Revisiting Jackson-Hunt calculations: Unified theoretical analysis for generic multi-phase growth in a multi-component system
RRevisiting Jackson-Hunt calculations: Unified theoretical analysis for genericmulti-phase growth in a multi-component system
Arka Lahiri a, ∗ , Abhik Choudhury a a Department of Materials Engineering, Indian Institute of Science, Bangalore - 560 012, India.
Abstract
A straight-forward extension of the Jackson-Hunt theory for directionally solidifying multi-phase growth where thenumber of components exceeds the number of solid phases becomes difficult on account of the absence of the requirednumber of equations to determine the boundary layer compositions ahead of the interface. In this paper, we thereforerevisit the Jackson-Hunt(JH) type calculations for any given situation of multi-phase growth in a multi-component systemand self-consistently derive the variations of the compositions of the solid phases as well as their volume fractions, whichgrow such that the composite solid-liquid interface is isothermal. This allows us to unify the (JH) calculation schemesfor both in-variant as well as multi-variant eutectic reactions. The derived analytical expressions are then utilized tostudy the effect of dissimilar solute diffusivities and interfacial energies on the undercoolings and the solidified fractions.We also perform phase field simulations to confirm our theoretical predictions and find a good agreement between ouranalytical calculations and model predictions for model symmetric alloys as well as for a particular Ni-Al-Zr alloy.
Keywords:
Phase-field; Jackson-Hunt; multi-component; multi-phase; multi-variant; in-variant; eutectic
1. Introduction
Eutectic solidification in a generic multi-component al-loy, where two or more solids exhibit coupled growth, canbe associated with degrees of freedom greater than or equalto zero. Experimentally, invariant (zero degrees of free-dom) eutectic reactions have been observed in [1, 2, 3, 4,5, 6, 7, 8, 9, 10, 11, 12, 13] Our theoretical understandingof invariant eutectic reactions is fairly advanced for bi-nary [14, 15, 16, 17] as well as for ternary systems [18, 19].In these studies, the solid phase fractions are assumed tobe the ones predicted by the equilibrium phase diagram atthe invariant temperature. This allows the determinationof the magnitude of the composition boundary layers at ∗ Corresponding author
Email addresses: [email protected] (Arka Lahiri), [email protected] (AbhikChoudhury) the eutectic front from the conditions of equality of under-coolings at different solid-liquid interfaces.In contrast to binary alloys, multi-component alloy sys-tems can display eutectic reactions which are not invariant(possessing degrees of freedom greater than zero). An ex-ample of such a reaction is the concurrent solidificationof two phases in ternary alloys, which possesses a singledegree of freedom, and is known as the monovariant (orunivariant) eutectic. This reaction exists over a range oftemperatures in the equilibrium phase diagram comparedto its invariant counterpart in binary. Experimental stud-ies in several multi-component systems report the exis-tence of such multivariant reactions [20, 21, 4, 22, 23, 24,25]. Multivariant eutectics are also susceptible to Mullins-Sekerka [26] like destabilization of the solidification frontin the presence of one or more impurity components lead-ing to the formation of eutectic cells or colonies as seen
Preprint submitted to Physica D October 8, 2018 a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec n [27, 28]. Eutectic cells have been studied theoreticallyand numerically in [29] and [30], respectively.The theoretical development for multi-variant eutecticreactions have mainly been centered around ternary mono-variant eutectics. An attempt to extend the JH-type calcu-lations to explain monovariant eutectics leads to an under-determined system where the number of unknowns (fordescribing the magnitude of the solute boundary layers)exceed the equations (the equality of interfacial undercool-ings at different solid-liquid interfaces). This marks a de-parture from the theories of invariant eutectics [17, 18, 19],where there are enough equations as unknowns to renderthe system consistent.McCartney et al. [31] are the first to circumvent thisdifficulty by introducing an additional constraint relat-ing the magnitude of the composition boundary layers ofthe two independent components assuming the solid phasefractions to be given by the equilibrium phase diagram.The under-determined nature of the problem of ternarymonovariant eutectics urge a reduction in the number ofthe unknowns in the problem by expressing them as func-tions of the solid phase volume fractions. Thus, the solidphase fractions are no longer determined by the equilib-rium phase diagram but by the growth dynamics. Thedynamic selection of solid phase fractions during growthbeing an experimental fact valid for invariant and non-invariant eutectics alike, prompted several theoretical stud-ies which attempt to understand eutectic solidification dy-namics as functions of solid phase fractions. Donaghey andTiller [32] calculate the composition fields in the liquid asfunctions of the solid phase fractions for binary as well asternary systems, which are then utilized by Ludwig andLeibbrandt [33] to demonstrate the dependence of interfa-cial undercoolings on solid phase fractions, but for binarysystems only. Magnin and Trivedi [34] consider differentdensities of the eutectic solid phases for a binary eutectic toderive the curvature at each point of the eutectic solidifica-tion front required to maintain an isothermal solid-liquid interface. These curvatures when averaged over the twoeutectic solids, lead to contact angles different from whatis predicted by the criterion of mechanical equilibrium atthe triple points, constituting a driving force for a dynamicselection of volume fractions of solid phases different fromwhat is predicted by the equilibrium phase diagram. Forternary monovariant eutectics, De Wilde et al. [35] treata particular solid phase fraction as a material parameterwhich is then varied to obtain the corresponding varia-tion in the growth dynamics at the extremum condition ofminimum undercooling of the eutectic front.A couple of recent theoretical studies have a gone a stepfurther by presenting a method to compute the solid phasefractions as a key step to determine non-invariant eutec-tic growth involving two solid phases in a generic multi-component alloy. The study by Catalina et al. [36] presentsa linearized theory in this regard without allowing for thechanges in composition in one of the solidifying phases.Here, the solid phase fractions are determined from thecriterion of equal undercoolings at the two solid-liquid in-terfaces. A more rigorous extension is provided recentlyby Senninger and Voorhees [37], where they take into ac-count the composition variations of both the solid phases.Although, our work shares a similar spirit in this aspect,we present an alternate derivation. One of the major dif-ferences is that we relate the deviations of the phase com-positions to the departures of the diffusion potentials andtemperature and thereby the functional dependence be-tween the variations of the solid and liquid compositionsis more elegantly retrieved. Secondly, our theory is appli-cable for any generic multi-phase eutectic growing with alamellar arrangement which is in contrast to the work ofSenninger and Voorhees [37], who limit themselves to two-phase growth. In addition, we verify our analytical cal-culations with phase-field simulations considering modelsymmetric alloys as well as a Ni-Al-Zr alloy. In all thestudies mentioned above, the effect of solute diffusivitiesin modifying the selection of solid phase fractions have not2een explored. We explore this aspect using our phase-field simulations as well as analytical calculations.
2. Analytical theory
In order to motivate our present work let us re-visitthe main results of the classical Jackson-Hunt analysis asdetailed in [17], for deriving the undercooling vs spacingrelationships for two-phase growth in a binary alloy. Thesituation is modeled by considering a repeating represen-tative unit of two phases α and β growing in a directionalsolidification set-up where the imposed temperature gra-dient (G) at the interface traverses with a velocity V, thatsets the rate of solidification. The undercooling at eachinterface can be written as,∆ T ν = − m νB ( (cid:102) c νB − c EB ) + Γ ν (cid:102) κ ν , (1)where, (cid:102) c νB represents the average composition in the liquidin local equilibrium with the ν -th phase and c EB representsthe eutectic composition. m νB is the liquidus slope. Γ ν and κ ν denote the Gibbs-Thomson coefficient and the in-terfacial curvature, respectively.We start by writing the composition profiles as a Fourierseries with amplitudes that are determined from the con-dition that the composition profiles obey both the gov-erning equation and the Stefan condition. A correspond-ing generic analysis for invariant eutectic growth in multi-component systems is laid out in [19], where expressionsfor all the amplitudes apart from the zeroth order mode(representing the boundary layer) can be determined fromthe inverse Fourier transform. In order to understand thedifficulty in determining the amplitude of the zeroth or-der, we first inspect the expression obtained by perform-ing an inverse Fourier transform of the same, written as, B = (cid:104) c αlB η α + c βlB (1 − η α ) (cid:105) − (cid:104) c αB η α + c βB (1 − η α ) (cid:105) , where c α/β,lB represents the liquid compositions in equilibrium with the α/β interfaces, η α is the volume fraction of the α phase and B denotes the boundary layer amplitude cor-responding to the component B . If one uses the volumefractions and compositions at the eutectic temperature (asfor the other Fourier modes) for determining the boundarylayer composition B (far-field composition is at the eutec-tic) it would result in zero and the corresponding under-coolings at the interface would not be equal. This calcula-tion, would also be physically incorrect, as the phase com-positions deviate from their values at the eutectic temper-ature. Jackson and Hunt in their analysis, treat this diffi-culty by keeping the B as an unknown which is fixed bythe condition that the undercoolings at both solid-liquidinterfaces are equal, while the volume fractions η α at theeutectic temperature are utilized for computing both theconstitutional and curvature undercoolings. In general,one can solve this problem of invariant growth for a multi-component system as in [19], where it has been shown toagree well with experiments as well as phase-field simula-tions.For the mono-variant reaction however, for instancein a two-phase growth in a ternary alloy, there would betwo boundary layer compositions, whereas the equality ofa common undercooling imparts only a single equation,thereby the system of equations become under-determined.The system of equations can only be made deterministic byinvoking the functional dependence of the boundary layercompositions on the phase compositions and the solid-fractions. This motivates our present derivation, which inspirit unifies the theories of in-variant and multi-varianteutectic growth. The following discussion is generic to a directionallysolidifying multi-component alloy of K components (with K − N solid phases, possessing a degree of free-dom given by F = K − N . Though, we present the theory3ssuming independent diffusion of solutes in the liquid (nodiffusion in the solid), it can be considered to be represen-tative of a system with non-zero off-diagonal terms in thediffusivity matrix, when such an analysis is carried out inthe basis system of the eigenvectors of the diffusivity ma-trix. In the following discussion, the indices i and j arereserved for solutes, while ν and p denote the solid phasesappearing due to eutectic solidification. Assuming a flatinterface, the composition variation in the liquid is of theform [19], c i = c ∞ i + n = ∞ (cid:88) n = −∞ I n e ˆ ik n x − q in z , (2)where, ˆ i = √−
1, and k n = 2 πn/λ , are wavenumbers char-acterizing the variation of solute concentrations in the liq-uid across a solid-liquid interface aligned along x with theeutectic solids growing in z . Conformity of Eq. 2 to thestationary form of the diffusion equation given below, V ∂c i ∂z + D ii ∇ c i = 0 , (3)leads to, q in = (cid:18) V D ii (cid:19) + (cid:115) k n + (cid:18) V D ii (cid:19) , (4)where D ii denotes the diffusivity of the i th component and V represents the sample pulling velocity. Following the dis-cussion in [19], a single wavelength of the eutectic consistsof M units ( M > = N ) of the eutectic solids with eachone of the M units corresponding to one of the N phases.The periodic variation starts at x = 0 and terminates at x M = 1 with the width of the ν -th unit being given by( x ν − x ν − ) λ ; the entire wavelength being ( x M − x ) λ .Thus, the volume fraction of a particular phase p , denotedby η p , can be calculated from a single wavelength of theeutectic lamellae as, η p = M − (cid:88) ν =0 ( x ν +1 − x ν ) δ νp , (5)where, δ νp = , if ν = p, , if ν (cid:54) = p. (6) The mass balance across a particular location at thesolid-liquid interface for the ν -th unit can be written as, V ∆ c νi = − D ii ∂c i ∂z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 ,x ∗ ν , (7)with ∆ c νi = c νli − c νi , where c νli and c νi denote the liquidand the solid compositions in local equilibrium at a loca-tion x ∗ ν on the ν − l interface, respectively. An expressionfor the Fourier constants I n is obtained by invoking the or-thogonality of the Fourier basis functions while integratingEq. 7 over an entire period ( λ ) of the eutectic, leading to, q in I n λ = 2 l i M − (cid:88) ν =0 (cid:90) x ν +1 λx ν λ exp( − ik n x )∆ c νi dx, (8)where, l i = 2 D ii /V , is the diffusion length associated withthe i -th component. For the mode corresponding to n = 0,Eq. 8 yields for the i -th component, I = 1 λ M − (cid:88) ν =0 (cid:90) x ν +1 λx ν λ ∆ c νi dx. (9)It is beneficial to define average compositions in front of aparticular phase p as,∆ c pi = (cid:80) M − ν =0 δ νp (cid:82) x ν +1 λx ν λ ∆ c νi dx (cid:80) M − ν =0 ( x ν +1 − x ν ) λδ νp , (10)which allows us to re-express Eq. 9 as, I = N (cid:88) p =1 η p ∆ c pi . (11)Similarly, the average interfacial composition in the solid( c pi ) and the liquid c pli ahead of it is defined as, c p,pli = (cid:80) M − ν =0 δ νp (cid:82) x ν +1 λx ν λ c ν,νli dx (cid:80) M − ν =0 ( x ν +1 − x ν ) λδ νp . (12)The theory in [19] following that of Jackson & Hunt [17]provides an expression for c pli , which has the form, c pli = c ∞ i + I + λη p l i f i (cid:16) P ( η , · · · , η N ) , · · · , P r ( η , · · · , η N ) , ∆ c i , · · · , ∆ c Ni (cid:17) , (13)where, each of one of the k infinite series’ P k ( η p ), k =1 , · · · , r , p = 1 , · · · , N , are composed of terms which are4rigonometric functions of η p . The value of r and the formof P k ( η p ) are determined by the number and repetitionsof solid phases in a single periodic unit of wavelength λ .It must be mentioned at this point that the term I rep-resents the principal term determining the liquid composi-tions c pli at the flat interface, with the secondary influencebeing due to that of the higher order modes averaged overthe lamellar widths denoted by the final term in the RHSof Eq. 13. An example of such a term for a ternary mono-variant eutectic [19], f i = 2 P ( η α ) (cid:16) ∆ c αi − ∆ c βi (cid:17) , (14) P ( η β ) = P ( η α ) = ∞ (cid:88) n =1 πn ) sin ( πnη α ) . (15)The average undercooling (∆ T p ) ahead of a particularsolid ( p )-liquid ( l ) interface is given by,∆ T p = T ∗ − T p = K − (cid:88) i =0 m pi (cid:16) c l, ∗ i − c pli (cid:17) + Γ p κ p , (16)where, m pi are the liquidus slopes, Γ p denotes the Gibbs-Thomson coefficient with the average curvature of the solid( p )-liquid interface ( κ p ) given by, κ p = 2 sin θ pm η p λ , (17)where, θ pm is the angle made by the tangent to the solid( p )-liquid( l ) interface and the horizontal towards the side ofthe p -th phase when located adjacent to the m -th phase,and averaged over all such contiguous arrangements of thesolid phases m and p in the entire period.The fact that the imposed thermal gradient has a lengthscale much larger than the lamellar width, implies thegrowth of all the eutectic solids at equal undercoolings,which can be expressed as,∆ T = ∆ T = · · · = ∆ T N = ∆ T . (18)Also, the sum of volume fractions of the phases in a singleperiod of the lamellae must be equal to unity, N (cid:88) p =1 η p = 1 . (19) Here, one needs to solve the Eqs. 11, 13, 16, 18 and19 simultaneously, in order to retrieve the phase composi-tions, the solid fractions and the undercooling of the eu-tectic growth front. For this, we need to describe the func-tional dependence between the solid compositions c pi andthe liquid compositions c pli . We do this by calling upon therelations of the phase compositions c p,pli (cid:16) µ pi , T (cid:17) , where µ pi corresponds to the diffusion potential of p − l equilibrium,averaged over all occurrences of the solid phase p in a pe-riodic unit of the eutectic. This then reduces the abovesystem, Eqs. 11, 13, 16, 18 and 19 in terms of µ pi , T , and η p . A point to note here is that, Senninger and Voorhees[37], replace the Eqs.11 with a mass conservation con-straint. Mass conservation is implicit in our set of equa-tions. This can be seen by considering only those N equa-tions out of the N ( K −
1) in Eqs.13 which represent thecomposition fields of a particular component i in the liq-uid in equilibrium with different solids ( p ). Summing overall such equations after multiplying both sides of eachof them with the respective volume fractions η p , gives I = (cid:80) p c pli η p − c ∞ i . This along with Eqs.11 implies that (cid:80) p c pi η p = c ∞ i , which is the mass conservation equationused by Senninger and Voorhees [37].However, given that the thermodynamical relations c p,pli (cid:16) µ pi , T (cid:17) are routinely non-linear, the resultant set ofequations become difficult to resolve. Given that most de-partures from equilibrium in case of eutectic reactions aresmall, we therefore linearize our set of equations about thechosen eutectic temperature T ∗ , the average diffusion po-tentials µ p, ∗ j and the solid phase fractions η ∗ p , correspond-ing to T ∗ . It must be noted that the equilibrium solid andliquid phase compositions corresponding to T ∗ , µ p, ∗ j and η ∗ p , are c p, ∗ i and c pl, ∗ i , respectively. This results in a set oflinear equations which can be solved for, consistently.5 .3. Linearized Theory We express the average compositions in the solid ( c pi )and the liquid ( c pli ) as functions of diffusion potentials ( µ pj )as, c pi = c p, ∗ i + K − (cid:88) j =1 (cid:20) ∂c pi ∂µ j (cid:21) µ p, ∗ j ∆ µ pj − ∂c pi ∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∗ ∆ T p , (20)and, c pli = c pl, ∗ i + K − (cid:88) j =1 (cid:20) ∂c li ∂µ j (cid:21) µ p, ∗ j ∆ µ pj − ∂c li ∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∗ ∆ T p , (21)where, ∆ µ pj = µ pj − µ p, ∗ j , denote the change in averagediffusion potentials from their values at the chosen eu-tectic point µ p, ∗ j , under an additional constraint of con-stant [ ∂c i /∂µ j ] matrices. The vector ∂c pi /∂T determinethe change in composition with temperature at constantdiffusion potentials.Employing Einstein’s indicial notation which conveyssummation over repeated indices (except for p in our anal-ysis, which denotes a particular phase), the above equa-tions can be written as, c pi = c p, ∗ i + χ pij ∆ µ pj − ζ pi ∆ T p , (22)and, c pli = c pl, ∗ i + χ lij ∆ µ pj − ζ li ∆ T p , (23)where, χ pij = ∂c pi ∂µ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ p, ∗ j , (24) χ lij = ∂c li ∂µ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ p, ∗ j , (25) ζ pi = ∂c pi ∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∗ , (26) ζ li = ∂c li ∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∗ . (27)Thus, the difference in the average compositions of the solid and the liquid as obtained from Eqs. 20, 21, 22 and 23,∆ c pi = c pli − c pi , = (cid:16) c pl, ∗ i − c p, ∗ i (cid:17) + K − (cid:88) j =1 (cid:18) ∂c li ∂µ j − ∂c pi ∂µ j (cid:19) µ p, ∗ j ∆ µ pj − (cid:18) ∂c li ∂T − ∂c pi ∂T (cid:19) T ∗ ∆ T p = ∆ c p, ∗ i + ∆ χ pij ∆ µ pj − ∆ ζ pi ∆ T p , (28)where, to obtain the last equality expressed in indicial no-tation, we have used,∆ c p, ∗ i = c pl, ∗ i − c p, ∗ i , (29)∆ χ pij = χ lij − χ pij , (30)∆ ζ pi = ζ li − ζ pi . (31)Using, Eqs. 22, 23 and 28, the 2 N ( K −
1) composition vari-ables have been expressed as functions of N ( K −
1) inten-sive variables in the form of change in diffusion potentials∆ µ pj , which can be further related to ∆ η p (= η p − η ∗ p ) and∆ T p by invoking equality of Eqs. 13 and 21 which pro-vide additional N ( K −
1) equations, stated in the indicialnotation (with no sum over p and i ) as, c pli = c pl, ∗ i + χ lij ∆ µ pj − ζ li ∆ T p = c ∞ i + I + λη p l i f i (cid:16) P ( η , · · · , η N ) , · · · , P r ( η , · · · , η N ) , ∆ c i , · · · , ∆ c Ni (cid:17) (32)The RHS of Eq. 32 (or Eq. 13), is in general non-linear in∆ µ pj , ∆ η p and ∆ T p . Thus, to express ∆ µ pj as an explicitfunction of ∆ η p and ∆ T p , we linearly expand each termin the RHS of Eq. 32 starting with I , given by, I = I ∗ + N (cid:88) m =1 ∂I ∂ ∆ c mi K − (cid:88) j =1 ∂ ∆ c mi ∂µ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ m, ∗ j ∆ µ mj − N (cid:88) m =1 ∂I ∂ ∆ c mi ∂ ∆ c mi ∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∗ ∆ T m + N (cid:88) m =1 ∂I ∂η m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η ∗ m ∆ η m , (33)6here, µ p, ∗ i , T ∗ and η ∗ p are the quantities corresponding tothe equilibrium in the phase diagram. The different termsin the RHS of Eq. 33 can be computed from Eqs. 11, 28and 31 as, ∂I ∂ ∆ c pi = η ∗ p , (34) ∂ ∆ c pi ∂µ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ p, ∗ j = (cid:18) ∂c li ∂µ j − ∂c pi ∂µ j (cid:19) µ p, ∗ j = ∆ χ pij , (35) ∂ ∆ c pi ∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∗ = (cid:18) ∂c li ∂T − ∂c pi ∂T (cid:19) T ∗ = ∆ ζ pi , (36) ∂I ∂η p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η ∗ p = ∆ c p, ∗ i . (37)At this point we introduce the following quantities (no sumover p ), ∆ χ pij = η ∗ p ∆ χ pij , ∆ ζ pi = η ∗ p ∆ ζ pi , (38)to express Eq. 33 in indicial notation as, I = I ∗ + ∆ χ mij ∆ µ mj − ∆ ζ mi ∆ T m + ∆ c m, ∗ i ∆ η m , (39)with m and j being the indices representing phases andcomponents respectively, which are summed over. The sec-ond term in the RHS of Eq. 32, being only a second-ordercorrection to the interfacial liquid composition is assumedto be a function of 1 /η p only, with all the other quantitiesevaluated at the conditions prevailing at the eutectic. Thissimplifying assumption is necessary to maintain tractabil-ity of the equations. Thus, writing, f i = f ∗ i (cid:16) P ( η ∗ , · · · , η ∗ N ) , · · · , P r ( η ∗ , · · · , η ∗ N ) , ∆ c , ∗ i , · · · , ∆ c N, ∗ i (cid:17) , (40)we re-write the linearized version of Eq. 32 indicially, as, c pli = c pl, ∗ i + χ lij ∆ µ pj − ζ li ∆ T p = c ∞ i + (cid:104) I ∗ + ∆ χ mij ∆ µ mj − ∆ ζ mi ∆ T m + ∆ c m, ∗ i ∆ η m (cid:105) + (cid:34)(cid:32) λη ∗ p − λη ∗ p ∆ η p (cid:33) ˜ f i (cid:35) , (41) where, ˜ f i = f ∗ i l i , (42)and, p is the index which represents a particular solidphase and is not summed over in Eq. 41 as well as inthe following equations. The quantities enclosed in squarebrackets in the RHS of Eq. 41 represent terms obtainedby linearizing the individual terms in the RHS of Eq. 32.Eq. 41 can be re-written to express the diffusion poten-tials as a function of the interfacial undercoolings and solidphase volume fractions as, (cid:104) χ lij − ∆ χ pij (cid:105) ∆ µ pj − ∆ χ m (cid:48) ij ∆ µ m (cid:48) j = (cid:18) c ∞ i + I ∗ + λη ∗ p ˜ f i − c pl, ∗ i (cid:19) − ∆ ζ m (cid:48) i ∆ T m (cid:48) − (cid:16) ∆ ζ pi − ζ li (cid:17) ∆ T p + ∆ c m (cid:48) , ∗ i ∆ η m (cid:48) + (cid:32) ∆ c p, ∗ i − λη ∗ p ˜ f i (cid:33) ∆ η p , (43)where, a summation over the index m (cid:48) runs from 1 , · · · , N leaving out p . So, Eq. 43 represents a system of N ( K −
1) equations which relates the N ( K −
1) ∆ µ pi ’s to ∆ η p ’sand ∆ T p ’s. To describe this dependence, we utilize thelinearity of Eq. 43 to write an explicit relation of the form,∆ µ pi = R pi + R p,T m i ∆ T m + R p,η m i ∆ η m , (44)where, R pi , R p,T m i , and R p,η m i are coefficients determinedby solving Eq. 43 with m being the lone index which issummed over in Eq. 44. These relationships will enablethe elimination of ∆ µ pli completely from the expressionsof the undercoolings ahead of every solid phase given inEq. 16. Substituting Eq. 44 into Eq. 39 we write in theindicial notation, I = I ∗ + ∆ χ mij (cid:16) R mj + R m,T v j ∆ T v + R m,η v j ∆ η v (cid:17) − ∆ ζ mi ∆ T m + ∆ c m, ∗ i ∆ η m , = (cid:0) I ∗ + ∆ χ mij R mj (cid:1) + (cid:16) ∆ χ mij R m,T v j ∆ T v − ∆ ζ mi ∆ T m (cid:17) + (cid:0) ∆ χ mij R m,η v j ∆ η v + ∆ c m, ∗ i ∆ η m (cid:1) , (45)7here v is an index running over the phases 1 , · · · , N and issummed over along with the other phase index m . The in-dex j denoting the components is also summed over while i continues to represent a particular component. Terms ofsimilar character are collected within the parentheses inEq. 45.The resulting set of equations in 16 ( N in number) arethen solved for the 2 N unknowns in ∆ T p and ∆ η p underthe constraints of equality of undercoolings ahead of thesolid phases given by Eq. 18 ( N − p ,but summming over i ,∆ T p = m pi (cid:16) c l, ∗ i − c pli (cid:17) + Γ p κ p = m pi (cid:20) c l, ∗ i − (cid:18) c ∞ i + I + λη p ˜ f i (cid:19)(cid:21) + Γ p κ p . (46)Linearizing the RHS of the above equation about equilib-rium quantities and employing Eq. 45 leads to,∆ T p = (cid:34) m pi (cid:18) c l, ∗ i − (cid:18) c ∞ i + I ∗ + ∆ χ mij R mj + λη ∗ p ˜ f i (cid:19)(cid:19) + Γ p κ ∗ p (cid:35) − (cid:104) m pi (cid:16) ∆ χ mij R m,T v j ∆ T v − ∆ ζ mi ∆ T m (cid:17)(cid:105) − (cid:34) m pi (cid:32) ∆ χ mij R m,η v j ∆ η v + ∆ c m, ∗ i ∆ η m − λη ∗ p ˜ f i ∆ η p (cid:33) + Γ p κ ∗ p η ∗ p ∆ η p (cid:35) , (47)where κ ∗ p is obtained by evaluating Eq. 17 for η ∗ p . The threeterms each enclosed in square brackets in the RHS of theabove equation contain the constants, and the terms linearin ∆ T m and ∆ η m respectively. We now impose Eq. 18 on Eq. 47 to re-express it in terms of ∆ T and ∆ η p ’s, as follows, (cid:34) m pi (cid:32) ∆ χ mij N (cid:88) v =1 R m,T v j − N (cid:88) m =1 ∆ ζ mi (cid:33)(cid:35) ∆ T + (cid:34) m pi (cid:32) ∆ χ mij R m,η p j + ∆ c p, ∗ i − λη ∗ p ˜ f i (cid:33) + Γ p κ ∗ p η ∗ p (cid:35) ∆ η p + (cid:104) m pi (cid:16) ∆ χ mij R m,η v (cid:48) j + ∆ c v (cid:48) , ∗ i (cid:17)(cid:105) ∆ η v (cid:48) = (cid:34) m pi (cid:18) c l, ∗ i − (cid:18) c ∞ i + I ∗ + ∆ χ mij R mj + λη ∗ p ˜ f i (cid:19)(cid:19) + Γ p κ ∗ p (cid:35) , (48)where, v (cid:48) is another phase index running from 1 to N ex-cept p and is summed over along with the other phaseindex m . The component indices i and j are also summedover in Eq. 48. At this stage, we will invoke Eq. 19 to elim-inate ∆ η N from the above set of equations. Now, Eq. 48represents a system of N linear equations containing thesame number of unknowns in ∆ T and ∆ η , · · · , ∆ η N − .Solving Eq. 48 to compute ∆ T and ∆ η , · · · , ∆ η N − , en-ables the calculation of ∆ µ j , · · · , ∆ µ Nj from Eq. 44. There-after, the phase compositions can be directly obtainedfrom Eqs. 20 and 21.To summarize, our analytical method involves the fol-lowing steps: • For every solid-liquid equilibrium in a single periodof the eutectic, the solid and the liquid compositionsaveraged over their corresponding lamellar widthsare expressed as linear functions of the correspondingchanges in diffusion potentials and undercoolings. • We also determine the average composition in theliquid in equilibrium with the different solid phasesfrom the Jackson-Hunt type analysis involving thesuperposition of multiple Fourier modes, which isagain linearized about the chosen eutectic point toobtain the liquid compositions as functions of changesin diffusion potentials, undercoolings and changes in8olid phase volume fractions. This also involves ex-pressing the boundary layer compositions in terms ofthe departure of diffusion potentials and phase frac-tions from their corresponding values at the eutecticalong with the undercooling at the solid-liquid inter-face. • Using the equality of the liquid compositions ob-tained from the earlier steps we derive expressionsof the diffusion potentials as functions of undercool-ings and changes in solid phase fractions. • Substituting for the liquid compositions using thelinearized version of the Fourier series representationin terms of undercoolings, diffusion potentials andsolid phase fractions into the expressions of the un-dercoolings at each solid-liquid interface, and enforc-ing the isothermal nature of the interface we com-pute the magnitude of the interfacial undercoolingand solid phase fractions. • Phase compositions get automatically determined dueto their explicit and implicit (due to diffusion po-tentials) dependence on undercooling alongside theirdependence on solid phase fractions.A point to note over here is while Eq. 44 relates thedeviations of the diffusion potentials ∆ µ pi with the devi-ations of the solid fractions and the undercoolings, onecan additionally invoke the condition of local thermody-namic equilibrium at the interface (including curvature)and thereby eliminate the undercoolings from the rela-tion in Eq. 44. A similar approach has been used bySenninger and Voorhees [37]. We have tried this out aswell and the results from both approaches are compara-ble. This completes our theoretical derivation of genericmulti-component multi-phase eutectic growth. In the fol-lowing section, we validate our theory against phase-fieldsimulations of invariant and mono-variant eutectic growth.
3. Phase field model
Following [38], the grand potential functional(Ω) canbe expressed as,Ω ( µ , T, φ ) = (cid:90) V (cid:34) Ψ ( µ , T, φ )+ (cid:18) (cid:15)a ( φ , ∇ φ ) + 1 (cid:15) w ( φ ) (cid:19) (cid:35) dV, (49)where φ = [ φ , φ , · · · , φ N ] are the phase-fields represent-ing the spatial arrangement of N phases and µ = [ µ , µ , · · · , µ K − ] are the diffusion potentials associ-ated with each one of the K − w and a represent the surface potential and thegradient energy density respectively. The minimization ofΩ leads to the evolution of the spatial arrangement of thephases ( φ ) denoted by, τ (cid:15) ∂φ p ∂t = (cid:15) (cid:18) ∇ · ∂a ( φ , ∇ φ ) ∂ ∇ φ p − ∂a ( φ , ∇ φ ) ∂φ p (cid:19) − (cid:15) ∂w ( φ ) ∂φ p − ∂ Ψ ( µ , T, φ ) ∂φ p − Λ , (50)where Λ is calculated to ensure (cid:80) Nm =1 φ m = 1 at everymesh point in the simulation domain. τ is the relaxationconstant with its value set based on the criterion statedin [38] and [39] to obtain a diffusion controlled interfacemotion.In this model, ∆Ψ mp = Ψ m − Ψ p , (51)represents the driving force for a transformation of phase m to p , with the grand-potentials of the individual phasesgiven by,Ψ p = f p ( c p ( µ , T ) , T ) − K − (cid:88) i =1 µ i c pi ( µ , T ) . (52)All the grand-potentials of the participating phases (Ψ p )’sat any particular point in the simulation domain are inter-polated to obtain Ψ as,Ψ ( µ , T, φ ) = p (cid:88) Ψ p ( T, µ ) h p ( φ ) , (53)9here, h p ( φ ) = φ p (3 − φ p ) + 2 φ p N,N (cid:88) m =1 ,n =1 ,m 1) independent compo-nents. [ · ] denotes a matrix of dimension (( K − × ( K − { · } represents a vector of dimension ( K − J at , i has a sense and magnitude which nullifies solute trapping at the solid-liquid interfaceand is determined by the expressions given in [39].The atomic mobility, M ij ( φ ) is obtained by interpo-lating the individual phase mobilities as, M ij ( φ ) = (cid:88) ν M pij g p ( φ ) , (58)where the individual phase mobilities are given by, (cid:2) M pij (cid:3) = [ D pik ] (cid:20) ∂c pk ( µ , T ) ∂µ j (cid:21) , (59)where D pij are the solute inter-diffusivities in the p -th phaseand g p ( φ ) are interpolants given as, g p ( φ ) = φ p (3 − φ p ) . (60)The composition fields are obtained as functions of µ and φ as, c i = (cid:88) p c pi ( µ , T ) h p ( φ ) ,c pi ( µ , T ) = − V m ∂ Ψ p ( φ , µ , T ) ∂µ i . (61)with the molar V m is taken to be a constant across all thecomponents. 4. Results: Two-solid phases in a ternary system In this section we employ both the analytical and phase-field models described above to study the solidification oftwo solid phases in a ternary alloy and compare the predic-tions from these two techniques for different solute inter-diffusivities in the liquid. Although, the analytical theoryand phase field model are general enough to describe thesolidification at off-eutectic compositions, we restrict ourstudy to eutectic compositions only.The solid phases in the ternary monovariant eutecticare anointed as α and β with the independent compo-nents constituting the ternary alloy being A and B . Allour studies are performed at a single sample pulling ve-locity of V = 0 . 01 under an imposed thermal gradient of G = 0 . η ∗ α and η ∗ β A three phase equilibrium ( α , β and liquid) in a ternarysystem is associated with a single degree of freedom as itcan exist over a range of temperatures. During directionalsolidification, the far-field liquid composition can be foundto correspond to a particular temperature ( T ∗ ) in the equi-librium phase diagram at which it is in equilibrium withtwo other solid phases. If such a liquid is assumed tosolidify at this temperature, the volume fractions of theresultant α and β phases are what we refer to as η ∗ α and η ∗ β , respectively. As there are two independent far-fieldcompositions in a ternary system, we invoke two artificialphase fractions η (cid:48) α and η (cid:48) β (cid:54) = (1 − η (cid:48) α ) to solve for, c ∞ A = c αA η (cid:48) α + c βA η (cid:48) β ,c ∞ B = c αB η (cid:48) α + c βB η (cid:48) β , (62)consistently. In general, η (cid:48) α + η (cid:48) β (cid:54) = 1 and we compute thenormalized volume fractions, η ∗ α = η (cid:48) α η (cid:48) α + η (cid:48) β ,η ∗ β = η (cid:48) β η (cid:48) α + η (cid:48) β , (63)obeying η ∗ α + η ∗ β = 1 and serving as values of the volumefractions about which linearization is performed. To isolate and understand the effect of differences insolute diffusivities on the eutectic growth dynamics, weselect a system where the equilibrium phase compositionsof solid phases are symmetric with respect to the liquidcomposition ( c αA = 0 . c αB = 0 . c βA = 0 . c βB =0 . c lA = 0 . c lB = 0 . m αA = 0 . m αB = 0, m βA = 0, m βB = 0 . α -liquid and β -liquidinterfacial energies which serves as a reference when weattempt to understand the dynamics of systems displayingdissimilar interfacial energies of the eutectic solids withliquid. α -liquid and β -liquid interfacial energies The equality of α -liquid and β -liquid interfacial ener-gies leads to θ αβ = θ βα = 30 ◦ with the Gibbs-Thomson co-efficients computed to be Γ α = Γ β = 0 . 77 for the particularthermodynamics employed. We present analytically cal-culated variations of interfacial undercoolings (∆ T ), solidphase volume fractions ( η α ) and compositions of the α andthe β phases with lamellar width( λ ) in Fig. 1 and comparethem against predictions obtained from phase field simu-lations. As can be seen from Fig. 1a, the analytical andphase-field calculations are in very good agreement as faras the predictions in λ min (the lamellar width correspond-ing to the minimum in ∆ T ) are concerned. The closeagreement between the analytical theory and the phasefield simulations are also evident from the variations of η α versus λ presented in Fig. 1b and the variation of the aver-age solid phase compositions in Figs. 1c and 1d. The devi-ations of analytically computed values of ∆ T (in Fig. 1a)from those obtained from phase field simulations can be at-tributed to the inherent assumptions in the Jackson-Huntcalculations, where a planar interface is used to approxi-mate the diffusion-field ahead of the solid interfaces, whichare in reality curved for a system with isotropic surface en-ergies. This mismatch has also been shown before [40], anda concomitant comparison of the phase-field method withcalculations based on the boundary-integral method haveproved that the predictions of the phase-field method aremore accurate in this regard.From Fig. 1a, it can be seen that lowering either of thesolute diffusivities leads to a reduction in the length scaleof the eutectic ( λ min ) with a consequent rise in ∆ T . This is11 ∆ T λ (T,2.0,2.0)(S,2.0,2.0)(T,2.0,1.0)(S,2.0,1.0)(T,1.0,2.0)(S,1.0,2.0) (a) η α λ (T,2.0,2.0)(S,2.0,2.0)(T,2.0,1.0)(S,2.0,1.0)(T,1.0,2.0)(S,1.0,2.0) (b) c o m po s i t i on o f α λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,1.0)(S,c A ,2.0,1.0)(T,c B ,2.0,1.0)(S,c B ,2.0,1.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (c) c o m po s i t i on o f β λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,1.0)(S,c A ,2.0,1.0)(T,c B ,2.0,1.0)(S,c B ,2.0,1.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (d)Figure 1: (Color online) Plots showing variations of (a)∆ T , (b) η α ,(c) α phase compositions and (d) β phase compositions, with λ ,during two phase growth in a model symmetric ternary alloy. Thefirst position in the figure legends in (a) and (b) indicates whether theplotted data comes from theory (T) or simulations (S); the secondand third positions represent values of D AA and D BB respectively.For (c) and (d), the first position in the figure legends conveys thesame as in (a) and (b), the second position indicates whether theplotted variation in compositions corresponds to A or B . The thirdand fourth positions in the figure legends in (c) and (d) representthe values of solutal interdiffusivities of D AA and D BB respectively. a result of a lowered effective diffusivity leading to a lowereffective diffusion length. As a reflection of the inherentsymmetry in the system, the theoretical calculations for∆ T vs λ are exactly identical for D AA = 2 . , D BB = 1 . D AA = 1 . , D BB = 2 . D AA = 2 . , D BB = 1 . α phase occupies a larger solidfraction ( η α > . 5) of the lamellar width ( λ ); the sameobservation being valid for β when D AA = 1 . , D BB = 2 . D AA in comparison to D BB would result in a lower undercooling ahead of the α − l interface than the β − l interface. To recover an isother-mal situation between the two interfaces, would requirethe α − l interface to assume an interfacial curvature thatis greater than that acquired by the α − l interface whenmechanical equilibrium is maintained at the trijunction.This departure from equilibrium acts as a driving force,where mechanical equilibrium is re-established through anincrease in the volume fractions of the phase α . This ex-plains the observed variation of η α with λ in Fig. 1b. Asimilar argument can be made with respect to the lower-ing of the value of D AA with respect to D BB (see Fig. 1b),where it must be noted that as a consequence of the un-derlying symmetry in the system that the η α vs λ curve for D AA = 1 . , D BB = 2 . η α = 0 . D AA = 2 . , D BB = 1 . 0. Ingeneral, any change(change in diffusivity, interfacial ener-gies etc.) which causes an increase in the undercooling ofa particular phase-liquid interface would need to be offsetthrough an appropriate decrease in the interfacial curva-ture which can be affected only through a departure of theangles at the triple-point to lower values than that existsat equilibrium, keeping the same phase fractions. This12eparture acts as a driving force to establish equilibriumwhich is achieved by a decrease in the volume fraction ofthis phase. α -liquid and β -liquid interfacial energies Here we probe the effect of unequal surface energieson the steady-state monovariant eutectic growth dynamicswhile retaining the symmetry in the phase compositionsand the liquidus slopes from the system just discussed.The interfacial energies are so chosen such that θ αβ = 30 ◦ and θ βα = 45 ◦ with the Gibbs-Thomson coefficients beingΓ α = 0 . 77 and Γ β = 0 . 94. Fig. 2 shows the variation in∆ T , η α and the solid phase compositions with λ obtainedfrom both analytical and phase field calculations.The variation of ∆ T with λ reported in Fig. 2a presentsa departure from the symmetry observed in Fig. 1a asthe curves corresponding to D AA = 2 . , D BB = 1 . D AA = 1 . , D BB = 2 . η α with λ are effectively captured for all the cases, the mag-nitude of variation between the simulations and theory islarger than the previous simulations with the symmetricinterface properties. The reason for this is the asymmet-ric nature of the interface shapes, where in the phase-fieldsimulation the β − l interface is ahead of the α − l interface(see Fig.3) and thereby clearly the interfacial undercool-ings of the two phases are not the same. Additionally, thedeparture from a planar interface is higher for the β − l interface compared to the α − l interface, which also im-plies that this brings in added asymmetry with respect toa mismatch with the analytical calculations which are per-formed for a planar interface. Thereby, now any changein the interface shape which reduces the curvature of the ∆ T λ (T,2.0,2.0)(S,2.0,2.0)(T,2.0,1.0)(S,2.0,1.0)(T,1.0,2.0)(S,1.0,2.0) (a) η α λ (T,2.0,2.0)(S,2.0,2.0)(T,2.0,1.0)(S,2.0,1.0)(T,1.0,2.0)(S,1.0,2.0) (b) c o m po s i t i on o f α λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,1.0)(S,c A ,2.0,1.0)(T,c B ,2.0,1.0)(S,c B ,2.0,1.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (c) c o m po s i t i on o f β λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,1.0)(S,c A ,2.0,1.0)(T,c B ,2.0,1.0)(S,c B ,2.0,1.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (d)Figure 2: (Color online) Plots showing variations of (a)∆ T , (b) η α ,(c) α phase compositions, and (d) β phase compositions, with λ , dur-ing two phase eutectic growth in a model symmetric ternary alloywith unequal interfacial energies. The figure legends can be inter-preted in the same way as described in the caption of Fig. 1. z / λ x/ λ α -liquid β -liquid Figure 3: Plot showing the locations of the α − l and the β − l interfaces. β − l interface decreases the disparity between the ana-lytical calculations and phase-field simulations and addi-tionally with increasing curvature differences between the α − l and β − l interfaces, the discrepancies between the the-oretical predictions and simulation results also increases.Thus, this brings to light a limitation of the analytical cal-culations, which work best when interfacial shapes of thesolid-liquid interfaces are similar. In this section we study the steady-state dynamics ofmonovariant eutectic growth in a Ni-Al-Zr alloy at thebackdrop of the insights developed in the previous section.The equilibrium phase compositions at the temperatureof interest are: c αA = 0 . , c αB = 0 . , c βA = 0 . , c βB =0 . , c lA = 0 . , c lB = 0 . 19 with the liquidus slopes being m αA = 0 . , m αB = 1 . , m βA = − . , m βB = − . 0. TheGibbs-Thomson coefficients are Γ α = 1 . , Γ β = 0 . 81 withthe contact angles θ αβ = θ βα = 30 ◦ . We present thevariations in ∆ T , η α and the solid phase compositions inFig. 4.From all the diffusivity combinations studied it can besaid that the equilibrium phase compositions and the liq-uidus slopes are such that a higher volume fraction of α is the preferred morphology. Furthermore, it can be seenfrom Figs. 4a and 4b, that the dynamics is much moresensitive to a change in D BB compared to a change in D AA . The undercoolings are found to be much higher for ∆ T λ (T,2.0,2.0)(S,2.0,2.0)(T,2.0,1.0)(S,2.0,1.0)(T,1.0,2.0)(S,1.0,2.0) (a) η α λ (T,2.0,2.0)(S,2.0,2.0)(T,2.0,1.0)(S,2.0,1.0)(T,1.0,2.0)(S,1.0,2.0) (b) c o m po s i t i on o f α λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,1.0)(S,c A ,2.0,1.0)(T,c B ,2.0,1.0)(S,c B ,2.0,1.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (c) -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 50 100 150 200 250 300 350 c o m po s i t i on o f β λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,1.0)(S,c A ,2.0,1.0)(T,c B ,2.0,1.0)(S,c B ,2.0,1.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (d)Figure 4: (Color online) Plots showing variations of (a)∆ T , (b) η α ,(c) α phase compositions, and (d) β phase compositions, with λ , dur-ing two-phase growth in a Ni-Al-Zr alloy. The figure legends can beinterpreted in the same way as described in the caption of Fig. 1. AA = 2 . , D BB = 1 . λ min )towards smaller values. Also, η α displays a steady risewith λ which is much steeper for D AA = 2 . , D BB = 1 . 5. Results: Three phases in a ternary system Following up from the previous studies on two-phasemono variant growth, in this section we investigate three-phase invariant growth. Contrary to two-phase growthwhere there is a single possibility for the lamellar arrange-ment of the phases, for the case of three-phase growth thereexist infinitely many configurations (e.g., αβγ, αβαγ . . . ).In the following discussion, we consider two such possi-bilities for study, using both analytical calculations andphase-field simulations. Here we conduct simulations forthe different choices of the diffusivity matrices and com-pare the predictions of the phase compositions and thevolume fractions between the phase-field simulations andthe theoretical predictions. The equilibrium phase com-positions at the temperature of the invariant eutectic are: c αA = 0 . , c αB = 0 . , c βA = 0 . , c βB = 0 . , c lA =0 . , c lB = 0 . 333 with the liquidus slopes being m αA =0 . , m αB = 0 . , m βA = 0 . , m βB = 0 . m γA = − . , m γB = − . 91. The Gibbs-Thomson coefficients are Γ α = Γ β =Γ γ = 1 . 558 with the contact angles θ αβ = θ βα = θ βγ = θ γβ = θ αγ = θ γα = 30 ◦ . The directional solidification con-ditions are kept the same as in the study of monovarianteutectic growth in ternary alloys.We first consider the simplest arrangement αβγ , wherefor the case of equal diagonal diffusivities, we get excellentagreement between our theory and phase-field simulationresults (see Fig. 5), which is reflected not only in thevariations of the undercooling with spacing, but also in thecompositions of the phases (see Fig. 5) and in the volume fractions which given the symmetry of the phase-diagramand the diffusivities remain at ( η α , η β , η γ ):(1/3,1/3,1/3). ∆ T λ (T,2.0,2.0)(S,2.0,2.0) (a) η α λ (T,2.0,2.0)(S,2.0,2.0) (b) η β λ (T,2.0,2.0)(S,2.0,2.0) (c)Figure 5: (Color online) Plots showing variations of (a)∆ T , (b) η α ,and (c) η β , with λ , during three phase eutectic growth in a modelsymmetric ternary alloy. A single wavelength of the eutectic solidshas the configuration: αβγ . The figure legends can be interpreted inthe same way as described in the caption of Fig. 1. However, for the case of unequal diffusivities, an infer-ence from the phase-field simulations can be seen in Fig. 6where we notice a tilt in the lamellar arrangement withrespect to the growth direction.We note that this tilt is not an ”instability” that occursbeyond a spacing as has been reported during two and15 c o m po s i t i on o f α λ (T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (d) c o m po s i t i on o f β λ (T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (e) c o m po s i t i on o f γ λ (T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (f)Figure 5: (Color online) Plots showing variations of (d) α , (e) β , and(f) γ phase compositions with λ , during three phase eutectic growthin a model symmetric ternary alloy. A single wavelength of the eu-tectic solids has the configuration: αβγ . The figure legends can beinterpreted in the same way as described in the caption of Fig. 1. z / λ x/ λ α - ββ - γγ - αα -l β -l γ -l (a) z / λ x/ λ α - ββ - γγ - αα -l β -l γ -l (b)Figure 6: Plots showing orientations of all the interfaces for,(a) D AA = D BB = 2 . 0, and (b) D AA = 2 . , D BB = 1 . 0, duringthree phase eutectic growth in a model symmetric ternary alloy. Asingle wavelength of the eutectic solids has the configuration αβγ . αβγ configurations to tilt isthat there are no symmetry planes once the volume frac-tions of the phases become unequal. The next arrange-ment αβαγ however, possesses, two mirror axes, passingthrough the β and the γ phases. Going by the symme-try arguments placed in a previous paper [19], a steady-state growth mode where the lamellae are aligned withthe growth direction, is therefore expected. We repeatthe simulation and analytical calculations for this config-uration for different diffusivities, and the results are re-ported in Figs.7. It is important to note that for thisconfiguration a short wavelength instability exists whichresults in the transformation of the αβαγ to αβγ occur-ring below a critical wavelength (see discussion in [19]).Thereby, we limit our analysis to only the stable lamel-lar states. For these spacings, we again derive an ex-cellent agreement for the undercooling vs spacing varia-tions, volume fractions and the phase compositions, be-tween phase-field simulations and theoretical predictions.Due to the variation in the stability regimes we have lim-ited our calculations for the case of only unequal diffusivi-ties D AA = 1 . , D BB = 2 . 0, as the stability region for thecontrary case of D AA = 2 . , D BB = 1 . 6. Conclusions In this study, we derive an analytical theory to deter-mine the interfacial undercoolings, volume fractions andcompositions of the solid phases in directionally solidifying ∆ T λ (T,2.0,2.0)(S,2.0,2.0)(T,1.0,2.0)(S,1.0,2.0) (a) η α λ (T,2.0,2.0)(S,2.0,2.0)(T,1.0,2.0)(S,1.0,2.0) (b) η β λ (T,2.0,2.0)(S,2.0,2.0)(T,1.0,2.0)(S,1.0,2.0) (c)Figure 7: (Color online) Plots showing variations of (a)∆ T , (b) η α ,and (c) η β , with λ , during three phase eutectic growth in a modelsymmetric ternary alloy. A single wavelength of the eutectic solidshas the configuration αβαγ . The figure legends can be interpretedin the same way as described in the caption of Fig. 1. c o m po s i t i on o f α λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (d) c o m po s i t i on o f β λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (e) c o m po s i t i on o f γ λ (T,c A ,1.0,2.0)(S,c A ,1.0,2.0)(T,c B ,1.0,2.0)(S,c B ,1.0,2.0)(T,c A ,2.0,2.0)(S,c A ,2.0,2.0)(T,c B ,2.0,2.0)(S,c B ,2.0,2.0) (f)Figure 7: (Color online) Plots showing variations of (d) α , (e) β , and(f) γ phase compositions with λ , in three phase eutectic growth ina model symmetric ternary alloy. A single wavelength of the eutec-tic solids has the configuration αβαγ . The figure legends can beinterpreted in the same way as described in the caption of Fig. 1. lamellar eutectics for a generic multi-component, multi-phase alloy. While our work bears similarities to the re-cently published work of Senninger and Voorhees [37] whichis particular for two-phase growth, our work gives a genericprescription for treating any given multi-phase, multi-componentalloy and in this respect can be seen as an extension of theprevious work in [19]. A principal point in our theoreticalcalculations is that we treat the multi-variant and invari-ant eutectic reactions alike, by expressing the boundarylayer compositions as functions of the respective state vari-ables, which for our derivation are the diffusion potentials,the phase fractions and the undercooling. This allows us tosolve the system of equations self-consistently for the un-dercoolings, phase fractions and the phase compositionsalong with the boundary layer compositions irrespectiveof the degrees of freedom in the system. Our derivation,thus unifies the method of theoretical calculations of theJackson-Hunt type for any given multi-variant/invarianteutectic growth.We also perform phase-field simulations to corroborateour theoretical predictions and they are found to be in rea-sonably good match with each other where we investigatethe case of monovariant two-phase and three-phase invari-ant growth. Both the phase-field and the analytical the-ory exhibit the same trends in the variation of interfacialundercooling, solid phase volume fractions and composi-tions with change in lamellar width. It is important tohighlight that the numerical differences in the predictionsobtained from the two techniques are attributed to the as-sumption of a flat interface in the analytical calculations.Particularly, asymmetry in the interfacial shapes broughtabout either by strongly different phase fractions or inter-facial energies result in asymmetric discrepancies betweenthe theoretical predictions and the phase-field predictions.Thus we expect the match between the two methods w.r.tthe predictions of the phase fractions and phase compo-sitions to be the best for situations where the interfacialshapes of the phases are similar. Furthermore, we note in18assing that while the theoretical expressions are genericin the spirit in which they have been derived, the exis-tence of a steady-state lamellar growth mode needs to beascertained through either phase-field simulations or ex-periments, before applying the results.Secondly our study clearly highlights the importanceof understanding the dependence of phase fractions on thediffusivity matrices. Changes in volume fractions can beassociated with microstructural changes during two-phasegrowth(lamellar to rod), and many further possibilitiesduring three-phase growth as seen in [43, 44]. Therefore,dependence of the volume fractions on the diffusivity ma-trices needs to be accounted for in order to derive a betterunderstanding of pattern formation during bulk eutecticgrowth in multi-phase systems. 7. Appendix The expression of the free energy of a solid phase( p ) isgiven by, f p ( c p , T ) = K − ,K − (cid:88) i =1 ,j =1 ,i< = j A pij c pi c pj + K − (cid:88) j =1 B pj ( T ) c pj + D p ( T ) , (64)and that of the liquid is given by, f l ( c l ) = K − ,K − (cid:88) i =1 ,j =1 ,i< = j A lij c li c lj , (65)where the constants A p,lij are set to obtain (cid:20) ∂c p,l ∂µ (cid:21) matri-ces while B pj and D p are determined from the equilibriumbetween solid and the liquid phases at a particular tem-perature as described in [45]. References [1] H. Kerr, A. Plumtree, W. Winegard, Structure of tin-lead-cadmium eutectic, Journal of the Institute of Metals 93 (2)(1964) 63.[2] D. Cooksey, A. Hellawell, The microstructures of ternary eutec-tic alloys in the systems cd-sn—pb, in, tl–, al-cu—mg, zn, ag–,and zn-sn-pb, J INST METALS 95 (6) (1967) 183–187. [3] H.-Q. Bao, F. Durand, Morphologie eutectique dans le syst`emecd-pb-sn, Journal of Crystal Growth 15 (4) (1972) 291–295.[4] M. Rinaldi, R. 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Ueyama, I. Ohnaka, Effects of mn and co addi-tion on morphology of unidirectionally solidified fesi2 eutecticalloys, Materials Science and Engineering: A 208 (1) (1996) 101– 107.[22] G. Garmong, The directional solidification of al- cu- mg mono-variant alloys, Metallurgical and Materials Transactions B 2 (8)(1971) 2025–2030.[23] P. Fehrenbach, H. Kerr, P. Niessen, Unidirectional solidifica-tion of monovariant eutectic cu-mg-ni alloys: I. planar interfacestability criterion, Journal of Crystal Growth 16 (3) (1972) 209–214.[24] F. Schnake, G. Varchavsky, Microstructures of unidirection-ally solidified al-rich cu-si alloys, Materials Characterization39 (2–5) (1997) 345 – 359.[25] S. Raj, I. Locci, Microstructural characterization of adirectionally-solidified ni–33 (at.%) al–31cr–3mo eutectic alloyas a function of withdrawal rate, Intermetallics 9 (3) (2001)217–227.[26] W. W. Mullins, R. Sekerka, Stability of a planar interface duringsolidification of a dilute binary alloy, Journal of applied physics35 (2) (1964) 444–451.[27] S. Akamatsu, G. Faivre, Traveling waves, two-phase fingers, andeutectic colonies in thin-sample directional solidification of aternary eutectic alloy, Physical Review E 61 (4) (2000) 3757.[28] S. Akamatsu, M. Perrut, S. Bottin-Rousseau, G. Faivre, Spi-ral two-phase dendrites, Physical review letters 104 (5) (2010)056101.[29] M. Plapp, A. Karma, Eutectic colony formation: A stabilityanalysis, Physical Review E 60 (6) (1999) 6865.[30] M. Plapp, A. Karma, Eutectic colony formation: A phase-fieldstudy, Physical Review E 66 (6) (2002) 061608.[31] D. G. Mccartney, J. D. Hunt, R. M. Jordan, The structuresexpected in a simple ternary eutectic system: Part 1. theory,Metallurgical Transactions A 11 (8) (1980) 1243–1249.[32] L. Donaghey, W. Tiller, On the diffusion of solute during theeutectoid and eutectic transformations, part i, Materials Scienceand Engineering 3 (4) (1968) 231 – 239.[33] A. Ludwig, S. Leibbrandt, Generalised ‘jackson–hunt’ model foreutectic solidification at low and large peclet numbers and any binary eutectic phase diagram, Materials Science and Engineer-ing: A 375–377 (2004) 540 – 546.[34] P. Magnin, R. Trivedi, Eutectic growth: A modification of thejackson and hunt theory, Acta metallurgica et materialia 39 (4)(1991) 453–467.[35] J. De Wilde, L. Froyen, V. T. Witusiewicz, U. Hecht, Two-phaseplanar and regular lamellar coupled growth along the univarianteutectic reaction in ternary alloys: An analytical approach andapplication to the al–cu–ag system, Journal of Applied Physics97 (11).[36] A. V. Catalina, P. W. Voorhees, R. K. Huff, A. L. Genau, Amodel for eutectic growth in multicomponent alloys, IOP Con-ference Series: Materials Science and Engineering 84 (1) (2015)012085.[37] O. Senninger, P. W. Voorhees, Eutectic growth in two-phasemulticomponent alloys, Acta Materialia 116 (2016) 308 – 320.[38] A. Choudhury, B. Nestler, Grand-potential formulationfor multicomponent phase transformations combined withthin-interface asymptotics of the double-obstacle potential,Phys.Rev.E 85 (2011) 021602.[39] A. Choudhury, Quantitative phase-field model for phase trans-formations in multi-component alloys, Vol. Band 21, KIT Sci-entific Publishing, 2012.[40] R. Folch, M. Plapp, Quantitative phase-field modeling of two-phase growth, Phys. Rev. E 72 (2005) 011602.[41] A. Karma, A. Sarkissian, Morphological instabilities of lamellareutectics, Metallurgical and Materials Transactions A 27 (3)(1996) 635–656.[42] S. Bottin-Rousseau, M. S¸erefo˘glu, S. Y¨ucet¨urk, G. Faivre,S. Akamatsu, Stability of three-phase ternary-eutectic growthpatterns in thin sample, Acta Materialia 109 (2016) 259–266.[43] A. Choudhury, Y. C. Yabansu, S. R. Kalidindi, A. Dennstedt,Quantification and classification of microstructures in ternaryeutectic alloys using 2-point spatial correlations and principalcomponent analyses, Acta Materialia 110 (2016) 131–141.[44] A. Lahiri, A. Choudhury, Effect of surface energy anisotropy onthe stability of growth fronts in multiphase alloys, Transactionsof the Indian Institute of Metals 68 (6) (2015) 1053–1057.[45] A. Choudhury, Pattern-formation during self-organization inthree-phase eutectic solidification, Transactions of the IndianInstitute of Metals 68 (6) (2015) 1137–1143.(al 2 cu)] planar growth and destabilisation along the univarianteutectic reaction in al–cu–ag alloys, Scripta materialia 51 (6)(2004) 533–538.[21] I. Yamauchi, S. Ueyama, I. Ohnaka, Effects of mn and co addi-tion on morphology of unidirectionally solidified fesi2 eutecticalloys, Materials Science and Engineering: A 208 (1) (1996) 101– 107.[22] G. Garmong, The directional solidification of al- cu- mg mono-variant alloys, Metallurgical and Materials Transactions B 2 (8)(1971) 2025–2030.[23] P. Fehrenbach, H. Kerr, P. Niessen, Unidirectional solidifica-tion of monovariant eutectic cu-mg-ni alloys: I. planar interfacestability criterion, Journal of Crystal Growth 16 (3) (1972) 209–214.[24] F. Schnake, G. Varchavsky, Microstructures of unidirection-ally solidified al-rich cu-si alloys, Materials Characterization39 (2–5) (1997) 345 – 359.[25] S. Raj, I. Locci, Microstructural characterization of adirectionally-solidified ni–33 (at.%) al–31cr–3mo eutectic alloyas a function of withdrawal rate, Intermetallics 9 (3) (2001)217–227.[26] W. W. Mullins, R. Sekerka, Stability of a planar interface duringsolidification of a dilute binary alloy, Journal of applied physics35 (2) (1964) 444–451.[27] S. Akamatsu, G. Faivre, Traveling waves, two-phase fingers, andeutectic colonies in thin-sample directional solidification of aternary eutectic alloy, Physical Review E 61 (4) (2000) 3757.[28] S. Akamatsu, M. Perrut, S. Bottin-Rousseau, G. Faivre, Spi-ral two-phase dendrites, Physical review letters 104 (5) (2010)056101.[29] M. Plapp, A. Karma, Eutectic colony formation: A stabilityanalysis, Physical Review E 60 (6) (1999) 6865.[30] M. Plapp, A. Karma, Eutectic colony formation: A phase-fieldstudy, Physical Review E 66 (6) (2002) 061608.[31] D. G. Mccartney, J. D. Hunt, R. M. Jordan, The structuresexpected in a simple ternary eutectic system: Part 1. theory,Metallurgical Transactions A 11 (8) (1980) 1243–1249.[32] L. Donaghey, W. Tiller, On the diffusion of solute during theeutectoid and eutectic transformations, part i, Materials Scienceand Engineering 3 (4) (1968) 231 – 239.[33] A. Ludwig, S. Leibbrandt, Generalised ‘jackson–hunt’ model foreutectic solidification at low and large peclet numbers and any binary eutectic phase diagram, Materials Science and Engineer-ing: A 375–377 (2004) 540 – 546.[34] P. Magnin, R. Trivedi, Eutectic growth: A modification of thejackson and hunt theory, Acta metallurgica et materialia 39 (4)(1991) 453–467.[35] J. De Wilde, L. Froyen, V. T. Witusiewicz, U. Hecht, Two-phaseplanar and regular lamellar coupled growth along the univarianteutectic reaction in ternary alloys: An analytical approach andapplication to the al–cu–ag system, Journal of Applied Physics97 (11).[36] A. V. Catalina, P. W. Voorhees, R. K. Huff, A. L. Genau, Amodel for eutectic growth in multicomponent alloys, IOP Con-ference Series: Materials Science and Engineering 84 (1) (2015)012085.[37] O. Senninger, P. W. Voorhees, Eutectic growth in two-phasemulticomponent alloys, Acta Materialia 116 (2016) 308 – 320.[38] A. Choudhury, B. Nestler, Grand-potential formulationfor multicomponent phase transformations combined withthin-interface asymptotics of the double-obstacle potential,Phys.Rev.E 85 (2011) 021602.[39] A. Choudhury, Quantitative phase-field model for phase trans-formations in multi-component alloys, Vol. Band 21, KIT Sci-entific Publishing, 2012.[40] R. Folch, M. Plapp, Quantitative phase-field modeling of two-phase growth, Phys. Rev. E 72 (2005) 011602.[41] A. Karma, A. Sarkissian, Morphological instabilities of lamellareutectics, Metallurgical and Materials Transactions A 27 (3)(1996) 635–656.[42] S. Bottin-Rousseau, M. S¸erefo˘glu, S. Y¨ucet¨urk, G. Faivre,S. Akamatsu, Stability of three-phase ternary-eutectic growthpatterns in thin sample, Acta Materialia 109 (2016) 259–266.[43] A. Choudhury, Y. C. Yabansu, S. R. Kalidindi, A. Dennstedt,Quantification and classification of microstructures in ternaryeutectic alloys using 2-point spatial correlations and principalcomponent analyses, Acta Materialia 110 (2016) 131–141.[44] A. Lahiri, A. Choudhury, Effect of surface energy anisotropy onthe stability of growth fronts in multiphase alloys, Transactionsof the Indian Institute of Metals 68 (6) (2015) 1053–1057.[45] A. Choudhury, Pattern-formation during self-organization inthree-phase eutectic solidification, Transactions of the IndianInstitute of Metals 68 (6) (2015) 1137–1143.