemperature of the bulk liquid reaches a minimumbefore recalescence

Based on a consideration of the relationship betweendendrite tip temperatures and growth rates for differenttemperature gradients Burden and Hunt 62 argued thatas solidication progresses the growth rate of theequiaxed crystals ahead of the columnar front rapidlyincreases and the growth rate of the columnar interfacedecreases A condition is eventually reached where thecolumnar front has almost stopped and the equiaxedgrains are growing rapidly leading to the CET

Witzke et al 63 modelled the composition andtemperature eld ahead of a vertical columnar front inthe presence of convection and suggested that the CET

occurs only if the liquid reaches a sufcient degree of constitutional supercooling and second if the volume of the undercooled zone is sufcient These two conditionsare required to generate the necessary amount of equiaxed crystals to hinder columnar growth Liptonet al 64 have also suggested that the CET is governed bythe undercooling of the columnar interface and thethickness of the undercooled boundary layer at theinterface This layer determines the growth rate of the equiaxed crystals For an organic analogue Liptonet al 65 observed that release of latent heat duringequiaxed growth caused a rise in temperature of theboundary layer that accompanied the CET Theyconsider that the rise in temperature is because of theoverlap of the thermal boundary layers of adjacentequiaxed grains (considered to be spheres) An alter-native criterion for the CET was therefore suggestedthat it will occur when an equiaxed grain has grown to acritical radius equal to one-tenth of the distance betweentwo equiaxed grains

Hunt32

using a probability approach to consider thegrowth interaction of columnar and equiaxed grainsderived another criterion for the CET He argued thatequiaxed growth will occur when the volume fraction of equiaxed grains (again considered as spheres) is greaterthan 0 49 (or an lsquoextendedrsquo volume fraction of 0 66)

Mahapatra and Weinberg 30 proposed that the colum-nar dendrite tips may become unstable when thetemperature gradient ahead of the tips falls below acritical value This might then lead to solute accumula-tion at the tips and restricted growth of the columnargrains The liquid ahead of the columnars would thencool and equiaxed grain nuclei would grow

Gandin66

applied a one-dimensional nite differenceheat transfer model to the studies of the non-steadydirectional freezing of 99 99 wt-Al and Alndash7 wt-Sidescribed elsewhere 37 It was observed that if the ingotsolidied fully columnar then the velocity of thecolumnar dendrite tips increased initially during thestage of superheat loss then decreased when nosubstantial gradient remained in the liquid ahead of the growing interface When applied to the freezing of AlndashSi alloys the maximum velocity was reached whenthe dendrite tip interface reached two-thirds of thelength of the ingot This position corresponded closelyto the position of the experimentally observed CETwhich led the author to propose that at this maximumvelocity dendrite arm remelting occurs resulting indestabilisation of the macroscopic interface Since thepredicted position of maximum velocity as a function of heat extraction rate the superheat and the alloycomposition is similar to that reported for the CET heproposes a CET criterion based on the position of themaximum velocity of the columnar dendritic interfaceHe comments that since the criterion is independent of the nucleation parameters of the equiaxed grains itshould only be applied in the absence of addedinoculants

Martorano et al 45

proposed a new mechanism forthe transition based on solutal interactions between theequiaxed grains and the advancing columnar front Thesolute blocking is achieved by basing the undercoolingthat drives dendrite tip growth on the average soluteconcentration of the liquid surrounding the grainenvelopes (extradendritic liquid) C l instead of the





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initial alloy composition When the solute rejected fromthe equiaxed grains is sufcient to dissipate the under-cooling at the columnar front such that C l has increasedto C l (where C l is the liquidus concentration given bythe phase diagram at temperature T ) the CET willoccur

Overall consideration of termination of columnar growthThe proposed mechanismscriteria for the termination of columnar growth aid understanding of the CET If all of the solidication phenomena and length scales asso-ciated with the CET could be incorporated into a singlecomputational model then the mechanisms of impor-tance would be naturally revealed Proposed criteria forthe termination have been incorporated into simpliedmodels of the CET The most widely used to date hasbeen that due to Hunt 32

Models for predicting CET Over the past 20 years modelling has played asignicant role in the continued interest in the CETTo simplify the problem the majority of these modelshave considered the unidirectional solidication of alloys As discussed earlier if true unidirectional freez-ing occurs in the absence of uid ow the mechanism of equiaxed grain formation that is usually assumed isheterogeneous nucleation in the supercooled liquidahead of the columnar front Models can attempt topredict the critical conditions at the columnar front thatgive rise to the CET or to match the predicted andexperimentally observed location of a CET for a specicalloy and set of processing conditions or to examine theeffect of different parameters on the transition

CET models can be classied as being eitherdeterministic or stochastic Stochastic models followthe nucleation and growth of each individual grainwhereas deterministic models rely on averaged quanti-ties and equations that are solved on a macroscopicscale 45

Stochastic models simulate the solidication of bothcolumnar and equiaxed grains and computed micro-graphs can be created as the grain structure evolves Nocriterion is required in such models for determining theposition at which the CET occurs A subjective judgement is made from examining the micrographsas to when columnar growth ceases

Some reported deterministic models of the CET onlyconsider the growth of the columnar grains and attemptto predict the conditions that exist at the locations of experimentally observed columnar to equiaxed transi-tions 303133 Many deterministic models involve con-sideration of the formation of the equiaxed grains andtheir competition with the advancing columnar frontSuch models require the inclusion of a criterion for thetermination of columnar growth They also attempt toincorporate the growth kinetics of the columnar and

equiaxed grains Often the growth of the dendrite tips inboth columnar and equiaxed grains is assumed to begoverned by a simple relationship between tip velocityand undercooling This undercooling has most oftenbeen expressed in terms of the initial alloy compositionC 0 However as discussed by Martorano et al 45 thisimplies for a given columnar dendrite tip temperature

that tip velocity is the same regardless of the presence of equiaxed grains growing ahead of the columnar frontThey therefore developed a model where the columnarand equiaxed dendrite growth velocities are a functionof a solutal undercooling proportional to the differencebetween the local liquidus concentration and the localaverage solute concentration in the extradendritic liquidsee the section lsquoProposed mechanismscriteria for thetermination of columnar growthrsquo above Deterministicmodels incorporating equiaxed grains most often con-sider that the grains are static although models are nowattempting to take into account equiaxed grain move-ment during solidication In some models nucleationoccurs at a xed undercooling (termed growth models inthis review) In this type of model the number of equiaxed grain nuclei is often chosen to give the bestagreement between predicted and experimental datarelating to the CET for the assumed undercoolingAlternatively best agreement may be obtained bydetermining the value of the undercooling assuming a

xed number of nuclei In other models nucleationkinetics are taken into account by considering theinuence of undercooling on nucleation rate (nucleationand growth models)

Some stochastic models also incorporate dendrite tipgrowth models 43 However other stochastic modelswhich simulate solidication at a microscopic scale andallow the solution of solute diffusion equations duringsolidication and the application of solidliquid interfacefront tracking methods to the surfaces of the individualdendritic grains 6768 have no need to include dendrite tipgrowth models

Although models are able to indicate probable effectsof alloy and process parameters on the CET at presentthey are unable to simulate the various mechanisms of equiaxed grain formation and modes of interactionbetween equiaxed and columnar grains found in omni-directionally solidied alloys

Deterministic models Analytical modelsA number of analytical models for the CET have beenproposed over the last half century that try to illustratethe interrelationship of the alloy and processing condi-tions governing the transition 293263

Tillerrsquos pioneering model 29 considered the unidirec-tional freezing of a semi-innite liquid from one end andcalculated the solute and temperature distributions inthe liquid with time assuming that the solidliquidinterface advances at a rate proportional to (time) 12 It was assumed that nucleation of equiaxed grainsoccurred in the constitutionally supercooled regionahead of the interface and that this was directlyresponsible for the CET The transition from columnarto equiaxed growth was assumed to occur at a criticalnucleation frequency corresponding to a critical max-imum undercooling value for a particular alloy Heargued that this critical condition is determined by G Ralone An expression was also derived for the ratio of equiaxed zone length to columnar zone length in a realnite ingot This predicts that the columnar zone willdecrease as superheat increases as the freezing rangeincreases as the number and catalytic activity of thenucleating centres increase and as stirring of the liquid





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increases The CET will again occur at a critical ratio of G R but will also depend on ingot length

Witzke et al 63 considered the thermal and chemicalelds ahead of the tips of dendrites growing as a verticalcolumnar front ahead of which liquid is owing underconditions of thermal laminar ow They derived anexpression for constitutional supercooling which asstated in the section lsquoProposed mechanismscriteria forthe termination of columnar growthrsquo above theyconsider is responsible for the CET This model againaccounts for the fact that development of the equiaxedzone is favoured by low superheat strong undercoolingat the dendrite tips high values of thermal to chemicaldiffusivity ratio and mould height

Hunt 32 developed a steady state directional solidica-tion model for the columnar and equiaxed growth of both dendrites and eutectic An analysis was presentedfor the growth of equiaxed grains in the supercooledliquid ahead of the columnar front in order to predictthe conditions necessary for a wholly equiaxed structureAs mentioned in the section lsquoProposed mechanismscriteria for the termination of columnar growthrsquo abovefully equiaxed growth ie the CET was considered tooccur when the extended volume fraction of equiaxedgrains exceeds 0 66 Hunt calculated that fully equiaxedgrowth would occur when the steady state temperaturegradient was below a critical value given by

G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)

where N 0 is the number of nucleiunit volume DT N isthe critical undercooling for nucleation and DT C is theundercooling at the columnar front

The model was applied to Alndash3 wt-Cu As the

volume fraction of equiaxed grains decreases mixedcolumnar-equiaxed and nally fully columnar structuresare anticipated The model predicts that at low growthvelocities equiaxed growth depends on the efciency of the nucleating substrates for the equiaxed grainswhereas at high gradients the number of nucleationsites is more important The columnar front under-cooling can be expressed in the form

DT C ~ f A0m(1 k )C 0V g1=2 (6)

It can therefore be seen that as the so-called lsquogrowthrestriction factorrsquo ndash m(1ndash k )C 0 increases the tendency toequiaxed growth increases

Simulation modelsIn more recent years a variety of approaches have beenemployed to model the conditions giving rise to theevolution of the CET for specic alloys by simulatingthe solidication process These models attempt toinclude to different extents the thermodynamics andkinetics of the transformation process

Growth modelsLipton et al 64 assume that the columnar front is

advancing with a velocity t ndash0 5

and that its undercoolingis governed by a velocity-undercooling relationshipA specic number of equiaxed particles (necessarilyconsidered as a free parameter) occupying xedpositions are considered to grow as spheres at a ratedetermined by the square root of the mean temperaturedifference between the particle and its surrounding melt

and of the mean dwell time in the melt The model doesnot take into account the latent heat released by thegrowing equiaxed grains As stated in the sectionlsquoProposed mechanismscriteria for the termination of columnar growthrsquo above they assumed that the criterionfor the CET was that an equiaxed grain had grown toone-tenth of the spacing between two adjacent equiaxedgrains and that convective ow equalises the tempera-ture in the bath and controls the heat transfer to thesolid shell The model showed that two parameterscontrolled the CET undercooling of the interface andthe thickness of the undercooled layer

Fredriksson and Olsson 61 have used a similarapproach assuming again a velocity t ndash0 5 relationshipfor the columnar front that the front undercooling isdependent on velocity that convective ow equalises thetemperature in the bath and that there are a speciednumber of growing equiaxed crystals They assume thatthe equiaxed grains grow with a spherical shape havingan internal solid fraction of 0 3 and that the growth rate

is dependent on the square of the undercooling Theirmodel accounts for the latent heat evolved by theequiaxed grains However the growth rate and under-cooling of the columnar interface are not affected by thegrowth of the equiaxed grains Best values were chosenfor a growth constant and the number of equiaxeddendrites per unit volume to give a best t between ameasured cooling curve recorded at the centre of a steelingot and a curve predicted from the model Both curvesdisplayed recalescence and as stated in the sectionlsquoProposed mechanismscriteria for the termination of columnar growthrsquo above they predicted that the CEToccurred at the time corresponding to the minimum

temperature preceding recalescence Their model pre-dicts an increase in the columnar zone the larger thesuperheat the smaller the number of free crystals thelower the solute content the higher the cooling rate andthe greater the height andor width of the ingot

Flood and Hunt 6970 were the rst to simultaneouslymodel the growth of both the columnar and equiaxedgrains This was achieved using a 1D nite differencethermal model The model takes into account bothconduction and convection in the bulk the latent heatliberated by the equiaxed grains and the thermalinteraction between the solidifying columnar andequiaxed grains In this way the velocity and under-cooling of the columnar front are calculated dynamicallythroughout the simulation and are not xed a priori Convection is treated with a simple boundary layerapproximation it being assumed that there is completemixing ahead of a conducting boundary layer and thatthe bulk is isothermal It is assumed that the velocities of the columnar and equiaxed grains are governed by aparabolic velocity undercooling relationship The liquidfraction in the columnar grains is assumed to be relatedto temperature by the Scheil equation 71 lsquotruncatedrsquo atan undercooling determined by the heat ow at thedendrite tips 69 The spherically shaped equiaxed grains

ahead of the columnar front either grow as soon as somecritical undercooling is exceeded or are nucleated at atemperature dependent rate The equiaxed grains areassumed to be isothermal and to also have a solidfraction governed by the Scheil equation Equiaxedgrain impingement is allowed for using an Avramitreatment and the CET is assumed to occur when the





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lsquoextendedrsquo volume fraction of equiaxed grains exceedsthe value specied by Hunt 32 (see the section lsquoProposedmechanismscriteria for the termination of columnargrowthrsquo above) Assuming that equiaxed grains grow assoon as the temperature falls below the liquidus themodel predicts that the columnar range decreases onincreasing the alloy composition and convection and onincreasing the initial superheat and rate of heatextraction The model also predicts that the equiaxedgrowth dominates columnar growth more than inpractice possibly pointing to factors that inuenceequiaxed grain formation in addition to growth suchas nucleation convective ow and sedimentation Theeffect of superheat was not as dramatic as that found inpractice which they conclude is probably as a resultof the model failing to consider the effect of superheaton the remeltingsurvival of nuclei eg as associatedwith the big-bang mechanism of equiaxed grain forma-tion The introduction of a temperature dependentnucleation rate increased the columnar zone whichcould be combated by increasing the cooling rate

Mahapatra and Weinberg 30 used a completely differ-ent approach to those so far described in that anumerical model was used to evaluate the CET observedin experimental ingots However the model did notinclude the formation of the equiaxed grains Smallcylindrical ingots of SnndashPb alloys were melted in situand directionally solidied from a water cooled chillTemperature measurements were taken throughoutsolidication from four thermocouples placed in themelt at different heights A 1D implicit nite differencemodel was then used to simulate the solidication of theingots and heat transfer coefcients at the chill interface

were obtained by iteration that gave best t betweenmodelled and measured temperatures The nite differ-ence model with the appropriate heat transfer coef-cient was then used to determine the conditions at theCET for each ingot They concluded that the CEToccurred at a critical temperature gradient of 0 1080101 and 0 126 K mm ndash1 respectively for alloys con-taining 5 10 and 15 wt-Pb They found that the CETwas independent of superheat and suggested a criterionfor the transition based on instability of the dendrite tipsbelow a critical temperature gradient (see the sectionlsquoProposed mechanismscriteria for the termination of columnar growth aboversquo) In a later paper Ziv andWeinberg 31 also concluded for an Alndash3 wt-Cu alloythat the CET occurred when the temperature gradientfell to 0 06 K mm ndash1 Applying a similar 1D nitedifference heat transfer analysis to the data of Mahapatra and Weinberg Spittle and Tadayon 72 couldnd no evidence to support the contention that the CEToccurs at a critical gradient

Suri et al 33 adopting a similar approach to that of Weinberg and co-workers above used a 1D heattransfer model to identify the locations of the CET indirectionally solidied Alndash4 5 wt-Cu alloys

Brown and Spittle 73 described a 2D implicit nite

difference model for simulating the solidication of experimental AlndashCu alloy castings in dry sand mouldsThe experiments demonstrated in line with the observa-tions of Tarshis et al 21 that in small ingots thetransition from wholly columnar to wholly equiaxedstructures occurs with only a very small increase in theparameter P In the model the growth rates of the

columnar and equiaxed grains are assumed to be equalthe columnar grains grow with a xed undercoolingof 10 K the equiaxed grains grow as spheres with axed internal solid fraction (as predicted by the Scheilequation for an undercooling of 0 5 K) equiaxed grainsdo not interact with each other or the growing columnarfront and latent heat generated from the equiaxed grainsis distributed uniformly to all nodes ahead of thecolumnar front The equiaxed grain density wasobtained from grain size measurements on the actualcastings The CET was assumed to occur for a givenvolume fraction of equiaxed grains A best matchbetween predicted and measured thermal analysis curveswas obtained to provide the metalmould heat transfercoefcient The model was able to simulate mixedcolumnar equiaxed structures and indicated that theCET can occur for a very low equiaxed grain density

Wang and Beckermann 7475 reported a novel techni-que for modelling the CET under conditions of diffusioncontrolled dendritic growth ie in the absence of melt

convection and transport of solid grains The techniqueinvolves the multiphase approach described by Wangand Beckermann 76 in which a control volume contain-ing either columnar or equiaxed grains is considered toconsist of three phases solid interdendritic liquid andextradendritic liquid A set of macroscopic equationsgoverning solute diffusion in the three-phase system isderived using a volume averaging procedure Theseequations are then coupled with the heat ow equationwhich is solved using a two time-step fully implicitcontrol volume-based nite difference method Themodel is able to simulate the CET taking into accountnucleation and growth of grains and dendrite morphol-

ogy (primary and secondary arm spacings) In the modelreported an average nuclei density was assumed in theequiaxed region with nucleation occurring instanta-neously at the liquidus temperature The CET isdetermined using the criterion due to Hunt 32 Theequiaxed nuclei density represents the only adjustableparameter A 1D model was applied to the simulationof the experiments performed by Mahapatra andWeinberg 30 and Ziv and Weinberg 31 on SnndashPb alloysand AlndashCu (see above) Applying the model to the Snndash Pb alloys it was observed that columnar zone lengthincreases as equiaxed grain density decreases as the heattransfer coefcient at the chill increases and as soluteconcentration decreases The effect of superheat wasonly pronounced at low superheats ( 10 K) colum-nar length increasing with superheat Assuming a singleequiaxed nuclei density for all experimental runs of 107 m ndash3 resulted in a match of experimental andpredicted CET positions to within 20 In the case of the Alndash3 wt-Cu alloy the CET position and the timewhen it occurs were fairly well predicted Although it isreasonable to compare the model with the SnndashPb andAlndashCu data where convection would have been minimalWang and Beckermann acknowledge that convectionand crystal fragmentationtransport need to be includedfor situations where diffusion is not dominating

In a similar manner to that used by Mahapatra andWeinberg 30 and Ziv and Weinberg 31 Gandin 66 applied aone-dimensional nite difference heat transfer model tothe non-steady columnar freezing of 99 99 wt-Al andAlndash7 wt-Si alloys (equiaxed grain formation was notconsidered in the model) The model indicated as





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discussed in the section lsquoProposed mechanismscriteriafor the termination of columnar growthrsquo above that theexperimentally observed CET occurred at the positionwhere the columnar front velocity was a maximum

Martorano et al 45

used a modied version of themultiphasemultiscale model of the CET proposed byWang and Beckermann 74 The model incorporated anew criterion for the CET as described in the sectionlsquoProposed mechanismscriteria for the termination of columnar growthrsquo above based on solutal interactionsbetween the equiaxed grains and the advancing colum-nar front The model was validated by predicting theCET in the three AlndashSi alloys previously experimentallystudied by Gandin 37 Equiaxed grains were assumed togrow at a xed undercooling and nucleation under-coolings were determined that provided best agreementbetween measured and calculated CET positions Foreach of the three compositions the nucleation under-cooling for the equiaxed grains was found to be aboutequal to the maximum tip undercooling deduced byGandin 66 This nding was insensitive to the equiaxedgrain density The authors say their data support thesuggestion by Gandin that the origin of the equiaxedgrains is breakdown or fragmentation of the columnardendrites rather than heterogeneous nucleation

Nucleation and growth modelsHunt 32 in addition to his analytical model (see thesubsection lsquoAnalytical modelsrsquo above) also developed amore rigorous heat ow model This did not requireequiaxed nucleation to occur at a xed undercooling andthe temperature gradient to remain constant during thegrowth of the equiaxed grains Heat ow through a unitarea box was considered within which nucleation of equiaxed grains took place at a rate dependent on theamount of supercooling The relationship between G andV for the CET was again determined for Alndash3 wt-Cuassuming that fully equiaxed growth occurred when theextended volume fraction of equiaxed grains at thecolumnar front reached 0 66 As can be seen in Fig 8there was very close agreement with the predictions of his analytical model

Cockroft et al 77

reported the development of amathematical model based on Huntrsquos CET model Tocalculate the columnar dendrite tip undercooling theKurz Giovanola and Trivedi (KGT) model for a binaryalloy was used 78 This calculates the undercooling interms of G R C 0 m and k The mathematical modelwas extended to multicomponent alloys by assuming

that the diffusion elds around the dendrite tipsassociated with each alloy species can be superimposedAn expression due to Rappaz 79 which assumes acontinuous nucleation distribution of the form of aGaussian distribution was used to calculate the densityof grains at a given undercooling The model was ttedto experimental data ( G and R conditions) for the CETtransition in a land turbine blade alloy CMSX-4 byadjusting the value of the centre of distributionnucleation parameter required in Rappazrsquos expressionAs with the Hunt model 32 an extended volume fractionof 066 for the equiaxed grains was assumed for theCET The value calculated was then used for generalapplication of the model to other casting geometries of the alloy

Gaumann et al 80 developed a numerical model as amodication of Huntrsquos models 32 which again considersthe growth of equiaxed grains ahead of the columnarfront The CET is assumed to occur for an actualvolume fraction of equiaxed grains of 0 5 ahead of the

columnars The model takes account of the compositionliquidus prole and the change in undercooling withdistance ahead of the columnar front at both low andhigh velocities The KGT dendrite model is used todetermine the columnar dendrite tip temperature Thenucleation and growth rates of the equiaxed grains aregoverned by the local undercooling which is dependenton the distance from the front The model was againapplied to Alndash3 wt-Cu and the data compared withthose of Hunt (Fig 9) At low temperatures the CEToccurs at lower velocities than predicted by Hunt This isthought to be due to the underestimate of tiptemperature by Hunt At high velocities they predict

an increase in stability of the columnar structurethought to be due to rising tip temperatures Theinvestigators contend that the model is more appropriatefor predicting the CET in a range of processes includingthe high cooling rates experienced in welding and rapidsolidication processes

Quested and Greer 2 have used a novel approach tomodel the grain structuresgrain size of directionallysolidied alloys which have been inoculated with a grainrener Previous studies of the CET have primarilyfocused on uninoculated melts Modelling is compli-cated by the fact that grain renement appears to bevery inefcient in alloy castings with in the case of Alndash TindashB reners 1 of the TiB 2 nucleant substratesactually nucleating grains In this kinetic model theinitiation and growth of equiaxed grains are consideredin order to determine nal equiaxed grain size underdirectional freezing conditions The authors use the freegrowth theory to describe initiation of grains 81 Theterm initiation is used since the onset of free growth isnot nucleation controlled Larger particles becomeactive nucleants at smaller undercoolings The onset of undercooling for free growth of a grain can be calculatedfrom the diameter of the inoculant particle Theinoculant particles are assumed to be randomly dis-

persed and static and are associated with a knownparticle size distribution in the master alloy Theimpingement of growing grains on inoculant particlesis thought to be the mechanism by which renementefciency is reduced and a key assumption of the modelis that the fraction of particles rendered inactive at anyinstant is equal to the fraction of the system affected by

8 Plot of growth velocity versus temperature gradient forAlndash3 wt-Cu showing columnar and equiaxed regionscalculated using approximate analysis and more accu-rate analysis 32





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impingement with growing grains Grain size was foundto vary with cooling rate variation of G or V having thesame effect This contrasts with the opposing effects of G and V on the CET The CET locus on a log V versuslog G plot was calculated using the authors dendritegrowth modelling procedure and Huntrsquos criterion 32 forblocking of columnar grain growth By plotting thecontours of equal grain size on the same axes it ispredicted that the equiaxed grain size at the CET couldvary over a wide range depending on the solidicationconditions (Fig 10)

Ludwig and Wu 82 have recently reported a three-phase Eulerian approach to model the CET where the

three phases are considered to be solidifying columnardendrites solidifying equiaxed grains and the parentmelt The phases are considered to be spatially inter-penetrating and interacting continua The nucleationrate of the equiaxed grains is modelled with a hetero-geneous nucleation law The conservation equations of mass momentum species and enthalpy for the phasesare solved with the commercial CFD software FLUENT61 The model includes mixed columnar and equiaxedsolidication the occurrence of the CET melt convec-tion and equiaxed grain sedimentation The model was

used to carry out an axisymmetric simulation of a Fendash 034 wt-C ingot 66 mm in diameter and 170 mm inheight The solidication sequence the sedimentation of the equiaxed grains the movement of the columnarfront and the nal phase distributions were found to tgenerally accepted explanations of experimental nd-ings However no quantitative evaluation was made

Stochastic modelsSimulating the solidication of individual grains pro-vides the potential for examining more precisely thelocal conditions controlling the CET Modelling at thisscale has in more recent models enabled the solution of solute diffusion equations and microscopic tracking of the movement of the solidliquid interface of the grainsThis has removed the need to include dendrite tipgrowth models for the grains and has resulted in thesimulation of dendritic grain morphologies that have arealistic appearance However assumptions are stillrequired regarding the nucleation of the grains Themodels are also severely limited by the grain numberdensities that can be handled in a simulation Thesignicant feature of the models is that they are capableof creating computed micrographs of grain structuresthat closely resemble those found in castings etcVarious researchers have reported the use of 2Dstochastic models to simulate the evolution of grainstructures and the CET during solidication

Brown and Spittle 83 using a Monte Carlo computersimulation technique previously applied to a number of solid-state processes such as recrystallisation and graingrowth were the rst to use the method for solidicationtransformations The technique employs a lattice of sites squares or triangles which are initially designatedas liquid The transformation of these sites to solid isthen simulated according to a set of rules governingnucleation and growth Individual grains are identiedand tracked during solidication and the nal grain

structure can be drawn as a micrograph Usingappropriate densities and locations of nuclei realisticpictures of CETs were generated (Fig 11) The modeldoes not use real temperatures actual phase diagramparameters or any thermophysical properties data Itwas found that the CET occurred in agreement with theprobabilistic argument of Hunt 32 when the actual

9 Comparison of data in Fig 8 with that predicted from numerical model of Gaumann et al where D T n is undercoolingat heterogeneous nucleation temperature N 0 is total number of heterogeneous nucleation sites per unit volume andC 0 is wt-Cu 80

10 Contours of equal grain size shown on plot of solidi-cation velocity versus thermal gradient together withlocus for CET calculated from Huntrsquos analysis 2





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volume fraction of equiaxed grains at the columnarinterface was 0 5 In later papers 84ndash86 the investigatorsdemonstrated the relative effects of superheat and mouldtemperature 84 C 0 and P 85 and equiaxed grain settling 86

on the CET The modelled effects were in agreementwith experimental observations see the sectionlsquoParameters inuencing the CETrsquo above These papersdemonstrated that such models are effective tools forqualitatively examining the CET

Rappaz and Gandin87

developed a 2D cellularautomaton method for simulating the solidication of dendritic alloys held at a uniform temperature Againthe growth of individual grains is tracked but an attemptis made to incorporate some of the physics of solidica-tion Nucleation is treated using temperature dependentnucleation site distributions The model also accountsfor the growth kinetics of the dendrite tips and thepreferred growth directions in cubic metals The modeldemonstrated that columnar length decreases as com-position and cooling rate increases The model is limitedby the assumption of uniform temperature

Stochastic modelling routines for tracking and com-putational illustration of grain structure evolution havebeen linked with nite differencenite element methodsfor solving the heat transfer equation andor the solutediffusion equation for actual alloys 4367688889

Zhu and Smith 89 developed an improved Monte Carlomodel removing some of the limitations of the modelreported by Brown and Spittle 83 in order that it could beapplied to a real system Again the growth of individualgrains was tracked but the number of nuclei was relatedto undercooling and a probability model of crystalgrowth was based on a lowest free energy changealgorithm The Monte Carlo procedures were alsocoupled with 2D nite difference formulations for heattransfer and solute distribution The model was appliedto the prediction of the multidirectional freezing of a 2Dingot of Alndash4 5Cu In examining the CET it wasobserved that if the number of growing nuclei ahead of the columnar interface is not large then the growinggrains will be trapped by the columnars which can still

continue to grow The CET will only occur if thenumber of growing nuclei in the bulk is large enough tomake them spread over the advancing interface It wasalso commented that in the presence of a steep gradientahead of the columnars equiaxed growth could beconned to a narrow zone and equiaxed grain formationwould then progress inwards in a gradual manner

Gandin and Rappaz 43 reported a coupled cellularautomaton-nite element (CA-FE) model for the 2Dsimulation of the directional solidication of an Alndash 7 wt-Si alloy The CA algorithm simulates the evolu-tion of the columnar and equiaxed grains duringsolidication and takes into account nucleation andgrowth kinetics and the preferred growth directions of the grains Temperatures predicted at nodal points usingan FE heat transfer model are used to calculate thenucleation and growth of grains with the CA model Thelatent heat released during solidication determinedfrom the nucleation and growth CA model is thenincorporated back into the FE heat transfer model A

1D implicit formulation was used to solve the heat owequation Nucleation in the bulk and at the surface of the casting is described by Gaussian distributions 79 asundercooling increases For the bulk liquid under-coolings randomly generated with a Gaussian distribu-tion are attributed to randomly chosen cells If the localundercooling of a cell exceeds the prescribed under-cooling a new grain forms A different Gaussiandistribution is used for the surface cells The nucleationand growth of the individual grains are tracked and theirnal shapes in the solidied alloy reproduced incomputed micrographs An Alndash7 wt-Si ingot wasexperimentally melted in situ and directionally solidied

from a cooled chill The temperatures during solidica-tion were measured from seven thermocouples along theingot length The parameters of the Gaussian distribu-tion for the nucleation of grains in the bulk wereadjusted so that model predictions matched the experi-mental cooling curves and grain structures Many of thefeatures of the structure seen in the actual alloy werereproduced In the early stages of freezing some grainsnucleated in the bulk liquid were encapsulated by theadvancing columnar grains in agreement with earlierpredictions 3289 At a later stage the density of grainsgrowing in the bulk is sufcient to stop columnar growthbut the grains continue to grow in an elongated fashionFurther up the ingot as the temperature gradientdecreases the density of grains in the bulk increasesand their shape becomes equiaxed This opens up adebate as to which transition constitutes the CET theone to elongated or the one to equiaxed grains Themodel has better descriptions of dendritic growth thanthe Zhu and Smith model 89

Nastac 67 developed a comprehensive model forsimulating the evolution of dendritic crystals duringthe solidication of binary alloys Stochastic procedureswere used to control the nucleation and growth of thedendrites and nite-difference schemes were used to

calculate the temperature and concentration elds inthe solid and liquid phases The author reportedinvestigation of the CET using 2D models for bothmultidirectional and unidirectional solidication of IN718ndash5 wt-Nb alloy In the unidirectional modelthe modelled geometry was 10 mm 6 20 mm and thenumber of cells was 500 6 1000 (ie a 20 mm mesh size)

11 Computer generated columnar to equiaxed transitionusing a probabilistic numerical model 83





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It was observed from the unidirectional model that theCET was sharp which was attributed to the fastergrowth of equiaxed dendrites that have nucleated in theundercooled liquid ahead of the columnar front

Dong and Lee 68 have also used a CA-FD model tosimulate the CET during the directional solidication of AlndashCu alloys The CA-FE model of Gandin and Rappaz[Gan04] only considers the thermal eld and assumesthat equiaxed grains are heterogeneously nucleated inundercooled liquid ahead of the growing columnarfront The Dong and Lee model based on an earlierreported model 90 but modied to include nucleationahead of the solidifying interface has been used topredict the inuence of solutal interactions within anadvancing columnar dendritic network on the nuclea-tion of equiaxed grains The inuence of thermalgradient pulling velocity alloy composition crystal-lographic orientation of the columnar dendrites andnucleation parameters on the CET was also investigatedThe modelled results were compared with previousmodels and experimental studies The CA-FD modeldoes not use an analytical solution 78 to determine thevelocity of the solidliquid interface common in otherstochastic models but solves the solute conservationequation subject to an equilibrium condition at the solidliquid interface The model determines the solute buildup ahead of the interface and the solutally undercooledregion This undercooling is used to stochasticallydetermine the nucleation of equiaxed grains using aGaussian nucleation distribution relating grain densityincrease to increased effective undercooling Calculated

numbers of nuclei are distributed randomly among cellsThermal proles for directional solidication wereassumed where a constant temperature gradient withhorizontal isothermals was imposed moving upwardswith a constant velocity Simulations were performed ona regular 2D square grid with a cell edge length of 5 mmThe domain size was 3 mm wide by 6 mm high

(600 6 1200 cells) Simulations were performed for Alndash Cu alloys and phase diagram data were generated fromMTDATA 91 Dong and Lee modelled the solidicationof an Alndash3 wt-Cu alloy with G 5 30 K mm ndash1 and V varying as a function of solidication time to give valuessimilar to those in the non-steady studies of Ares andSchvezov 38 and Siqueira et al 40 The nucleation under-cooling was assumed to be close to the maximumcolumnar tip undercooling and was set to 5 K Themaximum nucleation density was set at a value of 106 1012 m ndash3 as used in a previous study 40 Thepredicted evolution of the structure is shown inFig 12 An interesting prediction of the model is thatunlike all prior CET models where equiaxed grains areassumed to nucleate ahead of the advancing columnarfront equiaxed grains are predicted to nucleate both atthe front and in the grooves between primary dendritesThe authors predict that the grooves have the highestsolute-adjusted undercooling and are the favourednucleation location As velocity increases grains nucle-ate ahead of the front as well as between the dendritetips leading to solutal andor mechanical blocking andeventually to the CET The authors dened the CET asoccurring when more than half the columnar dendritesare blocked by equiaxed grains To investigate theinuence of incorporating solute diffusion in the CETmodel the investigators compared the model to oneusing the KGT model 78 that relates growth velocity totip undercooling It was found that by neglecting solutediffusion the predicted CET occurs at a lower velocitythan actually required for a given gradient The

inuence of the crystallographic orientation of thecolumnar dendrites on the CET was found to benegligible Using the value for maximum nucleationdensity given above a log V versus log G map wasdetermined for the CET in Alndash3 wt-Cu The CA-FDpredictions are shown in Fig 13 together with Huntrsquosanalytical prediction 32 and the experimental results of

12 Predicted microstructural evolution for Alndash3 wt-Cu alloy 68





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i c a

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L t d

Siqueira et al 40 The predictions agree well with Huntrsquos

analytical model at low velocities but differ signicantlyat high velocities The difference between the CA-FDpredictions and the experimental results is less than thatbetween Huntrsquos predictions and the experimentalobservations

Beckermann and co-workers are currently applying2D phase-eld simulations to coupled columnar andequiaxed growth Beckermann et al 92 reported a modelfor the unidirectional solidication of a binary alloyunder an imposed temperature gradient Competitionbetween columnar and equiaxed dendrites was investi-gated as a function of nucleation undercooling equiaxedgrain density imposed cooling rate and temperature

gradient

CET in weldsSeveral reviews have been published that consider thedetailed relationship between solidication parametersand weld structures 93ndash96 and one of these 96 concentrateson the CET It is recognised that equiaxed grainstructures at the centres of welds would be of benetin reducingeliminating susceptibility to centreline crack-ing They would also improve the ultrasonic inspectionof certain alloys eg austenitic stainless steels Inpractice columnar equiaxed transitions are rarelyobserved in welds and obtaining an equiaxed structureis often not achievable or difcult to optimise 96 Aswould be expected equiaxed regions are found if at allat the centreline where the solidication rates are fastestand the temperature gradients lowest The nature of welding processes and the cooling rates involved (10 2 ndash 107 K s ndash1 ) make quantitative experimental evaluation of the CET virtually impossible Also because of weld poolshape care must be taken to ensure that an apparentCET is real and that the equiaxed grains are notsimply transverse sections through columnar grains Themechanisms proposed to account for the CET (in the

absence of inoculants) are those previously suggested inthe section lsquoMechanisms of equiaxed grain formationrsquofor castings with the added possibility of grain detach-ment in the partially molten region (see the sectionlsquoOverall consideration of proposed mechanisms of equiaxed grain formationrsquo above) Since the fusion zoneboundary in a weld is at the freezing temperature weld

pool solidication usually proceeds by epitaxial growthfrom grains of the parent metal This thereforeeliminates mechanisms of equiaxed grain formationthat in casting may originate at the onset of freezingas a result of heterogeneous nucleation 1316 Equiaxedgrain formation because of dendrite remelting is oftensuggested owing to the considerable uid ow andthermal uctuations in the weld pool

Those parameters that are important in determiningthe CET in castings namely G R C 0 and columnarfront undercooling will also be of importance to weldstructures In welds that completely penetrate a thinsheet the solidication velocity R is related to thewelding velocity U by

R~ U cos h (7)

where h is the angle between the welding and localgrowth directions 96 h decreases towards the centrelineIn the absence of experimental data of the solidicationconditions that exist during the CET Huntrsquos model 32

see the section lsquoAnalytical modelsrsquo above remains thebasis for explaining the CET in welds 96 The modelcould only be used to attempt to predict the CET if the pool thermal conditions were known However therelationship between weld pool thermal conditions andthe controllable weld process parameters are notestablished In order to relate the parameters to thethermal conditions in the pool Clarke et al 97 havereported the development of a 3D nite elementthermouids model for simulating steady state full-penetration moving gas tungsten arc (GTA) weldsTemperature predictions were compared with measuredvalues in AlndashCu welds and the welding process para-

meters were incorporated in the simulations The modelpredicts that a CET is favoured by decreasing weldingspeed for a given current or using a high current andwelding speed combination

Kerr 96 summarises a number of techniques that havebeen used to increase the equiaxed fraction in weldsExcluding inoculation these include alloying to promoteconstitutional supercoolingdendrite fragmentation sur-face nucleation eg use of an argon jet on a pool surfaceahead of the arc in GTA welds of AlndashMg alloysimposed electromagnetic stirring in GTA and GMAwelds arc oscillation eg using mechanical vibrationand the use of pulsed and modulated currents

SummaryThe probable origins of the equiaxed grains associatedwith a CET (in the absence of added inoculants) are nowwell dened and accepted During solidication one ormore of them will operate and which ones become activeor dominant will depend on factors such as cooling ratefreezing time the temperature distribution ahead of thecolumnar front thermal and constitutional supercool-ing and the number and efciency of potential nucleantsin the melt In turn these are governed by the alloy

system the amount of solute present section size typeof solidication process (eg castingweldingdirectionalsolidication) and processing conditions (eg superheatmould temperatures extent of uid ow etc) It isevident that the manner of termination of columnargrowth will differ according to the number and rate of growth of the equiaxed grains and the degree of

13 Predicted CET process map lled circles ndash columnaropen circles ndashequiaxed CET predicted from Huntrsquosmodel with D T N of 0 75 K lled diamond and trianglendash experimentally determined CET conditions 40 for Alndash2 and Alndash4 wt-Cu respectively 68





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i c a

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L t d

sedimentation that occurs In small laboratory ingots alarge number of equiaxed grain nuclei may be formednear the mould wall at the time of pouring owing tothermal supercooling and growth restriction of thegrains caused by solute redistribution At low super-heats the supercooling may extend throughout the bulkliquid almost from the time of pouring Because of thehigh density of rapidly growing equiaxed grains thatmay exist throughout the liquid the CET will be sharpand occur over a large part of the columnar front at asimilar time With increasing ingot size and solidicationtime equiaxed grain settling will become more impor-tant in determining the CET and the shape of theequiaxed zone It is also unlikely that the equiaxedgrains now originate at the mould walls and othermechanisms such as dendrite fragmentation becomeimportant The CET will occur gradually over theadvancing columnar front either due to equiaxed grainsettling onto the lower part of the front or attachment of individual grains to the vertical faces An insufcientfraction of grains reachingattaching to the front maylead to entrapment of equiaxed grains in the columnarzone Other solidication processes such as directionalfreezing and welding are characterised by steep gradientsahead of the front however in the former there may belittle or no ow in the melt whereas in the latter it can beconsiderable

It is almost impossible to compare the various sets of experimental data obtained from directional freezingstudies of the CET because of the wide range of alloysystems and differing compositions investigated the factthat some studies have been performed under steadygrowth conditions (Bridgman growth) and others under

non-steady conditions and different mechanisms of equiaxed grain formation may have been operative inthe different studies Although certain generalisationscan be made ie that the CET is favoured by high V low G high solute levels and easier nucleation there isno consistency regarding the proposed interaction of thealloy and processing parameters and the criteria for thetransition The experiments do however provide neces-sary data required by modellers to validate models

Most of the models of the CET are 1D or 2Ddescriptions of the directional solidication of alloysthat assume that static equiaxed grains heterogeneouslynucleated ahead of the columnar front evolve until theyterminate columnar growth Except in the case of stochastic models where the growth of the individualgrains is tracked some criterion usually has to beassumed for the point of termination Of necessity allmodels require assumptions regarding the nucleation of the equiaxed grains These cannot accurately accountfor the origins of equiaxed grains their movement dueto uid ow and settling and their possible remeltingDeterministic models permit modelling at the scale of actual castings directionally solidied alloys etc Howeverthey do not permit the accurate modelling of thosemicrostructural factors that are likely to contribute to

the CET such as solute diffusion interdendritic convec-tion evolution of equiaxed grain morphology etcStochastic models offer the opportunity to examine theeffects of microstructural factors but are severely limitedby the sizes of the domains that can be modelled It isevident that the CET is governed by a complex set of interacting phenomena including the local thermal and

solute elds and the numbersize of the equiaxed grainsRecent models have attempted to simulate determinis-tically 45 and stochastically 68 the simultaneous growthand interaction of the columnar and equiaxed grains andthe associated solute and thermal elds leading to theCET Future models must continue in this way if understanding of the CET is to signicantly improveHowever as generally recognised a signicant factorlimiting the accuracy of models is the description of nucleation undercooling and the number density of equiaxed grain nuclei

Although most models are capable of qualitativelydemonstrating the inuence of various parameters onthe CET as stated by Hansen et al 98 the modellingprocedures must be classied at present as curve-ttingexercises to experimental data In their paper 98 theyconsidered whether it is possible to model the CET in aphysically realistic way They discussed the variousstages that would be involved in such a model byassuming that the equiaxed grains originate fromdendrite fragments generated in the columnar zoneThese would then be transported into the bulk liquid byuid ow where as a result of growth and sedimenta-tion they cause columnar blocking

It is evident that the CET and the factors inuencingit are qualitatively quite well understood Any improve-ment in the predictive capabilities of existing models of the CET will only be achieved by more accuratelysimulating the physical aspects of the formation of equiaxed grains their movement in the melt and theirinteraction with the columnar front

References1 J A Spittle Mater Sci Technol 2005 21 5462 T E Quested and A L Greer Acta Mater 2005 53 46433 J E Stead Engineering 1906 82 40544 H M Howe J Iron Steel Inst 1916 94 1815 R Genders J Inst Met 1926 35 2596 F R Hensel PhD thesis University of Berlin 19297 W A Tiller K A Jackson J Rutter and B Chalmers Acta

Metall 1953 1 4288 S C Flood and J D Hunt ASM Metal Handbook 1990 15 1309 S E Kisakurek lsquoFormation of the equiaxed zone in ingot

castingsrsquo Bogazici University Research Centre PublicationsNo 104 Istanbul 1981

10 J Hutt and D St John Int J Cast Met Res 1998 11 1311 L Northcott J Inst Met 1938ndash39 65 17312 W C Winegard and B Chalmers Trans ASM 1954 216 121413 B Chalmers Aust J Inst Met 1963 8 25514 K A Jackson J D Hunt D R Uhlmann and T P Seward III

Trans AIME 1966 236 14915 R T Southin Trans Met Soc AIME 1967 239 22016 A Ohno T Motegi and H Soda Trans ISIJ 1971 11 1817 B P Pearce and H W Kerr Metall Trans B 1981 12B 47918 J L Walker in lsquoPhysical chemistry of process metallurgy IIrsquo 845

1961 New York Interscience19 G S Cole and G F Bolling Trans AIME 1965 233 156820 H Biloni and B Chalmers J Mater Sci 1968 3 13921 L A Tarshis J L Walker and J W Rutter Metall Trans 1971

2 258922 J A Spittle and S Sadli Mater Sci Technol 1995 11 53323 R D Doherty P D Cooper M H Bradbury and F J Honey

Metall Trans A 1977 8A 397

24 M Abdel-Reihim N Hess W Reif and M E J Birch J MaterSci 1987 22 21325 J A Spittle J M Keeble and M Al Meshhedani in lsquoLight metals

1997rsquo (ed R Huglen) 795ndash800 2001 Warrendale PA TMS26 J A Spittle Met Sci 1977 11 57827 T S Plaskett and W C Winegard Trans ASM 1959 51 22228 R Elliott British Foundryman 1964 9 39829 W A Tiller Trans Met Soc AIME 1962 224 448





P u

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i c a

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L t d

30 R B Mahapatra and F Weinberg Metall Trans B 1987 18B 425

31 I Ziv and F Weinberg Metall Trans B 1989 20B 73132 J D Hunt Mater Sci Eng 1984 65 7533 V K Suri N El-Kaddah and J T Berry Trans AFS 1991 99

18734 S Kim and R N Grugel Metall Trans A 1992 23A 180735 L J Ledgard and D G McCartney Proc 4th Decennial Int

Conf on lsquoSolidification processingrsquo Sheffield UK 1997 22736 T M Pollock and W H Murphy Metall Mater Trans A 1996

27A 108137 Ch-A Gandin ISIJ Int 2000 40 97138 A E Ares and C E Schvezov Metall Mater Trans A 2000 31A

161139 M A Martorano and J D T Capocchi Int J Cast Met Res

2000 13 4940 C A Siqueira N Cheung and A Garcia Metall Mater Trans A

2002 33A 210741 M Vandyoussefi and A L Greer Acta Mater 2002 50 169342 Calcom SA Parc Scientifique EPFL Lausanne Switzerland43 Ch-A Gandin and M Rappaz Acta Metall Mater 1994 42

223344 L Sturz A Drevermann C Pickmann and G Zimmermann

Mater Sci Eng 2005 A 413ndash414 37945 M A Martorano C Beckermann and Ch-A Gandin Metall

Mater Trans A 2003 34A

165746 G Reinhart N Mangelinck-Noel H Nguyen-Thi T SchenkJ Gastaldi B Billia P Pino J Hartwig and J Baruchel MaterSci Eng A 2005 413-414 384

47 D R Uhlmann T P Seward and B Chalmers Trans AIME 1966 236 527

48 G S Cole and G F Bolling Trans AIME 1966 236 136649 G S Cole and G F Bolling Trans AIME 1967 239 182450 R Morando H Biloni G S Cole and G F Bolling Metall

Trans A 1970 1A 140551 G S Cole K W Casey and G F Bolling Metall Trans A 1970

1A 141352 R J McDonald and J D Hunt Trans AIME 1969 245 199353 P M Thomas and J A Spittle J Inst Met 1971 99 16754 G S Cole and G F Bolling Trans AIME 1968 243 156855 G S Cole and G F Bolling Trans AIME 1969 245 725

56 J A Spittle G W Delamore and R W Smith lsquoThe solidificationof metalsrsquo Iron and Steel Institute Publication 110 318 1968London Iron and Steel Institute

57 W D Griffiths and D G McCartney Mater Sci Eng 1996A216 47

58 B Willers S Eckert U Michel I Haase and G Zouhar MaterSci Eng 2005 A402 55

59 S Eckert B Willers P A Nikrityuk K Eckert U Michel andG Zouhar Mater Sci Eng 2005 A413-414 211

60 H Fredriksson and M Hillert Metall Trans 1972 3 56561 H Fredriksson and A Olsson Mater Sci Technol 1986 2 50862 M H Burden and J D Hunt Metall Trans A 1975 6A 24063 S Witzke J P Riquet and F Durand Acta Metall 1981 29 36564 J Lipton W Kurz and W Heinemann Concast Technol News

1983 22 465 J Lipton W Heinemann and W Kurz Arch Eisenhuttenwes

1984 55 19566 Ch-A Gandin Acta Mater 2000 48 2483

67 L Nastac Acta Mater 1999 47 425368 H B Dong and P D Lee Acta Mater 2005 53 65969 S C Flood and J D Hunt lsquoModeling of casting and welding

processes IIrsquo 207 1983 American Institute of MiningMetallurgical and Petroleum Engineers

70 S C Flood and J D Hunt J Cryst Growth 1987 82 55271 E Scheil Z Metallkd 1942 34 7072 J A Spittle and M R Tadayon Cast Met 1994 7 12373 S G R Brown and J A Spittle in lsquoModeling of casting welding

and advanced solidification processes Vrsquo 395 1991 WarrendalePA TMS

74 C Y Wang and C Beckermann Metall Mater Trans A 199425A 1081

75 C Beckermann and C Y Wang JOM 1994 4276 C Y Wang and C Beckermann Metall Trans A 1993 24A

278777 S L Cockcroft M Rappaz A Mitchell J Fernihough and A J

Schmalz in lsquoMaterials for advanced power engineering part IIrsquo(ed D Coutsouradis et al ) 1145 1994 Netherlands KluwerAcademic Publishers

78 W Kurz B Giovanola and T R Trivedi Acta Metall 1986 34 823

79 M Rappaz Int Mater Rev 1989 34 9380 M Gaumann R Trivedi and W Kurz Mater Sci Eng 1997

A226-228 763

81 A L Greer A M Bunn A Tronche P V Evans and D JBristow Acta Mater 2000 48 2823

82 A Ludwig and M Wu Mater Sci Eng 2005 A413-414 10983 S G R Brown and J A Spittle Mater Sci Technol 1989 5 36284 J A Spittle and S G R Brown J Mater Sci 1989 23 177785 J A Spittle and S G R Brown Acta Metall 1989 37 180386 S G R Brown and J A Spittle Cast Met 1990 3 1887 M Rappaz and Ch-A Gandin Acta Metall Mater 1993 41 34588 J A Spittle and S G R Brown in lsquoNumerical methods in thermal

problems volume VIrsquo (ed R W Lewis and K Morgan) 2911989 Swansea Pineridge Press

89 P Zhu and R W Smith Acta Metall Mater 1992 40 336990 R C Atwood and P D Lee Metall Mater Trans B 2002 33B

20991 R H Davies A T Dinsdale T G Chart T I Barry and M H

Rand High Temp Sci 1990 26

25192 C Beckermann A Badillo and J C Ramirez Invited LectureJOM 57(2) TMS Annual Meeting and Exhibition Program andGuide 2005 33

93 G J Davies and J G Garland Int Met Rev 1975 20 8394 S A David and J M Vitek Int Mater Rev 1989 34 21395 S A David and J M Vitek in lsquoInternational trends in welding

science and technologyrsquo (ed S A David and J M Vitek) 1471993 Materials Park OH ASM International

96 H W Kerr in lsquoInternational trends in welding science andtechnologyrsquo (ed S A David and J M Vitek) 157 1993 MaterialsPark OH ASM International

97 J Clarke D C Weckman and H W Kerr in lsquoModeling of casting welding and advanced solidification processes VIIIrsquo (edB G Thomas and C Beckermann) 391 1998 Warrendale PATMS

98 G Hansen A Hellawell S Z Lu and R S Steube Metall MaterTrans A 1996 27A 569




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i c a

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L t d

easily achieved experimentally and modelled since thenumberpotency of nucleant particles will be lowerHowever commercially the use of grain reners may bethe normal practice eg in the DC casting of Al alloysand one recent study has attempted to model the CETduring directional solidication of Al alloys with grainrener addition 2

Because of the problems associated with the manip-ulation of processing parameters and structural exam-ination on a commercial scale most qualitative andquantitative studies of the CET have been performed

experimentally on relatively small volumes of lowmelting point materials (usually aluminium alloys)Generally speaking most studies have involved one of two approaches either (i) investigation of the inuenceof various parameters on the as solidied grainstructures of simple shapes eg cylinders of eitherpoured castings or alloys melted and solidied lsquo in situ rsquo

or (ii) the directional solidication of alloys either underBridgman or non-steady (solidication from cooledchills) conditions Alternatively because of the opacityof metals the freezing of transparent non-metallicanalogues of alloy systems has been studied (egNH 4ClndashH 2O cyclohexanolndashphenol red)

Although the CET has been reported and discussed inmany early studies including those of Stead 3 Howe 4

Genders 5 and Hensel 6 signicant renewed interest arosefrom the initial quantitative derivation of the conditionsnecessary for the breakdown of a planar interface as aresult of lsquoconstitutional supercoolingrsquo (CS) by Tilleret al 7 [equation (1)] CS leading to the instability of planar growth will occur when

G =R v mC 0(1 k )=kD (1)

where G is the temperature gradient in the liquid R theinterface velocity (in this review R and V are used inter-changeably for interface velocity for ease of presentingreported data) m the liquidus slope C 0 the initial alloy

composition k the equilibrium distribution coefcientand D the solute diffusion coefcient in the liquidEquiaxed grain formation and the CET have been

extensively studied over the last half century and theimportant aspects that have been examined can besummarised as follows

(i) investigation of the mechanisms of equiaxedgrain formation (how and where do theequiaxed grains originate)

(ii) qualitative and quantitative experimental eva-luation of the parameters influencing the CET

(iii) investigation of the conditions causing thetermination of columnar freezing (what physi-cally causes columnar grains to stop growing)

(iv) development of deterministic and stochasticmodels for predicting the CET

The most recent comprehensive review of the CETgiving consideration to all of these aspects is that due toFlood and Hunt 8 In the intervening years there hasbeen additional experimental research and further modeldevelopment particularly the use of stochastic modelsfor simulating the nucleation and growth of individualcolumnar and equiaxed grains

Requirements and features of CET Unless otherwise stated the remainder of this reviewrelates to the grain structures of solid solution grains inalloys (usually binary alloys) Before embarking on adetailed consideration of the CET it is useful to state theobvious requirements for a CET and to list some simpleobservations pertinent to the discussion of how it isinuenced by different solidication parameters Toobtain a structure revealing a CET (ie showingcolumnar growth arrested by equiaxed grains) thefollowing are required

(i) the presence of supercooled liquid ahead of ornear the columnar front in which either (a)

equiaxed grains nucleate and grow or (b) towhich grainsfragments of grains nucleated else-where are transported and then grow In thelatter case during the early stages of solidifica-tion conditions must exist in the melt for thenucleationfragmentation of grains elsewhere andtheir transport and survival

1 Columnar to equiaxed (CET) grain transition in an Alndash4 wt-Cu alloy 56





P u

b l i s h e

d b y

M a n e y

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b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d















P ~ mC 0(1 k )=k (2)








P u

b l i s h e

d b y

M a n e y

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b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Q~ kP (3)













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d








G v 074V 064 (4)









alloys (Al60 Cr4

5 Co12

5 Hf0

16 Re6

3 Ta7





Ares and Schvezov38






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

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b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



modelsrsquo below43













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

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b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

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b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

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b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d















P ~ mC 0(1 k )=k (2)








P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Q~ kP (3)













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d








G v 074V 064 (4)









alloys (Al60 Cr4

5 Co12

5 Hf0

16 Re6

3 Ta7





Ares and Schvezov38






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



modelsrsquo below43













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d















P ~ mC 0(1 k )=k (2)








P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Q~ kP (3)













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d








G v 074V 064 (4)









alloys (Al60 Cr4

5 Co12

5 Hf0

16 Re6

3 Ta7





Ares and Schvezov38






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



modelsrsquo below43













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d















P ~ mC 0(1 k )=k (2)








P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Q~ kP (3)













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d








G v 074V 064 (4)









alloys (Al60 Cr4

5 Co12

5 Hf0

16 Re6

3 Ta7





Ares and Schvezov38






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



modelsrsquo below43













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Q~ kP (3)













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d








G v 074V 064 (4)









alloys (Al60 Cr4

5 Co12

5 Hf0

16 Re6

3 Ta7





Ares and Schvezov38






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



modelsrsquo below43













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d








G v 074V 064 (4)









alloys (Al60 Cr4

5 Co12

5 Hf0

16 Re6

3 Ta7





Ares and Schvezov38






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



modelsrsquo below43













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



modelsrsquo below43













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



modelsrsquo below43













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d













CET










P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d







solidication55











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Willers et al 58
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














Hunt32



Gandin66


Martorano et al 45






P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




G v 0617N 1=30 f1 (DT N )3=(DT C )3gDT C (5)




DT C ~ f A0m(1 k )C 0V g1=2 (6)
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


Martorano et al 45



Cockroft et al 77













P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d














P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d



Rappaz and Gandin87















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d











P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d




gradient





R~ U cos h (7)












P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d























P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763
















P u

b l i s h e

d b y

M a n e y

P u

b l i s h i n g

( c ) I O M

C o m m u n

i c a

t i o n s

L t d


































A226-228 763














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Review 2006 (Spittle)