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Fundamentals of Weld SolidificationJohn N. DuPont, Lehigh University

MICROSTRUCTURAL EVOLUTION dur-ing solidification of the fusion zone representsone of the most important considerations forcontrolling the properties of welds. A widerange of microstructural features can form inthe fusion zone, depending on the alloy compo-sition, welding parameters, and resultant solidi-fication conditions. The primary objective ofthis article is to review and apply fundamentalsolidification concepts for understanding micro-structural evolution in fusion welds.

Microstructural Features in FusionWelds

Figures 1 through 3 schematically demon-strate the important microstructural featuresthat must be considered during solidificationin fusion welds (Ref 1, 2). On a macroscopicscale, fusion welds can adopt a range of grainmorphologies similar to castings (Fig. 1), inwhich columnar and equiaxed grains can poten-tially form during solidification. The final grainstructure depends primarily on alloy composi-tion and the heat-source travel speed. Althoughsome of the concepts applicable to grain struc-ture formation in castings apply to welds, thereare also some unique differences. While thecolumnar and equiaxed zones can form inwelds, the fine-grained chill zone at the moldwall represented by the fusion line is rarelyobserved in welds. Fundamental concepts asso-ciated with nucleation are needed to understandthese differences. In addition, the solid/liquidinterface in welds is typically curved and itsrate of movement is controlled by the heat-source travel speed, which leads to differencesin formation of the columnar zone withinwelds. These differences can be understoodthough application of competitive grain growthprocesses that occur during solidification.On a microscopic scale, there can also be a

wide range of substructural morphologieswithin the grains (Fig. 2), including planar(i.e., no substructure), cellular, columnar den-dritic, and equiaxed dendritic. The type and rel-ative extent of each substructural region isgoverned by the process of constitutionalsupercooling in which the liquid becomes

cooled below its liquidus temperature due tocompositional gradients in the liquid. Theextent of constitutional supercooling in theweld is determined by the alloy composition,welding parameters, and resultant solidificationparameters. Lastly, the distribution of alloyingelements and relative phase fractions withinthe substructure (Fig. 3) are also importantmicrostructural features that strongly affectweld-metal properties. The particular exampleshown in Fig. 3 represents a case in whichextensive residual microsegregation of alloyingelements exists across a cellular substructureafter nonequilibrium solidification. This micro-segregation, in turn, produces a relatively highfraction of intercellular eutectic and associatedsecondary phase. The microsegregation behav-ior and concomitant amount of secondary phasethat forms can each be understood with soluteredistribution concepts. Lastly, dendrite tipundercooling can become important athigh solidification rates associated with high-energy-density welding processes. Tip under-cooling can lead to significant changes in theprimary solidification mode, distribution of sol-ute within the solid, and final phase fractionbalance. Rapid solidification concepts areneeded to understand these phenomena. Allof these fundamental solidification concepts(nucleation, competitive grain growth, constitu-tional supercooling, solute redistribution, and

Fig. 1 Types of grain morphologies that can form infusion welds

Fig. 2 Types of substructure morphologies that can form within the grains of fusion welds

ASM Handbook, Volume 6A, Welding Fundamentals and ProcessesT. Lienert, T. Siewert, S. Babu, and V. Acoff, editors

Copyright # 2011 ASM InternationalW

All rights reserved

www.asminternational.org

rapid solidification) depend on the solidificationparameters during welding. Thus, the importantsolidification parameters are briefly described,followed by detailed discussions and applica-tion of fundamental solidification concepts forunderstanding microstructural evolution.

Solidification Parameters

The temperature gradient (G), solid/liquidinterface growth rate (R), and cooling rate (e)are the important solidification parameters.These three parameters are related by:

e ¼ GR (Eq 1)

This simple relation is not always intuitive andcan be understood more clearly with the helpof Fig. 4. This figure shows a fixed temperaturegradient moving from right to left at a rate R sothat t1 > t2. At a fixed position x*, the temper-ature is reduced from T1 to T2 within the timet1 – t2. In other words, the time it takes for atemperature reduction (i.e., the cooling rate) isgoverned by the rate of movement (R) of thetemperature gradient (G) through space.The solidification parameters are not con-

trolled directly in fusion welding but are gov-erned by the welding parameters. As explainedin more detail subsequently, the growth rate isdetermined largely by the speed of the heatsource and shape of the weld pool. The coolingrate and temperature gradient are controlled pri-marily by the heat input (HI), which is defined as:

HI ¼ ZPS

(Eq 2)

where Z is the heat-source transfer efficiency, Pis the heat-source power, and S is the heat-source travel speed. The heat input representsthe amount of energy delivered per unit lengthof weld. It should be noted that e, G, and R can-not be represented by single values duringsolidification. Due to the complex nature ofheat flow in the weld pool, the spatial distribu-tion of temperature is not linear. Similarly, thevalue of R typically varies throughout the weldpool due to the change in growth directionsbetween the solid/liquid interface and heatsource. As a result, the values of e, G, and Rare functions of position and time. In general,the temperature gradient and cooling rate eachdecrease with increasing heat input. Detailedheat-flow equations can be used to quantifythe influence of welding parameters on solidifi-cation parameters. Heat flow in welding is dis-cussed in more detail in other articles in thisVolume (see, for example, the article “FactorsInfluencing Heat Flow in Fusion Welding”).

Nucleation Considerations in FusionWelding

Although the microstructures of castings andwelds share some similarities, there are alsosome significant differences. For example, asshown schematically in Fig. 5 (Ref 3), castingstypically exhibit a chill zone that consists offine equiaxed grains which form near the moldwall. This zone forms as a result of nucleationat the mold/casting interface. This region is typ-ically followed by a columnar zone and anotherequiaxed zone near the center of the casting.The columnar grain region and the centralequiaxed grain zones can also form in fusionwelds. However, the equiaxed grains associatedwith the chill zone generally do not form infusion welds. The differences in these featurescan be understood with the application of

nucleation theory, which is covered in thissection.

Nucleation Theory

Figure 6 compares examples of homoge-neous nucleation in a liquid (Fig. 6a, b) and het-erogeneous nucleation on a preexisting moldwall (Fig. 6c, d). The change in free energyassociated with homogeneous nucleation(△Ghom) is given by (Ref 4):

�Ghom ¼ �Vs�Gv þ ASLgSL (Eq 3)

where Vs is the volume of the nuclei, △Gv is thevolume free-energy change associated withnucleation, ASL is the solid/liquid interfacialarea, and gSL is the solid/liquid interfacialenergy. The volume free energy is the drivingforce for solidification and is shown schemati-cally in Fig. 7(a), which shows the variationin volume free energy for the solid and liquidas a function of temperature. Note that △Gv =0 at the melting point, so there is no drivingforce for solidification at T = Tm. Thus, under-cooling is generally required to drive the nucle-ation process. The interfacial energy is apositive contribution to the overall free-energychange and therefore works to oppose forma-tion of the nucleus. The change in volume freeenergy with undercooling (△T) is given by:

�Gv ¼ L�T

Tm(Eq 4)

where L is the latent heat of fusion. A sphericalnucleus is favored over other shapes becauseit provides the minimum surface area/volume ratio, thus providing the largest pos-sible reduction in △Ghom by maximizingthe negative Vs△Gv term and minimizingthe positive ASLgSL term. For a spherical

Fig. 3 Potential distribution of alloying elements andphase fractions that can form in fusion welds.

Example shown is for a simple eutectic system that formsprimary a phase and intercellular a/b eutectic underconditions of nonequilibrium solidification. Location ofcomposition trace is across primary and eutectic aphase, as shown by horizontal dotted line.

Fig. 4 Schematic illustration showing relation betweentemperature gradient (G), growth rate (R), and

cooling rate. The cooling rate is controlled by the rate ofmovement of the temperature gradient.

Fig. 5 Schematic illustration of grain structures thatcan form in castings, showing the chill zone

near the mold wall, the columnar zone, and theequiaxed zone in the center of the casting. Source: Ref 3

Fundamentals of Weld Solidification / 97

nucleus of radius r, the free-energy change forhomogeneous nucleation is given by:

�Ghom ¼ � 4

3pr3�Gv þ 4pr2gSL (Eq 5)

Equation 5 is plotted in Fig. 7(b), whichshows the individual contributions to △Ghom

and the overall change in △Ghom. The overallchange in △Ghom goes through a maximum,which describes the critical radius (r*) atwhich point the reduction in △Ghom due to�4/3pr3△Gv begins to become larger than theincrease in △Ghom associated with the 4pr2gSLterm. A nucleus that forms with a radius smal-ler than r* is unstable, because any furtherincrease in the nuclei radius will produce anincrease in △Ghom. Conversely, any nuclei thatforms with a radius larger than r* is stable,because further growth in the nuclei leads toan overall reduction in △Ghom. The maximumin △Ghom is denoted as △G* and representsthe activation energy associated with homoge-neous nucleation. This condition is given by:

d�Ghom

dr¼ �4pr2�Gv þ 8prgSL ¼ 0 at r� (Eq 6)

From which the following expressions can bederived for r* and △G*:

r� ¼ 2gSL�Gv

(Eq 7a)

�G� ¼ 16pg3SL3 �Gvð Þ2 (Eq 7b)

Equations 7 and 4 can be combined toreveal the influence of undercooling on r* and△G*:

r� ¼ 2gSLL

� �Tm�T

(Eq 8a)

�G� ¼ 16pg3SL3L2

� �Tm�T

� �2

(Eq 8b)

Note that, according to Eq 8(a) and (b), r* and△G* are infinite when △T = 0, indicating thatnucleation cannot occur without some under-cooling. The undercooling is needed so thatthe reduction in △Ghom due to the volumefree-energy change (by way of Eq 4) is largerthan that due to the increase in △Ghom asso-ciated with the interfacial energy term. Thus,r* and △G* each decrease with increasedundercooling.For heterogeneous nucleation on an existing

mold wall, the overall free-energy change isgiven by (Ref 4):

�Ghet¼�Vs�GvþASLgSL þ ASMðgSM � gLMÞ(Eq 9)

The first two terms are identical to those forhomogeneous nucleation. The third term(ASMgSM) represents the increase in overall freeenergy from formation of the solid/mold inter-face, while the fourth term (ASMgLM) representsthe reduction in overall free energy associated

with elimination of some of the liquid/moldinterface due to formation of the nucleus. Notethat the interfacial energy between two solids istypically less than that between a solid and aliquid. Thus, the ASM(gSM � gLM) term is typi-cally negative so the △Ghet < △Ghom. Thevalues of Vs, ASL, and ASM will depend on theshape of the nucleus as determined by the wet-ting angle, y (Fig. 6d). The value of y, in turn,is governed by the relative values of surfaceenergies and is given by a force balance in thehorizontal direction as:

gLM ¼ gSM þ gSL cos y (Eq 10a)

y ¼ cos�1gLM � gSM

gSL

� �(Eq 10b)

Assuming the nucleus forms as a sphericalcap, the values of ASL, ASM, and Vs are givenby:

ASL ¼ 2pr2ð1� cos yÞ (Eq 11a)

ASM ¼ pr2sin2 y (Eq 11b)

VS ¼ pr3ð2þ cos yÞ 1� cos yð Þ23

(Eq 11c)

Equations 9 to 11 can be combined for anexpression for △Ghet:

�Ghet ¼ � 4

3pr3�Gv þ 4pr2gSL

� �S yð Þ (Eq 12)

where:

S yð Þ ¼ 2þ cos yð Þ 1� cos yð Þ24

(Eq 13)

Note that 0� � y � 180� and 0 � S(y) � 1.Thus, it is apparent that △Ghet � △Ghom. Thisis shown schematically in Fig. 8. Low valuesof y are an indication that the solid/mold inter-facial energy is low so that the solid easily wetsthe mold wall. This leads to low values of S(y)and reduced values of △Ghet; that is, nucleationis made easier for reduced values of gSM.

Application to Fusion Welding

Fusion welding represents a unique casethat can be most easily understood bystarting with Eq 10(b) and noting that themold wall and the solid are identical, becausethe base metal acts as the mold in fusion weld-ing. Thus, it can immediately be noted thatfusion welding leads to the followingconditions:

gSM ¼ 0 (Eq 14a)

gLM ¼ gSL (Eq 14b)

θ

Fig. 6 Schematic illustrations of homogeneous (a andb) and heterogeneous (c and d) nucleation.

Figures on the left are for a temperature above themelting point. Figures on the right are for a temperaturebelow the melting point. The wetting angle, y, is shownin (d).

Fig. 7 (a) Variation in volume free-energy withtemperature for the solid and liquid. (b)

Variation in surface energy term, volume free-energyterm, and DG

homwith nucleus radius for homogeneous

nucleation

98 / Fundamentals of Fusion Welding

y ¼ cos�1gSLgSL

� �¼ 0 (Eq 14c)

Sðy ¼ 0Þ ¼ 0 (Eq 14d)

Thus, from Eq 12, the activation energy in fusionwelding (△Gfw) for nucleation is given simply by△Gfw = 0. This is compared to△Ghom and△Ghet

in Fig. 8. This can be interpreted by noting thatthe base-metal mold presents a perfect crystallo-graphicmatch for growth of the solid fusion zone.Thus, there is no solid/mold interface in fusionwelding due to this perfect crystallographicmatching. An example of this in a fusion weldmade with the electron beam process is shownin Fig. 9 (Ref 5). Note that there are no fineequiaxed grains at the fusion line, as oftenobserved in the chill zone of castings. Instead,the weld-metal grains grow directly from the pre-existing base-metal grains. As a result, there is nobarrier to formation of the solid. This condition isreferred to as epitaxial growth, because growthoccurs directly from the preexisting solid withoutthe need for nucleation. Therefore, there is noundercooling required to initiate solidificationat the fusion line, and solidification commencesat the liquidus temperature of the alloy. It shouldbe noted that undercooling can still occur near theweld centerline due to the process of constitu-tional supercooling, as explained in more detaillater. This can lead to the formation of the centralequiaxed zone often observed in fusion welds.Undercooling can also be required for nucleationof new phases during solidification.

Grain Structure of Fusion Welds

As described previously, the weld-metalgrains will grow epitaxially from the preexist-ing base-metal grains. However, not all of thesegrains will be favorably oriented for continuedgrowth. Two primary factors control thecontinued competitive growth of weld-metalgrains:

� The grains tend to grow in a direction anti-parallel to the maximum direction for heatextraction.

� The solid will grow in the “easy-growth”crystallographic directions.

The first criterion results from the need totransport the latent heat of solidification downthe temperature gradient into the cooler basemetal. Because the temperature gradient ishighest in a direction perpendicular to thesolid/liquid interface, the resultant heat-flowrate is also highest in this direction. Thus, thegrains tend to grow in a direction perpendicularto the solid/liquid interface. The second crite-rion results from the preferred crystallographicgrowth direction, which, for cubic metals, isalong the [100] directions. By combining thesetwo criteria, it can be seen that grains that havetheir easy-growth direction most closelyaligned to the solid/liquid interface normal willbe most favorably oriented to grow, thuscrowding out less-favorably-oriented grains.This phenomenon accounts for the columnargrain zone that is often observed in castings,shown schematically in Fig. 5. In this case,the grains that nucleated near the mold walland have their easy-growth direction alignednormal to the mold/casting interface outgrowthe less-favorably-oriented grains, leading tothe columnar region.The situation is slightly more complex in

fusion welding, because the pool shape pro-duces a curved solid/liquid interface that is con-stantly in motion as it follows the heat source.This is shown schematically in Fig. 10 (Ref6). Grains at the fusion line may initially be ori-ented in a favorable direction for growth, buttheir direction may become unfavorable as thecurved solid/liquid interface changes its posi-tion. These grains may then eventually beovergrown by other grains that exhibit morefavorable orientation for growth as the solid/liq-uid interface sweeps through the weld. Anexample of this is shown on a weld in nearlypure (99.96%) aluminum in Fig. 11(a) (Ref 7).

As may be expected, the pool shape can havea strong influence on competitive grain growthand the resultant grain structure of the weld.In turn, the pool shape can be influenced bythe welding parameters. At low-to-moderateheat-source travel speeds, the pool shape is gen-erally elliptical and typically produces the grainstructure pattern shown in Fig. 11(a). However,at higher travel speeds, the pool shape becomeselongated into a teardrop shape in which thesolid/liquid interface is straight. This elongatedshape is attributed to the low thermal gradientand high growth rate that exist at the weld cen-terline. The release of latent heat is proportionalto the growth rate. Because the growth rate ishighest at the weld centerline, the release rateof latent heat is also highest at the weld center-line. However, the temperature gradient is at a

Fig. 9 Example of epitaxial growth from the fusion linein an electron beam weld of alloy C103.

Original magnification: 400�. Source: Ref 5

Fig. 10 Schematic illustrations of competitive graingrowth in welds. (a) Early growth of grains

near the fusion line. (b) Continued growth of favorably-orientated grains at a later time. Source: Ref 6

Fig. 8 Comparison of free-energy changes associatedwith homogeneous nucleation, heterogeneous

nucleation, and fusion welding

Fundamentals of Weld Solidification / 99

minimum at the weld centerline, so it is difficultto transport the latent heat away from the poolto permit solidification. This causes elongationof the pool near the weld centerline and leadsto the teardrop shape. In this case, the directionof grain growth does not change (because thesolid/liquid interface is no longer curved), andthe grains grow straight toward the weld center-line until grains growing from each side of theweld intersect. This process typically leads toa centerline grain boundary, as shown inFig. 11(b) (Ref 7).Axial grains that grow along the direction of

heat-source travel can also occasionally beobserved in fusion welds. The various types ofgrain morphologies are summarized in Fig. 12(Ref 6). Examples of grain structures producedwith elliptical and teardrop-shaped weld poolswere shown in Fig. 11. Figures 12(c) and (d) rep-resent conditions in which an axial grain growsalong the direction of the heat-source travel.These grains form in the region where the solid/liquid interface is generally perpendicular to thedirection of heat-source travel, so that it becomesfavorable for one or more grains to grow in thisdirection. The width of this zone can depend onthe pool shape. The region of the interface thatis perpendicular to the heat-source direction isrelatively small in an elongated weld pool, sothe width of axial grains will also be small. Bycomparison, this perpendicular region is rela-tively larger for an elliptical pool, so the axialgrain region can also be larger.The large columnar grains and the potential

presence of centerline grain boundaries are gen-erally undesirable from a weldability andmechanical property point of view. Centerlinegrain boundaries can often lead to solidificationcracking associated with solidification shrinkageand low-melting-point films that become concen-trated at the centerline. Fine, equiaxed grains aredesired over coarse columnar grains for improve-ments in both cracking resistance andmechanicalproperties (at low temperature). One effectivemeans for minimizing or eliminating the coarsecolumnar grains is through manipulation of thepool shape. Figure 13 shows an example of aweld in which the arc was oscillated at a fre-quency of 1 Hz in a direction normal to theheat-source travel (Ref 8). In this case, the contin-uously changing direction of the solid/liquidinterface makes it difficult for the columnargrains to extend over large distances, thusproviding a degree of grain refinement. Grainsize reduction can also be achieved through theuse of inoculants. This process takes advantageof heterogeneous nucleation (discussed previ-ously) and liquid undercooling that occur due toconstitutional supercooling. This topic isdescribed in more detail in the next section.

Substructure Formation in FusionWelds

As shown previously in Fig. 2, grains inwelds typically exhibit various substructural

morphologies within the grains that can be cel-lular, columnar dendritic, or equiaxed dendritic.Cellular and columnar dendritic morphologiesdevelop due to breakdown of the initially planarsolid/liquid interface that forms at the fusionline, while equiaxed dendrites form by nucle-ation of solid in undercooled liquid, typically

near the weld centerline. Formation of thesefeatures can be understood with the concept ofconstitutional supercooling. The basics of thistopic are described first, followed by applica-tion of the theory to understanding the substruc-ture formation in fusion welds.

Constitutional Supercooling

As shown by the phase diagram in Fig. 14(a),formation of a solid leads to rejection of soluteinto the liquid. The extent of solute enrichmentin the liquid progresses as solidification pro-ceeds and the liquid composition follows theliquidus line. The solute rejected by the solidat the solid/liquid interface must be transportedaway from the interface by diffusion and/orconvection in the liquid. If the solid/liquidinterface growth rate is relatively high (whichleads to a high rate of solute rejection) and/orthe transport of solute into the liquid by diffu-sion or convection is low, then a solute bound-ary layer can develop in the liquid near thesolid/liquid interface. Because solute enrich-ment leads to a reduction in the liquidus tem-perature (for an element that partitions to theliquid), it follows that the presence of a solute

Fig. 12 Summary of various grain morphologies that can form from weld pools of different shapes. Source: Ref 6

Fig. 13 Grain structure in a fusion weld of alloy 2014made with transverse arc oscillation. Source:

Ref 8

Fig. 11 Examples of (a) competitive grain growth and(b) a centerline grain boundary forming on a

weld in 99.96 % Al. The weld in (a) was made at awelding speed of 250 mm/min (10 in./min). The weld in(b) was made at a welding speed of 1000 mm/min (40in./min). Source: Ref 7

100 / Fundamentals of Fusion Welding

boundary layer leads to a gradient in the liqui-dus temperature near the solid/liquid interface.This is shown schematically in Fig. 14(b) and(c), where the liquidus temperature is relativelylow at the solid/liquid interface due to the largeamount of solute in the liquid at that point. Theliquidus temperature gradually increases awayfrom the interface as the solute concentrationdecreases. The liquidus temperature does notchange outside the solute boundary layer.Next, consider how the relative magnitudes

of the liquidus temperature (TL) gradientdescribed previously (dTL/dx) and the actualtemperature (Ta) gradient (dTa/dx) in the liquidaffect the stability of a planar solid/liquid inter-face that forms as the alloy initially starts tosolidify. Figure 15(a) shows the condition fora relatively steep temperature gradient such thatdTa/dx > dTL/dx in the liquid. The planar inter-face is moving to the right and develops protru-sions during growth. Such protrusions canoccur in practical situations due to interfacepinning effects from inclusions in the liquidand/or differences in the rate of growth betweenneighboring grains that exhibit different crystal-lographic orientations relative to the solid/liq-uid interface (as described previously in thesection about competitive grain growth). Forthis case in which dTa/dx > dTL/dx, the tip ofthe protrusion encounters liquid that is at a

temperature above the liquidus temperature ofthe alloy. Thus, the solid protrusion is not stableand melts back so that the planar interfaceremains stable. Under this condition, no sub-structure forms.Figure 15(b) shows the case for a relatively

low-temperature gradient in which dTa/dx <dTL/dx in the liquid. In this case, the tips of theprotrusions encounter liquid that is below theliquidus temperature of the alloy. This conditionis referred to as constitutional supercoolingbecause the liquid is cooled below its liquidustemperature due to constitutional variationswithin the liquid. Under this condition, the solidat the tips is stable because it is growing into liq-uid that is cooled below the liquidus temperature.In addition, because the growth rate of the solid isproportional to the undercooling, the growth rateat the tips will be higher than at the roots of theprotrusions. As a result, the growth rate at the tipsis faster than that at the roots, and the planar inter-face breaks down into a cellular interface. Notethat solute redistribution in this case now occursin three dimensions, and solute can build upat the cell boundaries as neighboring cellsconverge. Depending on the cooling rate andsolute diffusivity in the solid, this local soluteenrichment may persist after solidification (seethe section “Solute Redistribution during Solidi-fication” in this article for more details). Thepresence of the cell walls reflects differences insolute concentration that are revealed by etching.The condition for breakdown of the planar solid/liquid interface and concomitant formation of acellular substructure is thus given by the condi-tion at which dTa/dx < dTL/dx. This can be deter-mined quantitatively by first developing anexpression for the solute concentration gradientin the liquid (dCL/dx) and then converting dCL/dx into dTL/dx via the phase diagram.Assuming a linear solute gradient in the liq-

uid, a mass balance about the solid/liquid inter-face is given by:

RðCL � CoÞ ¼ �DL

dCL

dx

� �(Eq 15)

where CL is the liquid composition at the solid/liquid interface, Co is the nominal alloy compo-sition, and DL is the solute diffusivity in the liq-uid. The left side of Eq 15 represents the soluteflux into the interface due to solute rejection bythe solid, while the right side of Eq 15 repre-sents the flux of solute down the gradient inthe liquid (assuming a linear concentration gra-dient). At steady state, these two fluxes balance.For a phase diagram with a linear liquidus line,a given value of dCL/dx produces acorresponding change in the liquidus tempera-ture dTL/dx that is given by:

dTLdx¼ mL

dCL

dx

� �(Eq 16)

where mL is the liquidus slope. Combination ofEq 15 and 16 provides an expression for theconstitutional criterion:

dTadx

<� RmLðCL � C0ÞDL

(Eq 17)

The value of dTa/dx is typically noted simply asG. At steady state, the value of CL is given byCL = Co/k (Ref 9), where k is the distributioncoefficient given by k = Cs/CL, where Cs isthe solid composition. Thus, the expression forconstitutional supercooling can be expressed as:

G

R<� mLCo 1� kð Þ

kDL

(Eq 18)

The parameters on the left are solidificationparameters, while those on the right are para-meters controlled by the alloy. The expressionmLCo(1 � k)/k represents the equilibrium solid-ification range of the alloy (△Teq), so that theconstitutional supercooling condition can alsobe written as:

G

R<��Teq

DL

(Eq 19)

This condition represents the onset of under-cooling and formation of a cellular substruc-ture. Other substructural zones can form withincreased undercooling. For example, the pla-nar regions separating neighboring cells canundergo constitutional supercooling betweenthe cells, thus leading to formation of columnardendrites. With further undercooling, the liquidahead of the dendritic interface can be super-cooled to the point where nucleation can occurwithin the liquid, thus leading to formation ofthe equiaxed zone often observed in welds andcastings and shown previously in Fig. 1(c) andFig. 5. The equiaxed zone forms near the centerof the weld or casting because the temperaturegradient is the lowest at this location.Examination of Eq 19 and reference to Fig.

15 indicate that supercooling is induced bylow values of G and high values of R. The influ-ence of G is readily apparent in Fig. 15, inwhich the extent of undercooling in the liquidincreases as G decreases. The effect of R is toincrease the concentration gradient in theliquid (Eq 15), which, in turn, increases theliquidus temperature gradient in the liquid.Thus, initial breakdown of the planar interfaceinto a cellular one occurs at relatively highvalues of G/R. As G/R decreases, the substruc-tural morphology can change from cellular tocolumnar dendritic to equiaxed dendritic. Thisis often summarized on a substructural diagram,as shown in Fig. 16.

Application to Fusion Welds

Fusion welds represent an interesting case ofsubstructure formation because the values of Gand R change continuously and in opposingdirections around the solid/liquid interfacewithin the weld pool. The value of G is rela-tively high at the fusion line because the moltenpool is in contact with relatively cool basemetal that has not been melted. The value of

Fig. 14 Schematic illustration showing (a) soluteenrichment in liquid during solidification,

(b) formation of a solute boundary layer in the liquid,and (c) variation in liquidus temperature near the solid/liquid interface due to formation of the solute boundarylayer

Fundamentals of Weld Solidification / 101

G decreases as the weld centerline isapproached because the molten pool is in con-tact with resolidified weld metal that wasalready heated well above the liquidus tempera-ture. An opposite trend exists for the growthrate. Figure 17 shows the relation betweenthe heat-source travel speed (S) and growthrate (R) in terms of the angles a and b, where

a represents the angle between the weldingdirection and normal to the solid/liquid inter-face, and b represents the angle betweenthe welding direction and [100] growth

direction (for a cubic metal). The growth rateR is given by:

R ¼ S cosacos a� bð Þ (Eq 20)

In many cases, the value of (a – b) is smallso that cos(a – b) � 1, and the relation betweenthe growth rate and heat-source travel speed isgiven simply as:

R ¼ S cosa (Eq 21)

From this it can be seen that R � 0 at the fusionline and R � S at the weld centerline. Thus, thevalue of G/R is very high at the fusion line anddecreases appreciably as the weld centerline isapproached. This can be portrayed on the G-Rdiagram as shown in Fig. 16. As a result,the fusion zone initially forms as a planar inter-face, which breaks down into cellular andcolumnar dendritic morphologies as the center-line is approached. If enough undercooling isobtained, an equiaxed zone can form in the cen-terline of the weld. The relative positions ofthese substructural morphologies are shownschematically in Fig. 2. An example of theequiaxed zone is shown in Fig. 18 (Ref 2).It is interesting to note that the planar zone

typically occupies a very small fraction of thefusion zone. This suggests that the planar

Fig. 15 Schematic illustrations showing (a) stability of a planar interface for a condition in which the actual temperature gradient in the liquid is greater than the liquidustemperature gradient and (b) breakdown of a planar interface for a condition in which the actual temperature gradient in the liquid is lower than the liquidus

temperature gradient

Fig. 16 Schematic illustration of a G-R diagramshowing regions of various substructural

morphologies Fig. 17 Schematic illustration showing the relationbetween the heat-source travel speed (S) and

growth rate (R) in terms of the angles a and b, where arepresents the angle between the welding direction andnormal to the solid/liquid interface, and b represents theangle between the welding direction and [100] growthdirection (for a cubic metal)

102 / Fundamentals of Fusion Welding

solid/liquid interface is inherently unstableunder most solidification conditions. This canbe verified by direct application of Eq 18, andresults of this for the aluminum-copper systemare shown in Table 1. These results show thecritical value of R required for breakdown ofthe solid/liquid interface for various values ofG and Co that may be encountered for typicalG values representative of casting, arc welding,and laser welding. For these calculations,values of mL (��3.3 �C/wt%) and k (�0.17)were obtained directly from the aluminum-cop-per phase diagram, and the value for solute dif-fusivity was taken as DL � 10�3 mm2/s. Notethat the critical growth rates for constitutionalsupercooling are significantly lower than actualgrowth rates expected during solidification, thusverifying that a planar interface is inherentlyunstable. This implies that the initially planarinterface growing from the fusion line willbreak down into a cellular one very quickly asthe value of G/R decreases, which is indeedobserved experimentally.

Columnar-to-Equiaxed Transition

The transition from cellular or columnar den-dritic growth at the fusion line to heterogeneousnucleation near the weld centerline is oftenreferred to as the columnar-to-equiaxed transi-tion (CET). As discussed subsequently, theCET occurs due to the relatively large

constitutional undercooling near the weld cen-terline. In welds of polycrystalline alloys, theCET can be exploited for grain refinement. Thisis typically accomplished by adding inoculantsto the weld to promote heterogeneous nucle-ation sites. The most effective inoculants arethose that minimize the activation energy(△Ghet*) required for heterogeneous nucle-ation, so that nucleation can occur with mini-mal undercooling. Equation 9 can be used toprovide some insight into effective inoculants,which, for heterogeneous nucleation on aninoculant, can be written as:

�Ghet ¼ �Vs�Gv þ ASLgSL þ ASIðgSI � gLIÞ (Eq 22)

This is identical to Eq 9, except that it is writtenfor heterogeneous nucleation on an inoculant (I)rather than a mold wall (M). As previously dis-cussed, the first two terms in Eq 22 are similarto those for homogeneous nucleation. Thefree-energy change associated with heteroge-neous nucleation on an inoculant can be mini-mized by minimizing the interfacial energybetween the solid and inoculant (gSI), so thatthe ASI(gSI � gLI) term is as large and negativeas possible. The inoculant/solid interfacialenergy can be minimized by optimizing thecrystallographic matching between the inocu-lant and solid. Common examples of inoculantsinclude the use of titanium and zirconium inaluminum welds (Ref 10) and TiN in ferriticstainless steels (Ref 11).

Fusion Welding of Single Crystals

Nickel-base superalloy single crystals areoften welded to rejuvenate worn blades or torepair miscast blades. Successful repair of thesealloys requires preservation of the single-crystalstructure during solidification of the weld poolby avoiding the CET. Constitutional supercool-ing has been supported as the CET mechanismby a wide range of experimental observationsin studies conducted on single-crystal alloysusing a variety of techniques, including autoge-nous laser surface melting (Ref 12–14), lasercladding (Ref 15), and electron beam welding(Ref 16). For example, studies on electronbeam and pulsed laser beam welds of a single-crystal nickel-base superalloy PWA 1480 byDavid et al. showed that it was difficult tomaintain the single-crystal structure duringwelding due to the formation of equiaxed straygrains. In contrast, similar experiments con-ducted on a relatively pure austenitic Fe-15Cr-15Ni stainless steel single crystal (Ref 17)demonstrated nearly perfect retention of thesingle-crystal nature in electron beam welds.The difference in these results can generallybe attributed to the solute content of the differ-ent alloys. Although chromium and nickel seg-regate to the liquid during solidification, itis well known (Ref 18) that the equilibriumpartition coefficients (k) for both chromiumand nickel are close to unity, and they do notsegregate strongly during solidification. This,

combined with the low level of other elementsin the Fe-15Cr-15Ni alloy, minimizes thebuildup of solute in the liquid during solidifica-tion. As a result, the liquidus temperaturevariation is minimized and provides a concom-itant reduction in the level of constitutionalsupercooling in the liquid. Thus, the reducedlevel of solute segregation makes it easier toavoid the CET. In contrast, commercial singlecrystals require the addition of substitutionaland interstitial alloying elements, many ofwhich partition strongly during solidification.This leads to higher concentration gradients inthe liquid and enhanced constitutionalsupercooling.Hunt (Ref 19) was the first to develop an

approximate analytical expression thatdescribed the CET. In that work, expressionsfor the nucleation rate were combined with den-drite growth undercooling equations to estab-lish the condition for the CET. The type ofgrowth that occurred was assumed to dependon the volume fraction of equiaxed grains thatformed in the undercooled region when theadvancing columnar front met the equiaxedgrains. The structure was assumed to be fullycolumnar, and thus the CET was avoided, whenthe volume fraction of equiaxed grains was lessthan 0.0066. This led to the following conditionfor avoiding the CET:

G > 0:617ð100NoÞ1=3 1� ð�TNÞ3

ð�TcÞ3" #

�Tc (Eq 23)

where G is the temperature gradient in the liq-uid, No is the total number of heterogeneoussites available for nucleation per unit volume,△TN is the associated undercooling requiredfor nucleation, and △Tc is the undercooling atthe solid/liquid interface, which depends onthe temperature gradient and growth rate. Thevalue of △Tc can be calculated using dendritegrowth undercooling models (Ref 20, 21). Thepractical difficulty in the application of Eq 23lies in the ability to determine appropriatevalues of △TN and No. Nevertheless, the modelis useful because it was shown to correctly cap-ture the observed effects of various factors onthe CET, such as growth rate, temperature gra-dient, and alloy composition.More recently, Gauman et al. (Ref 22)

extended the analysis proposed by Hunt. Thecomposition profile in the liquid was calculateddirectly using the appropriate solution of thediffusion equations for an isolated dendrite witha parabolic tip geometry. This was then used todetermine the liquidus temperature profile (Tz).The actual local temperature profile in the liq-uid (Tq,z) was considered to be controlled byheat extraction through the solid and was deter-mined through knowledge of the temperaturegradient and dendrite tip temperature as deter-mined by the Kurz, Giovanola, Trivedi (KGT)model (Ref 21). The actual undercooling(△Tz) at any location within the liquid is thengiven by:

Fig. 18 Example of equiaxed zone in the centerline ofa weld made with the gas tungsten arc

welding process on 6061 aluminum. Source: Ref 2

Table 1 Constitutional supercoolingcalculations for the aluminum-coppersystem showing the critical growth ratesrequired for breakdown of the solid/liquidinterface for various values of G and Co thatmay be encountered in casting, arc welding,and laser welding

Co,wt% Cu

G = 20 �C/mm (casting)

G = 200 �C/mm(arc welding)

G = 2000 �C/mm(laser welding)

0.5 0.0024 0.024 0.242 0.0006 0.006 0.064 0.0003 0.003 0.03

Fundamentals of Weld Solidification / 103

�Tz ¼ Tz � Tq;z (Eq 24)

Equiaxed grains will nucleate anywherein this undercooled region where the actualundercooling is more than that required fornucleation, △Tz > △TN. The critical volumefraction required for a fully equiaxed structureoriginally proposed by Hunt was used as thecritical CET value.Figure 19 shows an example of a microstruc-

ture selection map that was generated for thenickel-base single-crystal CMSX-4 using theapproach described previously (Ref 12). Thesolid continuous line in the plot represents thetransition between values of the solid/liquidinterface growth rate and temperature gradientin the liquid that lead to the CET. A G-R com-bination below this line will result in columnarsingle-crystal growth, while combinationsabove this line lead to equiaxed growth and lossof the single-crystal structure. Material para-meters required for calculation of the map weredetermined using a multicomponent thermody-namic database. The values of No and △TNwere assumed to be No = 2 � 1015/m3 and△TN = 2.5 �C. An increase in the value of No

or a decrease in the value of △TN will widenthe range where equiaxed growth occurs. Thismap is useful in a practical sense because itidentifies combinations of G and R that permitretention of the single-crystal structure duringweld repair. The use of heat-flow equationscan then be used to link R and G to the weldprocessing parameters, such as heat-sourcepower, travel speed, and preheat temperature,in order to develop process-microstructuremaps for successful weld repair.Gauman et al. (Ref 12, 15) developed a sim-

plified relationship between the temperaturegradient, growth velocity, volume fraction ofequiaxed grains (j), and nuclei density (No) as:

Gn

R¼ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4pNo

3 lnð1� jÞ3

s 1

nþ 1

" #n(Eq 25)

where a and n are material constants that aredetermined by fitting calculations of the consti-tutional tip undercooling from the KGT model

to an equation of the form △T = (aR)1/n. ForCMSX-4, these values are a = 1.25 � 106

K3.4/ms and n = 3.4 This relation is valid underhigh-temperature gradient conditions in whichthe value of No is most important forcontrolling nucleation and the value of △TNcan be ignored. Welds can be prepared undervarious values of G and R, and the resultantvolume fraction of equiaxed grains (j) can bedirectly measured on the weld cross sections.In this case, No is the only unknown in Eq 25and can thus be determined experimentally byfitting Eq 25 to the measured values of j. ForCMSX-4, this results in No = 2 � 1015/m3.When the original condition for a fully colum-nar microstructure proposed by Hunt is invoked(j = 0.0066), all values on the right side of Eq25 are known and lead to the following condi-tion for avoiding the CET:

Gn

R> K (Eq 26)

where K is a material constant that depends onNo, j, a, and n. For CMSX-4, K = 2.7 � 1024

K3.4m�4.4s. This approximate condition isshown as the dotted line in Fig. 19, and it canbe seen that this approximation is more restric-tive than the results obtained by the detailedcalculations. However, Eq 25 is useful becauseit permits straightforward coupling of R and Gto the weld processing parameters.Figure 20 shows an example of a microstruc-

ture selection map for three different welds onalloy CMSX-4 (labeled “A,” “B,” and “C”)prepared under different processing conditions.This plot shows the variation in the Gn/R ratioas a function of depth in the weld pool. Asexpected, the Gn/R ratio is highest at the fusionline (bottom of the weld) and decreases as thetop of the weld is approached. The criticalvalue for the CET of CMSX-4 is superimposedon the plot. Welds prepared under conditions inwhich the critical value of Gn/R remains belowthis critical value everywhere in the weld areexpected to retain the single-crystal structure(e.g., weld A), while welds prepared withregions less than this value (e.g., welds B andC) will undergo the CET and lose the single-

crystal structure. Experimental identification ofstray grains showed good agreement with thepredictions of Fig. 20.Figure 21 shows a process-microstructure

map that was proposed to reveal semiquantita-tive relations between the important processingparameters of heat-source travel speed (Vb),power (P), and preheat temperature (T0). Thismap was calculated using a single, integratedaverage of the Gn/R ratio to represent the varia-tion in G and R that occurs with position in themelt pool. The region of high Vb and low Prepresents very low heat-input conditions thatare insufficient to cause melting. At any travelspeed, a reduction in power is beneficial, andthis can be attributed to an increase in the tem-perature gradient. The results suggest that theeffect of heat-source travel speed depends onthe level of heat-source power. At low powers(i.e., <�550 W), the travel speed has no signif-icant effect. Apparently, at this low powerlevel, the gradient is high enough to avoidreaching the level of undercooling required fornucleation, regardless of the travel speed andresultant growth rate. At higher powers, the cal-culated results suggest that an increase in travelspeed is deleterious to preserving the single-crystal columnar zone.Experimental results generated to date gener-

ally confirm the expected influence of powerdescribed previously, but not travel speed. Forexample, Yoshihiro et al. (Ref 23) recentlyinvestigated the microstructure of welds onalloy CMSX-4 prepared over a wide range ofpowers and travel speeds using both the laserand gas tungsten arc (GTA) heat sources.A summary of their experimental results isshown in Fig. 22. They identified three typesof morphologies: single crystals with direc-tional dendrites that only grow in the [001]direction from the bottom of the weld, singlecrystals with disoriented dendrites, and weldswith stray grains. The single-crystal welds withdisoriented dendrites simply indicate the pres-ence of dendrites that grew in directions orthog-onal to the [001] direction. These resultsdemonstrate that a reduction in power andincrease in travel speed are always beneficialfor preserving the single-crystal structure

B

A

C

10–1

10–2

10–3

10–4

10–5

103 104 105

G, K/m

G3.4

RK=

R, m

/s

106 107

Equiaxeddendritic

Columnardendritic

Fig. 19 Processing map for alloy CMSX-4 showingrange of G-R conditions that produce

columnar dendritic and equiaxed dendritic growth.Adapted from Ref 12

Equiaxed

B

A

C

z, m

m

0

–0.2

–0.4

–0.6

–0.8

1020 1022 1024

G3.4/R, K3.4m–4.4s

1026

G3.4

R2.7 × 1024=

Columnar

Fig. 20 Microstructure selection map for threedifferent welds on alloy CMSX-4 (labeled

“A,” “B,” and “C”) prepared under different processingconditions. Source: Ref 12

Equiaxed(PX)

Columnar(SX)

Db = cst

T0 = 20�C (68 �F)T0 = 200 �C

(390 �F)

5001

10

Vb,

mm

/s

100

750 1000

P, W1250

No remelting

Fig. 21 Process-microstructure map for alloy CMSX-4that reveals the semiquantitative relations

between heat-source travel speed (Vb), power (P),

preheat temperature (To), and the type of dendrite

growth. Source: Ref 12

104 / Fundamentals of Fusion Welding

within the range of parameters investigated.The processing window for the GTA welds isslightly smaller than the laser welds. This isprobably associated with the higher-intensityheat source of the laser that produces a highertemperature gradient.Vitek (Ref 24) improved upon the model

developed by Gauman et al. (Ref 12) that per-mitted a more in-depth analysis of the effectof travel speed. In the early model, the Gn/Rratio was used as an indicator of stray grain for-mation, and a simple Gn/R value was calculatedat the centerline of the weld and averagedthrough the thickness. This neglects orientationeffects of the solidification front and does notprovide an accurate representation of straygrain tendency, because the fraction of straygrains does not vary linearly with Gn/R. Withthe newer approach, the fraction of stray grainswas determined directly at discrete positions inthe weld pool and used to determine an area-weighted average of stray grains as an indicatorof stray grain tendency. This improves theaccuracy by accounting for the pool shape andvariations in G and R around the pool.Figure 23 shows the calculated variation in

the weighted area fraction of stray grains inthe weld ðfcÞ as a function of welding speedfor different weld powers. The tendency forstray grain formation decreases ðfc decreases)with a decrease in power and an increase inwelding speed. Increasing travel speed is partic-ularly advantageous. The only minor exceptionto this trend is observed at the lowest powerand travel speed, where an increase in travelspeed causes a small increase in fc initiallybefore fc then decreases with increasing travelspeed. These results indicate that, within thisregime, the potential beneficial effect of theincrease in temperature gradient produced byincreasing travel speed is outweighed by thedetrimental effect of an increase in growthrate that occurs with increasing travel speed.This can be understood by noting that theformation of stray grains depends on theGn/R ratio (where n = 3.4 for CMSX-4, for

example). Thus, stray grain formation is moresensitive to G than R. The anomalous effect oftravel speed at low power has been attributed tochanges in weld pool shape. At low powers andtravel speeds, theweld pool shape is one inwhichthe area susceptible to stray grain formation is arelatively large fraction of the total weld poolarea. However, this trend is quickly diminishedwith further increases in travel speed.Factors Affecting Substructural Scale.

Dendrite spacing (l) can have an importantinfluence on the mechanical properties andtime required for postweld homogenizationtreatments and therefore deserves some con-sideration. Kurz and Fisher (Ref 25) have pro-posed a geometrical model for primarydendrite spacing that leads to a relationship ofthe form:

� / G�0:5R�0:25 (Eq 27)

which suggests that R and G have differentfunctional relationships on l. Recall that Gand R are related to the cooling rate throughEq 1. In most cases, dendrite spacing is relatedsemiempirically to the cooling rate through anequation of the form:

� ¼ Ae�n (Eq 28)

where A and n are material constants and typi-cally 0.3 � n � 0.5. Thus, dendrite spacingdecreases with increasing cooling rate. Thisconcept can be added to the G-R diagramshown previously in Fig. 16. Note that theratio of G/R controls the type of substructure,while the quantityGR (= e) controls the substruc-tural scale. As discussed previously, the coolingrate is inversely proportional to the heat input.Thus, high heat inputs lead to low cooling ratesand large dendrite spacings in the weld; this hasbeen observed experimentally in a number ofalloy systems (Ref 26–28). It should also be notedthat the cooling rate will vary throughout theweld due to changes inG andR, so that variationsin l within the fusion zone due to these changesare also expected.

Solute Redistribution duringSolidification

Binary Models. Solute redistribution is animportant topic because it controls both the dis-tribution of alloying elements across the cellu-lar/dendritic substructure and the type/amountof phases that form in the fusion zone duringsolidification. For many applications, soluteredistribution can be effectively assessed withthe aid of several simple models developed forbinary alloys. These are reviewed first, followedbymodels developed for ternary alloys. Exampleapplication of the models to multicomponentalloys is also described, followed by a discussionon the application of thermodynamic modelsdeveloped for multicomponent alloys.There are various solute redistribution mod-

els available for binary alloys that account forsuch factors as solute diffusivity in the liquidand solid, dendrite tip undercooling, and coars-ening. A good review on the subject is availablein Ref 29. For many fusion welding applica-tions, a large extent of solute redistributionbehavior can be understood with the fairly sim-ple equilibrium and nonequilibrium (Scheil)(Ref 30) models that account for the extremecases of solute redistribution:

Equilibrium lever law :

Cs ¼ kCo

ð1� kÞfL þ k(Eq 29a)

CL ¼ Co

ð1� kÞfL þ k(Eq 29b)

Nonequilibrium :

Cs ¼ kCo½1� fsk�1 (Eq 30a)

CL ¼ Co½fLk�1 (Eq 30b)

2500

2000

1500

1000

Lase

r po

wer

, W

500 Not melted Not melted

103 102 10

Laser scanning speed, mm/s

1

Single-crystallized withdirectional dendrite

Single-crystallized withdisoriented dendrite

Polycrystallized withstray crystal

Single-crystallized withdirectional dendrite

Single-crystallized withdisoriented dendrite

Polycrystallized withstray crystal

10–1 10–20

(a)

1500

1000

Wel

ding

hea

t inp

ut, W

500

102 10

Welding speed, mm/s

1 10–10

(b)

Fig. 22 Influence of heat-source power and travel speed on stray grain formation for alloy CMSX-4 for (a) laser weldsand (b) gas tungsten arc welds. Source: Ref 23

Fig. 23 Calculated variation in the weighted areafraction of stray grains in the weld as a

function of welding speed for three different weldpowers. Source: Ref 24

Fundamentals of Weld Solidification / 105

where Cs and CL are the solid and liquid com-positions at the solid/liquid interface, Co is thenominal alloy composition, fs and fL are thefraction solid and fraction liquid, and k is theequilibrium distribution coefficient, which isgiven by k = Cs/CL. These expressions assumelinear solidus and liquidus lines so that k is con-stant throughout solidification. The value of k isan important parameter because it describes theextent to which a particular element partitionsbetween the solid and liquid. For k < 1, the sol-ute partitions to the liquid, and the smallerthe value of k, the more aggressive the parti-tioning to the liquid. For elements in whichk > 1, the solute partitions to the solid duringsolidification.The equilibrium lever law assumes complete

diffusion in the liquid and solid during solidifi-cation, equilibrium at the solid/liquid interface,and no undercooling during growth. The non-equilibrium lever law (often referred to as theScheil equation) carries similar conditions,except that diffusion in the solid is assumed tobe negligible. These two cases represent theextreme conditions of residual microsegrega-tion after solidification. The equilibrium leverlaw represents the case where there are no con-centration gradients in the liquid or solid duringsolidification, and there is no residual microse-gregation in the solid after solidification. Incontrast, nonequilibrium conditions representthe most severe case of residual microsegrega-tion in the solid after solidification becausesolid diffusivity is negligible.As an example, consider a binary eutectic A-

B system, shown in Fig. 24(a), that exhibits lin-ear solidus and liquidus lines, a value of k =0.02 (for solute element B), a eutectic composi-tion of 20 wt% B, and a maximum solubilitylimit of 4 wt% B. Figures 24(b) and (c) showthe variation in liquid composition during solid-ification under equilibrium and nonequilibriumconditions for two alloys: one below the maxi-mum solid solubility with Co = 2 wt% B andone above the maximum solid solubility withCo = 5 wt% B. For the 2 wt% B alloy, the solid-ification conditions under each extreme arequite different. Under equilibrium conditions,the liquid composition never becomes enrichedto the eutectic composition because solute inthe solid is uniformly distributed and thereforecapable of dissolving all the solute before theeutectic point is reached in the liquid. Notefrom Eq 29(b) that the maximum solute enrich-ment in the liquid for the equilibrium conditionis given as Co/k, which occurs when fL = 0. Inthis case, Co/k < Ce (the eutectic composition).The resultant microstructure directly after solid-ification would simply consist of primary awith a uniform distribution of B. For the non-equilibrium case, the liquid composition willalways become enriched to the eutectic point.Thus, directly below the eutectic temperature,the 2 wt% B alloy exhibits primary a with aconcentration gradient and 0.06 weight fractionof the a/b eutectic when solidified under non-equilibrium conditions.

The two extreme solute redistribution beha-viors for the 5 wt% B alloy are compared inFig. 24(c). For the equilibrium case, the liquidcomposition will become enriched to the eutec-tic point because the nominal composition isabove the maximum solid solubility. In otherwords, Co/k > Ce. This eutectic reaction occurswhen there is 0.06 weight fraction remainingliquid. Thus, the solidification microstructuredirectly after equilibrium solidification consistsof primary a with a uniform distribution of Bat the maximum solid solubility of 4 wt% and0.06 weight fraction of the a/b eutectic. Forthe nonequilibrium case, the liquid compositionis always higher at any stage during solidifica-tion (i.e., any particular value of fL), becausethe solid does not dissolve as much solute. Asa result, more liquid remains when the eutecticcomposition is reached (0.18 weight fraction),and more of the eutectic constituent forms inthe solidification microstructure. The final alloyhere exhibits primary a with a concentrationgradient and 0.18 weight fraction of the a/beutectic (three times the weight fraction ofeutectic that formed for the equilibrium case).Figure 24(d) shows the corresponding soluteprofiles in the a solid phase after solidificationfor the 2 wt% B alloy. Under equilibrium con-ditions, there is simply primary a with auniform distribution of 2 wt% B. For nonequi-librium conditions, the primary a phase exhibitsa concentration gradient with a minimum ofkCo = 0.4 wt% B and a maximum at the solubil-ity limit of 4 wt% B. The portion of solid thatexhibits a uniform composition of 4 wt% Brepresents the eutectic a (the composition ofthe eutectic b is not shown).Equations 30(a) and (b) have the interesting

property that CL! 1 as fL! 0 and Cs! 1as fs! 1 (for k < 1), which indicates that thesolid will always be enriched to the maximumsolid solubility, while the liquid will alwaysbe enriched to the eutectic composition undernonequilibrium conditions. This can be attribu-ted to the lack of diffusion in the primary aphase, which leads to the inability of all the sol-ute to be incorporated into the primary phase.This can be understood by direct inspectionof Fig. 24(d) and noting that the dissolved sol-ute in the solid is given by the area under theCs – fs curve. For the equilibrium case, the totaldissolved solute is obviously 2 wt% B, which isthe nominal value. However, the dissolved sol-ute for the nonequilibrium case is always lessthan this due to the regions in the solid whereCs < Co. In this case, the excess solute mustbe accommodated by formation of the eutecticconstituent that contains the B-rich b phase.Strictly speaking, it is important to note that

the solute redistribution equations described inthis section do not account for undercoolingeffects that can occur at the cell/dendrite tipduring nonplanar solidification. Under high-energy-density welding processes that are oper-ated at high travel speeds, this undercoolingeffect may become significant. This subject isdiscussed in more detail in the section “Rapid

Solidification Considerations” in this article.However, this effect is typically not significantunder many moderate cooling-rate conditionstypical of arc welding and high-energy-densityprocesses operating at low heat-source travelspeeds. In these cases, the models describedfor planar solidification can be applied on alocal scale within a small volume element thatencompasses a planar interface. An example

1200

(a)

(b)

(c)

(d)

Tem

pera

ture

, K

1000

800Liquid

0

25

20 Nonequilibrium

CO = 2 wt% B

Equilibrium

Fraction liquid

15

10

Liqu

id c

ompo

sitio

n, w

t% B

5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5 10 15 20

Composition, wt% B

25 30 35 40 45 50

600

400

α

200 α + β

0

25

20 Nonequilibrium

Nonequilibrium

CO = 2 wt% B

Equilibrium

Equilibrium

Fraction liquid

15

10

Liqu

id c

ompo

sitio

n, w

t% B

5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fraction Solid, fs

Sol

id c

ompo

sitio

n, C

s

00

1

2

3

4

5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 24 Example of solute redistribution calculationsfor equilibrium and nonequilibrium

conditions. (a) Hypothetical phase diagram. (b) Variationin liquid composition for Co = 2 wt%. (c) Variation inliquid composition for Co = 5 wt%. (d) Variation in solidcomposition for Co = 2 wt%

106 / Fundamentals of Fusion Welding

of this is shown in Fig. 25 for cellular solidifi-cation. Here, the solidification process can berepresented by the enclosed region shown andnoting that solidification starts at the cell core(where fs = 0) and finishes at the cell boundarywhen two cells meet (fs = 1). The soluteredistribution then occurs locally within thegiven volume element that exhibits an essen-tially planar interface between the liquid andsolid.The only difference that separates these

two extreme cases of solute redistribution dur-ing solidification is the solute diffusivity inthe solid. Thus, it is useful to consider theextent of solute diffusivity expected for a givenset of parameters and resultant cooling rate.The Brody-Flemings model was the firstattempt at taking back-diffusion into the solidinto account during solidification and is givenas (Ref 31):

Cs ¼ kCo 1� fs1þ ak

� �k�1(Eq 31a)

CL ¼ Co 1� 1� fL1þ ak

� �k�1(Eq 31b)

a ¼ DstfL2

(Eq 31c)

The a parameter in Eq 31(c) is a dimension-less diffusion parameter, while Ds is the diffu-sivity of solute in the solid, tf is thesolidification time (cooling time between theliquidus and terminal solidus), and L is halfthe dendrite arm spacing. The Dstf term in thenumerator of Eq 31(c) essentially representsthe distance that solute can diffuse in the solidduring solidification, while the half dendritearm spacing, L, represents the length of the con-centration gradient. Thus, when Dstf << L2, thesolute is able to diffuse only a small fraction ofthe total gradient length, and solid-state diffu-sion will be insignificant. This represents thecase in which a � 0 and Eq 30 reduces to theScheil equation. Clyne and Kurz (Ref 32)pointed out that Eq 31(a) and (b) do not reduceto the lever law as a ! 1. They proposed analternate form of a that is given by:

a0 ¼ a 1� exp � 1

a

� �� �� 0:5 exp � 1

2a

� �(Eq 32)

Although this is not based on a physical model,use of a0 correctly reduces Eq 31 to the leverlaw when a = 1 and to the Scheil equationwhen a = 0. It should be noted that Kobayashiprovided a more comprehensive model of sol-ute diffusivity in the solid that can be usedwhen more detailed calculations are required(Ref 33).The potential for back-diffusion in the solid

during fusion welding can be assessed by deter-mination of the simple a parameter. The tf andL terms in Eq 31(c) will each depend on thecooling rate of the weld (e) via:

tf ¼ �Te (Eq 33)

L ¼ �2¼ Ae�n

2(Eq 34)

where △T is the solidification temperaturerange, l is the dendrite spacing, and A and nare material constants. (Equation 33 assumes alinear cooling rate through the solidificationtemperature range.) Thus, by knowing thesolidification temperature range, cooling rate,and dendrite spacing/cooling rate relationship,the a parameter can be directly estimated as afunction of cooling rate and the potential influ-ence of solid-state diffusion can be determined.Many studies have used these simple equa-

tions to explore the solidification behavior ofwelds in austenitic alloys. These studies havedemonstrated that the solute redistributionbehavior and resultant weld-metal microstruc-ture are controlled primarily by the pertinentvalues of k and Ds for the alloying elementsof interest. The value of Ds will control thepotential for back-diffusion in the solid duringsolidification. Elements with very low k valuescan produce steep concentration gradientsacross the cellular or dendritic substructure ofthe weld. Table 2 shows an example that sum-marizes diffusivity data for many alloying ele-ments in nickel (Ref 34). Upper bound valuesof a for each element can be calculated bydetermining Ds at the liquidus temperature,approximately 1350 �C (2460 �F) for manynickel alloys (Ref 35, 36), using a large but rep-resentative value of △T of 200 �C (360 �F) andreported values of A = 32 and n = 0.31 (Ref 37).Using these values, a can be calculated asa function of cooling rate and is shown inFig. 26 for the substitutional alloying elementslisted in Table 2. The values of A and n usedhere were for alloy 713. However, the valuesdo not vary significantly among many nickel-base alloys (Ref 38). The a value decreaseswith increasing cooling rate due to the decreasein solidification time (tf) with increasing cool-ing rate. More importantly, note that a << 1for all the elements considered under a widerange of cooling rates. The use of variousvalues of △T, A, and n representative of othernickel-base alloys does not change this resultsignificantly. It is also important to note thatan upper bound value of Ds was calculated ata typical liquidus temperature of 1350 �C(2460 �F). The value of a can only decreasewith the use of Ds values calculated at lowersolidification temperatures.These results clearly demonstrate that solid-

state diffusion of substitutional alloying ele-ments in austenitic alloys is insignificant duringsolidification of fusion welds. Experimentalevidence has been published that supports thesecalculations for a variety of elements, includingiron, chromium, niobium, molybdenum, andsilicon (Ref 39, 40). In that work, dendrite corecompositions were experimentally measured in

samples that were solidified under a wide rangeof cooling rates, including samples quenchedduring solidification and samples cooled at0.2 �C/s and 650 �C/s. The dendrite core com-positions were identical in all cases, indicatingthat solid-state diffusion is indeed negligible.In these cases, the simple form of Eq 30 canbe used for quantitative calculations of finalsolute profiles and phase fractions. An exampleis provided later in this section.Back-diffusion during weld solidification can

be significant in body-centered cubic systemsand for the diffusion of interstitial alloying

Fig. 25 Volume element for modeling soluteredistribution during cellular solidification.

The solidification process can be represented by theenclosed region shown and noting that solidificationstarts at the cell core (where fs = 0) and finishes at thecell boundary when two cells meet (fs = 1).

Table 2 Diffusion data for various alloyingelements in nickel

ElementDiffusion coefficient

(Do), m2/s

Activation energy (Q),kJ/mol

Fe 8.0 � 10�5 255Cr 1.1 � 10�4 272Co 1.4 � 10�4 275Nb 7.5 � 10�5 264Mo 1.0 � 10�4 275W 2.0 � 10�4 299Al 1.9 � 10�4 268Ti 4.1 � 10�4 275Cu 5.7 � 10�5 258C 8.0 � 10�6 135

Fundamentals of Weld Solidification / 107

elements. For example, calculation of the aparameter for carbon in nickel will yield valuesthat are significantly greater than unity. This isto be expected, because carbon diffuses by aninterstitial mechanism and therefore exhibitsdiffusion rates that are orders of magnitudehigher than the substitutional alloying elements.Evidence for this is reflected in the activationenergy term (Q) for diffusion of carbon innickel shown in Table 2. Note that Q for thesubstitutional alloying elements varies over afairly narrow range of 255 to 299 kJ/mol, whilethe value for carbon is approximately half thisat 135 kJ/mol.Figure 27(a) shows calculated results for sol-

ute redistribution of carbon in nickel (Ref 39).In this figure, the solute redistribution behaviorwas calculated with Eq 31 and 32 using thetemperature-dependent diffusion rate of carbonin nickel. Comparison is made between thenonequilibrium Scheil equation and the leverlaw. Note that the detailed results from theClyne-Kurz model are essentially identical tothat of the lever law, indicating that completesolid-state diffusion of carbon is expected innickel-base alloys during solidification. A simi-lar effect can be expected for nitrogen in nickel.This calculation was conducted for a coolingrate of 650 �C/s (1170 �F/s) through the solidi-fication temperature range. Higher cooling ratestypical of high-energy-density welding mayalter this result and begin to limit carbon

diffusion in the solid. Aside from this possibil-ity, these results demonstrate that carbon (andnitrogen) can be expected to exhibit completediffusion in the solid during most welding con-ditions. Figure 27(b) shows a similar calcula-tion for substitutional diffusion of titanium ina body-centered cubic Fe-10Al-5Cr alloy thatwas calculated using the detailed model pro-posed by Kobayashi (Ref 33, 41). These resultsare somewhat similar to those for carbon innickel in that solid-state diffusion is essentiallycomplete at moderate cooling rates. However,diffusion may become insignificant at very highcooling rates typical of high-energy-densityprocesses. These results are meant to serve asexample calculations that can be applied tounderstanding solute redistribution behavior infusion welds of any alloy system when the per-tinent alloy parameters are known.For conditions in which solid-state diffusion

is negligible, microsegregation will persist inthe as-solidified weld (except for the case inwhich k = 1). An example of this is shown inFig. 28 for fusion welds in a niobium-bearingsuperalloy (Ref 36, 42). The final degree ofmicrosegregation can be assessed by directdetermination of the k value for the element ofinterest, where the degree of microsegregationwill increase with decreasing k value (for kvalues < 1). For example, the lowest concentra-tion will occur at the dendrite core where solid-ification initiates. Direct inspection of Eq 30

indicates that, at the start of solidification whenfs = 0, the dendrite core composition (which isthe first solid to form) is given by kCo.Although the models described earlier were

strictly developed for binary alloys, they canbe used in a quantitative manner for multicom-ponent engineering alloys when the alloy prop-erly mimics the solidification behavior of abinary system. An example is provided hereby application of the simple binary Scheil equa-tion to Ni-Cr-Mo-Gd alloys. A typical as-solidi-fied microstructure of an alloy is shown inFig. 29(a). These alloys exhibit g dendritesand an interdendritic g/Ni5Gd eutectic-typeconstituent. Research (Ref 43, 44) has shownthat gadolinium controls the solidificationbehavior of these alloys. In particular, thesolidification temperature range and amountof terminal eutectic-type constituent thatforms at the end of solidification are essentiallydominated by the gadolinium concentration.Solidification of these alloys initiates at theliquidus temperature by the formation of pri-mary g-austenite. As solidification proceeds,the liquid becomes increasingly enrichedin gadolinium until the liquid ! g + Ni5Gdeutectic-type reaction is reached, at which pointsolidification is terminated.This reaction sequence and temperature

range is similar to that expected in the binarynickel-gadolinium system. Simple binary

Fig. 26 Dimensionless a parameter as a function of cooling rate for a wide range of alloying elements in nickel

0.200

(a)

(b)

0.150

0.100

Clyne-Kurz model

Scheil equation

Lever law

Fraction solid, fs

0.050

0.0000.0

3.0

2.0

1.0

Ti c

once

ntra

tion

in s

olid

, wt%

0.00.0 0.2 0.4

Fraction solid, fs

0.6 0.8 1.0

0.2

30,000 �C/s

Scheil

Lever law

30 �C/s0.1 �C/s

Sol

id c

ompo

sitio

n, w

t% C

0.4 0.6 0.8 1.0

Fig. 27 Comparison of solute redistribution behavior.(a) Carbon in a nickel-base superalloy

calculated using the lever law, Scheil equation, andClyne-Kurz model. (b) Titanium in an Fe-10Al-5Cr-1.5Ti-0.4C alloy with varying cooling rates (calculated usingthe Kobayashi model) compared to the lever law andScheil cases. Source: Ref 39

108 / Fundamentals of Fusion Welding

nickel-gadolinium alloys with less than approx-imately 13 wt% Gd exhibit a similar two-stepsolidification sequence consisting of primaryaustenite formation followed by a terminaleutectic reaction involving the Ni17Gd2 inter-metallic at 1275 �C (2327 �F). By comparison,the multicomponent Ni-Cr-Mo-Gd alloys com-plete solidification at �1258 �C (�2295 �F)by a terminal eutectic-type reaction involvingthe Ni5Gd intermetallic. Thus, although the sec-ondary phase within the terminal eutectic con-stituent is different in each case, the terminalreaction temperatures are very similar. Asshown in Fig. 29(b), a pseudo-binary solidifica-tion diagram can be developed for this systemthat is similar to the phase diagram of a binaryeutectic alloy. In this case, the solvent is repre-sented by the Ni-Cr-Mo solid-solution g-austen-ite phase, and gadolinium is treated as thesolute element. The similarity of this g-gadolin-ium binary system to a binary eutectic system isreadily evident in several ways:

� The as-solidified microstructure consists ofprimary g dendrites surrounded by an inter-dendritic eutectic-type constituent in whichthe secondary phase in the eutectic is soluterich.

� The amount of eutectic-type constituentincreases with increasing solute content.

� The proportional amount of each phasewithin the eutectic constituent is relativelyinsensitive to nominal solute content.

It was also observed that the eutectic tempera-ture is not strongly dependent on the nominalgadolinium concentration. Key points of the dia-gram shown in Fig. 29(b) were determined witha combination of thermal analysis and quantita-tive microstructural characterization techniques.Figure 29(c) shows a comparison of themeasured and calculated g/Ni5Gd fractioneutectic from fusion welds made on alloys withvarious gadolinium concentrations. In this plot,a comparison is made with the calculated

eutectic fraction using the simple Scheil equa-tion. The Scheil equation can be used for eutec-tic fraction calculations by noting that, whenCL = Ce (the eutectic composition), the remain-ing fraction liquid (fL) transforms to fractioneutectic (fe), so that:

fe ¼ Ce

Co

� �1=k�1(Eq 35)

Good agreement is observed betweenthe measured and calculated values. This sup-ports the use of a pseudo-binary analog formodeling the solidification behavior of thesealloys.Ternary Models. The expressions derived

earlier for binary alloys can be used quantita-tively in multicomponent alloys when thealloy behaves like a binary system. However,many engineering alloys exhibit multiple reac-tions during solidification that occur over arange of temperatures and exhibit more thanone eutectic constituent. Thus, they typicallycannot be treated in a quantitative fashionwith the simpler models described earlier.In this case, models for ternary alloys can beuseful, and relatively simple solidificationpath equations can be derived for limiting casesof solute redistribution in ternary alloys. Twosets of solute redistribution equations areneeded to fully describe the solidification pathsof ternary alloys. The first set describes the var-iation in liquid composition and fractionliquid during the primary stage of solidification.These expressions can be used to identifyif the liquid composition is enriched to a mono-variant eutectic-type reaction and, if so,what type of reaction will occur and the frac-tion of total eutectic constituent that willform in the microstructure. The second set ofexpressions describes the variation in liquidcomposition and fraction liquid during themonovariant eutectic reaction. These expres-sions can be used to determine if the liquidcomposition is enriched to the ternary eutecticreaction and the fractions of both the monovar-iant and ternary eutectic constituents. Only theprimary solidification path expressions are dis-cussed here. More detailed information on thefull ternary model can be found elsewhere(Ref 45).For ternary solidification, three limiting cases

can be identified based on the diffusivity of sol-ute in the solid phases:

� Negligible diffusion of each solute in thesolid phases, referred to as nonequilibriumsolidification

� Negligible diffusion of one solute in thesolid phases and infinitely fast diffusion ofthe other solute in the solid phases, referredto here as intermediate equilibrium.

� Infinite diffusion of each solute in the solidphases (equilibrium)

Expressions for the primary solidification pathsfor these three conditions are given by:

20(a)

(b)

15

10

5

0 10 20

Distance, μm

Dendritecore

Interdendriticregion

Nb

Si

30 40 50

Com

posi

tion,

wt%

0

Fig. 28 Example of microsegregation in a weld of aniobium-bearing nickel-base superalloy. (a)

Micrograph showing position of composition trace. (b)Corresponding electron probe microanalysis resultsshowing niobium microsegregation. Source: Ref 36, 42

Fig. 29 (a) Microstructure of fusion weld on a Ni-Cr-Mo-Gd alloy. (b) Pseudo-binary phase

diagram for the g-gadolinium system. (c) Comparison ofthe measured and calculated g/Ni5Gd fraction eutecticfrom fusion welds made on alloys with variousgadolinium concentrations. Source: Ref 44

Fundamentals of Weld Solidification / 109

Equilibrium:

CLA ¼ CoA

1�kaA1�kaB

� �CoB�kaBCLB

CLB

� �þ kaA

(Eq 36a)

Intermediate equilibrium:

CLA ¼ CoA

CoB � kaBCLB

ð1� kaBÞCLB

� �kaA�1(Eq 36b)

Nonequilibrium: CLA ¼ CoA

CLB

CoB

� �kaA�1kaB�1

(Eq 36c)

where fL is the fraction liquid, Coj is the nomi-nal concentration of element j, CLj is the con-centration of element j in the liquid, and kij isthe equilibrium distribution coefficient for ele-ment j in phase i. When the liquid compositiongiven by Eq 36 intersects a monovariant eutec-tic line, the remaining fraction liquid transformsto a binary-type eutectic and possibly the ter-nary eutectic. Thus, the value of fL at the inter-section point of the primary solidification pathand monovariant eutectic line defines the totalamount of eutectic in the microstructure.Details on the calculation procedure for deter-mining this value are provided elsewhere (Ref45). The difference among the three cases con-sidered here is governed by the diffusivity ofsolutes in the solid phases. Thus, the conditionthat most closely describes the solidificationbehavior for an actual application can be deter-mined by calculation of the a parameter foreach of the solute elements of importance.Figure 30 shows an example of primary

solidification path calculations made for fusionwelds on multicomponent superalloys that formthe g/NbC and g/Laves eutectic-type constitu-ents at the end of solidification (Ref 46).Although these alloys contain multiple ele-ments, they can be treated as a pseudo-ternaryg-Nb-C system. Niobium and carbon are treatedas the important solute elements here becauseeach element partitions aggressively to the liq-uid during solidification and leads to the forma-tion of the niobium- and carbon-rich NbC phaseand the niobium-rich Laves phase. For this sys-tem, niobium exhibits negligible diffusion,while carbon diffuses infinitely fast. Thus,Eq 36(b) was used for these calculations. Cal-culations are shown for alloys with similar nio-bium concentrations and various amounts ofcarbon. Note that the addition of carbon pushesthe primary solidification path up into the car-bon-rich side of the liquidus projection. The liq-uid composition must then “travel” a longdistance down the g/NbC eutectic line as theg/NbC constituent forms. This accounts for theobserved influence of carbon that leads to largeamounts of the g/NbC constituent in thesealloys. It was also observed that carbon addi-tions increase the start temperature of the L !g + NbC reaction. This effect is not intuitive,because solute additions in which k < 1 (such

as carbon) typically lower reaction tempera-tures. The calculations demonstrate the reasonfor this effect. The dotted arrow near the g/NbC eutectic line represents the direction ofdecreasing temperature (as determined throughthermal analysis). Note that carbon additionsdrive the solidification path to the carbon-richregion of the diagram where the L ! g +NbC reaction is relatively high, thus accountingfor the observed effect. Reasonable agreementwas obtained between the measured and calcu-lated volume fractions using these expressions(Ref 46).Modeling of Multicomponent Alloys. The

relatively simple models described previouslyare useful for assessing the solidifcation behav-ior of alloys that behave in a manner analogousto binary or ternary alloys. However, suchapproaches may be limited in some engineeringalloys. Multicomponent thermodynamic andkinetic software is now available that can alsobe used for understanding solidifcation behav-ior in such systems, and a simple example isprovided here. Figure 31(a) shows a light opti-cal micrograph of a weld between a superauste-nitic stainless steel (CN3MN) with �6 wt% Moand a nickel-base filler metal (IN686) with �14wt% Mo (Ref 47). This weld was prepared atthe 21% dilution level. Welds in these alloyssolidify with a primary L ! g solidifcationmode and then terminate solidifcation by a L! g + s reaction. Thus, the final weld micro-structure consists of primary g dendrites andinterdendrtic g/s eutectic. The s phase is amolybdenum-rich phase. Welds made at lowdilution levels (i.e., high molybdenum levels)would be expected to form relatively large frac-tions of the s phase. However, as shown inFig. 31(b), the s-phase content does not changesignificantly with weld dilution level and asso-ciated nominal molybdenum concentration ofthe alloy. This immediately suggests that otherfactors besides the nominal concentration ofmolybdenum are also affecting the s-phasecontent.

The s-phase content is controlled by theamount of eutectic constituent that forms dur-ing the end of solidification. The fraction ofeutectic is, in turn, controlled by the nominalalloy composition (Co), eutectic composition(Ce), and distribution coefficient (k) for molyb-denum (Eq 35). Note that the nickel and ironcontents in the weld change appreciably withchanges in dilution, and these changes in nickeland iron may affect the values of k and Ce.Figure 32(a) shows the variation in liquid com-position during solidification using a multicom-ponent Scheil simulation with Thermocalc (Ref48). The Scheil simulation is justified herebecause, as shown in Fig. 26, all the substitu-tional elements of interest are known to exhibitinsignificant diffusion rates during solidificationin austenite. Results are shown for welds at var-ious dilution levels. Solidification starts at thenominal composition (Co), which is controlledby the dilution level. The eutectic composition(Ce) is given by the inflection point where theeutectic reaction L! g + s begins. It is appar-ent that the eutectic composition decreasesappreciably with increasing dilution. Figure 32(b) shows the variation in the molybdenum par-tition coefficient (k) for the same dilution levelsof interest. The partition coefficient for a givendilution does not vary significantly duringsolidification, but it increases slightly (from

Fig. 30 Example of primary solidification pathcalculations made for fusion welds

on multicomponent superalloys that form the g/NbC andg/Laves eutectic-type constituents at the end ofsolidification. Source: Ref 46

Dendrite CoreInterdendritic Region

20 μm

5

(a)

(b)

4.5

4

3.5

3

2.5

2

1.5

1

0.5

00 20 40 60

Dilution, %

Vol

ume

frac

tion

of s

igm

a ph

ase,

%

80 100

Fig. 31 (a) Light optical micrograph showing primaryaustenite and interdendritic s phase that forms

in a dissimilar weld between a superaustenitic stainlesssteel base metal and nickel-base filler metal. (b) Variation ins-phase content with dilution. Source: Ref 47

110 / Fundamentals of Fusion Welding

�0.67 to 0.84) with decreasing dilution.Figure 32(c) summarizes the variations in Co,Ce, and k as a function of dilution. An increasein Co and a decrease in Ce and k will increasethe fraction eutectic and resultant amount ofsecondary phase. These results demonstratethere are offsetting effects in these parametersthat keep the s-phase content relatively inde-pendent of the weld-metal composition. Inother words, Ce and k each decrease withincreasing dilution, which would, in itself, leadto an increase in fraction eutectic and amountof s phase. However, these changes are offsetby the decrease in nominal molybdenum con-centration that occurs with increasing dilution.This accounts for the relatively constant s-phase content observed in these welds.The final s-phase content will also depend

on the maximum solid solubility of molybde-num in the austenite and s phases. All ofthese factors can be accounted for with a com-plete multicomponent Scheil simulation. Table 3shows the amount of s phase calculated for allthe dilution levels of interest. There is reason-able agreement between the measured and cal-culated amounts of s phase. These resultsalso carry important practical implications,because IN686 filler metal can be used at vari-ous dilution levels with minimal changes tothe s-phase content in the weld, which is bene-ficial from a solidification cracking and tough-ness standpoint.

Rapid Solidification Considerations

The conditions described previously ignorethe effects of the nonplanar interface and solutebuildup in the liquid on dendrite tip undercool-ing. Such factors can become important at highsolidification rates associated with high-energy-density welding processes. These undercoolingeffects are shown schematically in Fig. 33.The liquidus and solidus lines for a planar inter-face (tip radius, r = 1) are shown by the solidlines. Solute enrichment in the liquid at thesolid/liquid interface results from solute rejec-tion directly from the tip and diffusion downthe solute gradient that exists between the cells.This local enrichment produces a reduction inthe tip temperature, △TC, as shown in Fig. 33.Undercooling is also produced by the surfaceenergy effects due to the tip with a finite radius.In this case, the solidus and liquidus linesare depressed relative to those for an infinite

30

(a)

Liq

uid

co

mp

osi

tio

n, w

t% M

o

25

20

1.0 0.9 0.8

Fe, Ni, Cr, Mo, C, Mn, P, Si, Cu, N

100% Dilution

83% Dilution

0% Dilution

21% Dilution37% Dilution52% Dilution

0.7 0.6

Fraction liquid

0.5 0.4 0.3 0.2 0.1 0.0

15

10

5

0

1.00

(b)

k va

lue

0.90

0.80

0.00 0.10 0.20

Fe, Ni, Cr, Mo, C, Mn, P, Si, Cu, N

0.30 0.40

Fraction solid

0.50 0.60 0.70 0.80 0.90 1.00

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

0% Dilution21% Dilution

100% Dilution

83% Dilution

52% Dilution37% Dilution

1.0

(c)

Par

titi

on

co

effi

cien

t, kM

o

Co

mp

osi

tio

n, w

t%

0.8

0 20 40

Dilution, %

60 80 100

0.6

0.4

0.2

0.0

30

20

kMo

CoMo

CeMo

10

0

Fig. 32 Multicomponent Scheil calculations for fusion welds made between a superaustenitic stainless steel alloyand a nickel-base filler metal showing (a) variation in liquid composition with fraction liquid, (b) variation

in kMo with fraction solid, and (c) variation in Co, Ce, and kMo with dilution

Table 3 Comparison of the percent svalues calculated using ThermoCalc andmeasured experimentally

Dilution, % Calculated s phase, % Measured s phase, %

100 1.9 1.4 þ� 0.283 2.4 1.7 þ� 0.352 2.3 1.6 þ� 0.237 1.9 2.0 þ� 0.421 1.4 2.1 þ� 0.4

Fundamentals of Weld Solidification / 111

radius, as shown by the dotted lines. Thecorresponding undercooling associated with afinite tip radius is given by the △Tr term inFig. 33. As a result of these two effects, the liq-uid temperature and composition can bechanged significantly from (To, Co) to (Tt, Ct),as shown in the figure. Also note that this pro-duces an increase in the dendrite core composi-tion from kCo to kCt. This change in theoperating point of the dendrite tip can have asignificant effect on the stability of the primarysolidification phase, the final distribution of sol-ute in the solid, and the final phase distribution.Different approaches have been used to solve

the dendrite undercooling problem in order todetermine the local tip temperature and compo-sition. For example, Burden and Hunt (Ref 49)assumed that the dendrite tips grow with aradius that minimizes undercooling. Thisassumption was used to solve directly for thetip radius, which then provided a direct solutionfor the undercooling and resultant tip composi-tion. However, this assumption does not capturethe observed behavior in which a dendriticinterface eventually reverts to a cellular andthen planar interface at high solidification velo-cities. Note also that this reversion to a planarinterface is not captured within the classicalconstitutional supercooling criteria (describedpreviously), because the model neglects surfacetension effects. At high solidification velocities,the cell spacing is decreased, which leads to anincrease in the solid/liquid interfacial area.The increase in surface area helps accommo-date the rapid rate of solute rejection requiredat high solidification velocities but also leadsto increased surface energy. At very high velo-cities, the cellular interface is no longer ener-getically favorable, and a planar interfacereappears.The dendrite tip radius and solute redistribu-

tion around the tip are actually coupled, so thatthe assumption of a cell tip growing to mini-mize undercooling is not completely accurate.This condition has been considered in greaterdetail by Kurz et al. (Ref 21). In the KGTmodel, the tip radius is related to the tempera-ture gradient and growth rate through the Pecletnumber (Pe) by:

R2 p2�Pe2D2

l

� �þR

mlCoð1� kÞDl 1� ð1� kÞIðPeÞ½ � �

þG ¼ 0 (Eq 37)

where G is the Gibbs-Thomson parameter (ratioof specific solid/liquid interfacial energy tomelting entropy), Pe is the Peclet number givenby Pe = Rr/2Dl, and I(Pe) is given by I(Pe) =Peexp(Pe)E1(Pe), where E1 is the exponentialintegral function.Direct solution of Eq 37 for the dendrite tip

radius is not possible because Pe is defined interms of both R and r. Thus, the expressionmust be solved numerically in which a widerange of Pe values are selected to determineI(Pe), and then the quadratic Eq 37 is solvedfor R numerically. Once R is known, the den-drite radius is given by r = 2PeDl/R. The

resultant tip temperature and composition arethen given by:

C�l ¼Co

1� ð1� kÞIðPeÞ (Eq 38)

Tt ¼ To þ mlC�l �

2�

r(Eq 39)

As shown schematically in Fig. 33, this new“operating point” of the dendrite tip (i.e., thechange in tip composition and temperature from(To, Co) to (Tt, Ct)) also affects the subsequentsolute redistribution because the new core com-position is changed from kCo to kCt. Sarreal etal. (Ref 50) addressed this problem by assumingthat undercooling at the dendrite tip is dissipatedby forming a certain fraction of primary solidphase, f os , that is given by the lever law at theundercooled temperature:

f os ¼Co � C�lC�l ðk � 1Þ (Eq 40)

Under conditions in which solute diffusion inthe solid is negligible (which is often the casefor substitutional alloying elements at highsolidification velocities), the final fraction ofeutectic that forms is then given by the modi-fied form of the Scheil equation:

fe ¼ ð1� f os ÞCe

C�l

� � 1k�1

(Eq 41)

Various levels of complexity can beaccounted for with the KGT model. For exam-ple, Dl and ml can each vary with temperature,while k can vary with both temperature andsolidification velocity. At high solidificationvelocities, the value of k ! 1 due to solutetrapping in the solid. This velocity dependencecan be accounted for with the model proposedby Aziz (Ref 51) in which the effective distri-bution coefficient, k0, is given by:

k0 ¼ k þ aiRD

1þ aiRD

(Eq 42)

where ai is related to the interatomic distanceand is typically between 0.5 and 5 nm. Equation42 has the form in which k0= k at low values ofR and k0 = 1 at high R values.Figure 34 shows example calculations of

the KGT model for an Ag-5wt%Cu alloy.The results demonstrate how the tip radius(denoted as R in Fig. 34a), temperature, andcore composition (Cs* in Fig. 34c) change withincreasing solidification velocity. The curves inFig. 34(a) and (c) demonstrate how variousassumptions about Dl, ml, and k affect theresults. Note that the dendrite radius initiallydecreases with increasing velocity but then rap-idly increases and becomes infinite at highgrowth velocities. This is the condition inwhich a planar interface is restabilized. The

Composition

Tem

pera

ture

Solidus for r = rt

Solidus for r = ∞

Liquidus for r = ∞

Liquidus for r = rt

kCo kCt Ct

ΔTc

Tt

To

ΔTr

Co

Fig. 33 Schematic illustration showing dendrite tipundercooling

10–3

10–2

(a)

10–4

10–5

10–6

10–7

10–2 10–1 1 10

V, cm/s

Ag – 5 wt% Cu

G = 105 K/cm

D, k

o, m

= c

onst

.

D (

T )

k o, m

= c

onst

.

D (

T )

, k (

V )

k o, m

= c

onst

.

D (T ),

k (V,T ),

m (T )

D (T ),

k (V ) a

nd k o,

m = const. D (T

) an

d k o

, m =

con

st.

D, ko, m

= c

onst

.

R, c

m

102 103

1200

(b)

1180

1160

1140

112010–2 10–1 1 10

V, cm/s

Ag – 5 wt% Cu

G = 105 K/cm

Tt,

K

Tt

Tt

Tt – ΔTt

102 103

5

(c)

4

3

2

110–2 10–1 1 10

V, cm/s

Ag – 5 wt% Cu

G = 105 K/cm

C* s

, wt %

102 103

Fig. 34 Results from Kurz, Giovanola, Trivedi (KGT)model calculations showing variation in (a)

dendrite radius, (b) dendrite tip temperature, and (c) cellcore composition as a function of growth rate. Source:Ref 21

112 / Fundamentals of Fusion Welding

extent of undercooling at high velocities can bequite significant and can lead to subsequentenrichment in the cell core composition relativeto that under ideal Scheil conditions (Fig. 34c).This effect provides one mechanism for reduc-ing the extent of residual microsegregation thatoccurs in many alloys that exhibit low solutediffusivity in the solid.Stainless steels represent an important

group of engineering alloys in which dendritetip undercooling can have a significant in-fluence on the primary phase that forms. Thecomposition of many commercial stainlesssteels is adjusted to induce primary delta-ferrite(d) solidification. The delta-ferrite phase isdesirable as the first solidification phasebecause it exhibits a higher solubility for trampelements, such as phosphorus and sulfur. Thus,with primary ferrite solidification, more of thephosphorus and sulfur are kept in solution andavoid the low-melting-point phosphorus- andsulfur-rich phases that are known to aggravatesolidification cracking. In contrast, the solubil-ity for these tramp elements is relatively lowin austenite (g), and stainless steels that solidifyas primary austenite are well known to be rela-tively sensitive to this form of cracking.Although some stainless steels have a nominalalloy composition that is designed to solidifyas ferrite (under low solidification velocities),the nominal composition can lie close to thetransition in primary solidification mode so thatdendrite tip undercooling can cause a shift inthe primary solidification mode at higher cool-ing rates.

Fukumoto and Kurz (Ref 52) used a multi-component form of the KGT model to predicthow solidification velocity and resultant tipundercooling affects primary phase stability instainless steels. In this work, interface responsefunctions were developed by calculating the tiptemperature of the d and g phases as a functionof solidification velocity. An example calcula-tion for an Fe-18Cr-11.3Ni (wt%) alloy solidi-fied with a temperature gradient of 400 K/mmis shown in Fig. 35. The primary solidificationphase is identified by noting that the phase withthe highest tip temperature is the one that willsolidify first. Note that, for this alloy and tem-perature gradient, a shift in the primary solidifi-cation mode is expected at a solidification rateof 3.2 � 10-2 m/s, and such shifts have beenobserved experimentally.

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1720

1725

1730

1715

1710

1705

1700

Velocity, m/s

Fe-18.0%Cr-11.3%NiG = 400 K/mm

Tem

pera

ture

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10–110–210–310–410–5 1

Ts(δ)

Ts(γ)γ

γγ

δ

δ

TL(δ)

TL(γ)

Fig. 35 Phase stability calculations for an Fe-18Cr-11.3Ni alloy solidified with a temperature gradient of 400 K/mm(290 �F/in.). The primary solidification phase is identified by noting that the phase with the highest tip

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114 / Fundamentals of Fusion Welding

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