Multigrain and Multiphase Mathematical Model for Equiaxed Solidification. Marcelo Aquino Martorano, Davi Teves Aguiar & Juan Marcelo Rojas Arango

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1 Multigrain and Multiphase Mathematical Model for Equiaxed Solidification Marcelo Aquino Martorano, Davi Teves Aguiar & Juan Marcelo Rojas Arango Metallurgical and Materials Transactions A ISSN Volume 46 Number 1 Metall and Mat Trans A (2015) 46: DOI /s

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3 Multigrain and Multiphase Mathematical Model for Equiaxed Solidification MARCELO AQUINO MARTORANO, DAVI TEVES AGUIAR, and JUAN MARCELO ROJAS ARANGO A deterministic multigrain and multiphase model of equiaxed solidification of binary alloys is proposed, implemented, and analyzed. An important feature of the present model is the creation of classes of dendritic and globulitic grains according to their instantaneous sizes during solidification. Globulitic and dendritic grain growth, coarsening of secondary dendrite arms, distribution of nucleation undercoolings, and equiaxed eutectic growth are consistently included in the model equations. Important model assumptions are uniform temperature, negligible liquid convection, and negligible grain movement. Calculated cooling curves, solid fraction evolution, average grain sizes, and eutectic fractions agree well with predictions of previous models for dendritic and globulitic equiaxed grains. Predicted grain sizes decrease with an increase in cooling rate for an Al-2.12 pct Cu alloy and with an increase in Si concentration up to 3 pct for Al-Si alloys, agreeing quantitatively with experimental results. Simulations for an Al-7 pct Si alloy predict that an increase in grain size correlates with an increase in the magnitude of the recalescence observed in cooling curves. These calculations agree well with experimental results when the transition from a globulitic to a dendritic morphology occurs in the model before the minimum temperature of recalescence is reached. DOI: /s Ó The Minerals, Metals & Materials Society and ASM International 2014 I. INTRODUCTION CASTINGS with fine equiaxed grains are frequently pursued by the casting industry. Mathematical models have been proposed to elucidate the phenomena underlying the solidification of equiaxed grains. Oldfield [1] presented one of the first deterministic models of equiaxed solidification, especially developed for the solidification of eutectic cells that were assumed as solid spheres. Rappaz et al. [2] extended Oldfield s [1] model considering that the critical undercooling for heterogeneous nucleation on substrate particles within the melt follows a normal distribution, predicting the final average density or size of eutectic cells. Maxwell and Hellawell [3] and later Greer et al. [4] proposed models in which grains of the primary phase were solid spheres, i.e., globulitic, subdivided into classes of different grain sizes, nucleated at different undercoolings. These models are valid before globulitic grains change into dendritic and before interactions occur between neighboring MARCELO AQUINO MARTORANO, Associate Professor, is with the Department of Metallurgical and Materials Engineering, University of São Paulo, Av. Prof. Mello Moraes, 2463, São Paulo, SP , Brazil. Contact martoran@usp.br DAVI TEVES AGUIAR, Materials Engineer, is with the Materials Department, Petróleo Brasileiro S.A., Av. Henrique Valadares, 28, Rio de Janeiro, RJ Brazil. JUAN MARCELO ROJAS ARANGO, Professor, formerly with the Department of Metallurgical and Materials Engineering, University of Sa o Paulo, is now with the Department of Metallurgical and Materials Engineering, University of Antioquia, Medellín, Colombia. Manuscript submitted February 6, Article published online October 25, 2014 grains. These conditions usually exist until the end of the nucleation period, when the final average grain size can be predicted. Rappaz and The voz [5,6] developed the first deterministic model for the solidification of equiaxed dendritic grains. They assumed that around each grain an imaginary spherical envelope existed, including internal solid anquid, rather than only solid as in previous models, predicting grain sizes in agreement with experiments. [7,8] The undercooling for grain nucleation was defined by a normal distribution, and all grains were locally assumed to have the same size, rather than being subdivided into classes of sizes. The composition of the external liquid surrounding grain envelopes was assumed constant and unaffected by solute rejection from grain envelopes, disregarding an important source of interaction between grains. Wang and Beckermann [9,10] used the volume averaging technique to calculate the average solute concentration in the external liquid. They also adopted an effective diffusion length to obtain the solute fluxes into the solid and external liquid. The final grain size could not be predicted, however, because an instantaneous nucleation model was adopted. Gandin et al. [11] presented a deterministic mathematical model to simulate the equiaxed dendritic solidification of one grain in a levitated spherical droplet of a binary alloy. The model has many of the features of previous models, including the assumption of instantaneous nucleation, precluding its application to predict the final equiaxed grain size in castings. However, the eutectic solidification was modeled more carefully than in the preceding models. Wu and Ludwig [12] developed a comprehensive METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 46A, JANUARY

4 model of equiaxed solidification including liquid convection, grain movement, globulitic and dendritic grain morphologies, and the variation in the external liquid composition. As in the model by Rappaz and The voz, [5,6] they assumed a normal distribution of nucleation undercoolings and a locally uniform grain size. Eutectic fractions and grain sizes were calculated with the model, but only eutectic fractions were compared with experimental measurements, showing good agreement. The objective of the present work was to propose, implement, and validate a mathematical model for the equiaxed solidification of binary alloys of eutectic systems. Uniform temperature is assumed, as in previous models. [1,3,4,9,11] Nevertheless, several important effects that have never been considered before in the same model are included coherently and consistently. The temperature range in which nucleation of primary grains occurs is subdivided into smaller intervals. All grains that nucleate within the same interval are assumed to be of equal size, have the same growth velocity, and belong to the same grain class. Thus, the grain size and growth velocity can be calculated individually for each class, as in the models by Maxwell and Hellawell [3] and Greer et al. [4] This enables the consideration of different growth velocities for globulitic and dendritic grains growing simultaneously. The solute concentration in the external liquid is computed considering solute fluxes out of all grain envelopes of different grain classes. In eutectic solidification, nucleation and growth undercoolings are considered, as done by Gandin et al. [11] The proposed model can be used to predict the average equiaxed grain size, the distribution of grain sizes, the cooling curve during the complete primary and eutectic solidification, and the volume fraction of eutectic. These predictions are carefully compared with results available in the literature from previous models and from experiments. II. MATHEMATICAL MODEL OF EQUIAXED SOLIDIFICATION A. Simplifying Hypotheses A deterministic mathematical model for the solidification of equiaxed grains is proposed in the present work. The simplifying hypotheses are (1) the alloy is binary with a eutectic reaction, grains are motionless and equiaxed, and densities of all phases are equal and constant; (2) liquid convection is neglected; (3) temperature is uniform and varies with time, limiting the model application to relatively small castings or small parts of castings, representing a first step toward the development of a multidimensional model; (4) grains of different sizes, which nucleated at different undercoolings, are divided into N classes of index i, each class consisting of identical grains of an average size for that class, [3,4] as shown in Figure 1(a); (5) each grain is enclosed with an imaginary spherical envelope [5,6, 9] and can be dendritic or globulitic (Figure 1(b)); (6) before the eutectic reaction, inside the envelopes of a grain class i there can be the interdendritic liquid (di) and the primary solid (si), and all envelopes are immersed in the same external liquid (l); (7) the primary solid has no contact with the external liquid when grains are dendritic, but is completely in contact with this liquid when grains are globulitic; (8) the interdendritic liquid has uniform composition; (9) in the external liquid, there are randomly distributed substrate particles on which at most one grain of the primary solid can nucleate when the critical undercooling defined by a distribution is reached; (10) all eutectic cells (e) nucleate in the form of solid spheres randomly located throughout the whole liquid (interdendritic and external liquid), as observed experimentally in hypoeutectic Al-Si alloys, [13 16] at a predefined undercooling in relation to the eutectic temperature, stopping primary solidification; (11) the solid eutectic grows according to a kinetic law into the interdendritic and external liquid, which is denoted as the external eutectic (ee), with an average composition equal to that of the liquid; and (12) the eutectic does not grow into or out of the primary solid, i.e., eutectic dissolution or precipitation at the interface with the primary solid is neglected. B. Main Equations of the Model The entire model domain, with uniform temperature and grains of different sizes, is defined as the representative elementary volume (REV) shown in Figure 1(a). The main equations are derived from the principles of mass, energy, and species conservation applied to this REV. Application of energy conservation to the REV gives the following equation used in the present and many other models [2,5,9,17,18] dt c p L d ð e s þ e e Þ ¼ A 0 Q; where T is the temperature; t is the time; L is the volumetric latent heat; e s and e e are the total volume fractions of primary solid and eutectic, respectively, within the REV; c p =[e s c ps + e e c pe +(1 e s e e )c pl ]is the average specific heat, where c ps, c pl, and c pe are the volumetric specific heats, respectively, of primary solid, liquid, and eutectic constituents; and A 0 are the volume and boundary area of the REV; and Q is the average heat flux out of the REV, across its boundary. The composition of the liquid and solid constituents is generally non-uniform, requiring the definition of an intrinsic volume average of the solute mass fraction in constituent k, namely hc k i k ¼ ð1=v k Þ R V k C k dv, [19] where C k is the field of solute mass fraction within k, and V k is the volume of k in the REV. The volume fraction of k is e k = V k /. The total primary solid in all grains of class i is as an example of one type of constituent of the present model, implying that k = si. To complete the main part of the model, the following equations derived in the Appendix are added to Eq. [1] de si ð1 KÞC liq ¼ e dc liq gi e si þ ð1 KÞC liq S lsi w lsi S sdi þ D s KC liq hc si i si S ldi þ Dl C liq hc l d sdi for t ni <t<t ne ½2Š ½1Š 378 VOLUME 46A, JANUARY 2015 METALLURGICAL AND MATERIALS TRANSACTIONS A

5 d e si hc si i si de si ¼ KC liq S lsi w lsi þ KC liq;gi S lsi w lsi S sdi þ D s KC liq hc si i si d sdi þ D s S lsi d lsi KC liq;gi hc si i si for tni <t<t ne ½8Š Fig. 1 Schematic view of (a) the solidification domain (representative elementary volume REV) with grains of different sizes grouped in different classes (i, i + 1, and i + 2), immersed in the external liquid; and of (b) the details of dendritic, globulitic, and eutectic grains with an indication of all possible types of constituents (si, di, l, e, and ee) and their interfacial areas (A sdi, A ldi, A lsi, A die, A sie, A le ). de gi ¼ S ldi w ldi þ S lsi w lsi for t ni <t<t ne ½3Š de e ¼ S le þ XN S die!w e for t t ne ½4Š de ee ¼ S le w e for t t ne ½5Š e s ¼ XN e si e l ¼ 1 e ee XN e gi ½6Š ½7Š d e l hc l ¼ C liq de l þ XN þ D l þ D l X N X N S ldi S lsi C liq C liq;gi Slsi w lsi þ! C liq hc l C liq;gi hc l for t<t ne where N is the number of existing classes of grain sizes, which increases during solidification as new grains nucleate; e si and e gi are the volume fractions of primary solid and grains, respectively, in class i; P N e gi is the total volume fraction of grains; e l is the volume fraction of the external liquid; e ee is the volume fraction of the external eutectic, i.e., the eutectic outside grain envelopes; t ne is the time of nucleation of eutectic grains and t ni is the time of nucleation of primary solid grains of class i (Section II C 1); C liq and C liq,gi (Section II C 2) are, respectively, the concentration of the liquidus line of the phase diagram for temperature T and the concentration of the liquidus line corrected for the solid liquid interface curvature of the globulitic grains in class i; hc si i si and hc l are, respectively, the average composition of the primary solid of grains in class i and the average composition of the external liquid; S die, S le, S ldi, S lsi, S sdi (Section II C 5) are the area concentrations (area per unit volume) of the interfaces between the constituents indicated by the subscripts; w e, w ldi, and w lsi (Section II C 3) are the average normal growth velocities of, respectively, the eutectic, dendritic, and globulitic grains in class i; d sdi and d lsi (Section II C 4) are the thicknesses of the effective diffusion layers in the primary solid belonging to class i, at its interface with, respectively, the interdendritic liquid (dendritic grains) and the external liquid (globulitic grains); (Section II C 4) is the thickness of the effective diffusion layer in the external liquid, at its interface with the dendritic or globulitic grain envelopes of class i; K is the solute partition coefficient; D s and D l are the diffusion coefficients of solute in the primary solid anquid, respectively. The initial conditions are T = T 0 and hc l i l ¼ C 0 at t = 0, where T 0 and C 0 are the initial temperature (liquidus temperature of the alloy) and solute concentration (average alloy composition), respectively; ; ½9Š METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 46A, JANUARY

6 e gi = e si = 0 and hc si i si ¼ KhC l at t = t ni ; and e e = e ee =0at t = t ne. Equations [1] through [9] are solved for nine main unknowns: T, e s, e si, e gi, e l, e e, e ee, hc si i si,andhc l. The remaining unknowns are obtained from microscopic models described in the next subsection. Equation [3] is used to calculate the volume fraction of all grains in class i. The first and second terms on the right-hand side represent dendritic and globulitic grain growth, respectively. Equation [2] is used to calculate the fraction of primary solid in all grains of class i. When S lsi = 0, these equations (Eqs. [2] and [3]) reduce to those necessary to simulate dendritic grains, whereas when e gi = e si and S sdi = S ldi = 0, they automatically reduce to those for globulitic grains, namely, de gi = ¼ de si = ¼ S lsi w lsi. The first and second terms on the right-hand side of Eq. [2] represent the increase in primary solid fraction due to cooling and globulitic growth, respectively. The third and fourth terms represent the effect of back-diffusion into the solid and solute transfer into the external liquid, respectively, on the increase of solid fraction. The average composition of the primary solid of the grains in class i is obtained from Eq. [8]. The first term on its right-hand side accounts for the formation of a new solid layer at the solid liquid interface of dendritic grains, while the second is the solid layer formation on globulitic grains. The two last terms are the effect of back-diffusion fluxes into dendritic and globulitic grains, respectively. The average composition of the external liquid is obtained from Eq. [9]. The first two terms on the right-hand side of this equation are the effects of envelope growth into the external liquid, the second term being a correction for the presence of globulitic grains. The last two terms are the effects of solute transferred from dendritic and globulitic grains, respectively, into the external liquid. The present main model equations reduce exactly to those of Wang and Beckermann s [10] model when (a) globulitic growth is not considered (S lsi = 0); (b) all grains nucleate at the same undercooling (instantaneous nucleation) and, therefore, have the same size (N = 1); (c) the kinetic equations for eutectic growth, Eqs. [4]and [5], are not used; and (d) the correction for grain impingement is not applied. When only globulitic grains are considered (S sdi = S ldi = 0 and e gi = e si ), Eq. [2] reduces to Eq. [3], and these equations are analogous to the one used by Greer et al., [4] who neither calculated the average compositions of the solid anquid nor the eutectic volume fraction. When only eutectic solidification is considered, only Eqs. [1] and [4] are important. In this case, the equations are equivalent to those adopted by Rappaz et al. [2] and Oldfield [1] to model random nucleation and growth of spherical eutectic cells. C. Microscopic Models 1. Grain nucleation Grains of the primary phase nucleate only at the external liquid, because the interdendritic liquid cannot be undercooled in the present model. The undercooling of the external liquid is defined by DT = T f + m l hc l T, where T f is the melting point of the pure metal and m l is the slope of the liquidus line, assumed straight. When this undercooling reaches a predefined value DT ni for the first time (t ni ), n i grains per unit volume nucleate instantaneously and randomly in the external liquid, creating the class i of grains. The number density of nucleated grains is n i ¼ e l n ext i ½10Š n ext i ¼ DT ni þdt c =2 DT ni DT c =2 dn ddt ddt; ½11Š where n ext i is the extended number density of grain nuclei in class i, which includes the grains that would have nucleated in the liquid if no other grain existed ( phantom nuclei); e l is the volume fraction of external liquid at the moment of nucleation, t ni ; DT ni =(i 0.5)DT c is the undercooling of the external liquid at which all grains in class i nucleate; DT c = DT nucl /N max is the undercooling interval of integration to define the number density of grains in each class; DT nucl is the undercooling range in which all nucleation of the primary phase is expected to occur, which can be at most the difference between the liquidus and the eutectic temperature; N max is the maximum number of possible grain classes, used to subdivide DT nucl ; and dn/ddt is the density of the distribution of nucleation undercoolings for the grains of the primary solid. The undercooling distribution can be given by the instantaneous nucleation, normal, [2] or log-normal models. [4] The final number density of primary solid grains is n ¼ P N n i, which is related below to the final average grain size ( l) measured by the mean linear intercept method at a plane section through a sample volume [4] l ¼ ð0:5=nþ 1=3 : ½12Š When the undercooling in relation to the eutectic temperature, T e T (where T e is the eutectic temperature), reaches a predefined undercooling DT ne, the time is t ne and the number density of eutectic grains that nucleate instantaneously and randomly in the total liquid (interdendritic plus external liquid) is given by n e ¼ ð1 e s Þn ext e ; ½13Š where n ext e is the extended number density of eutectic cells and (1 e s ) is the volume fraction of total liquid at the moment of nucleation, t ne. 2. Local equilibrium and curvature at the solid liquid interface Local thermodynamic equilibrium is assumed at any solid liquid interface, implying that the concentration in the liquid adjacent to this interface can be calculated for dendritic grains as follows: C liq ¼ T T f ½14Š m l and for globulitic grains, considering the curvature of the solid liquid interface, as given below 380 VOLUME 46A, JANUARY 2015 METALLURGICAL AND MATERIALS TRANSACTIONS A

7 C liq;gi ¼ T T f C þ 2 m l m l R ext ; ½15Š i where C is the Gibbs Thomson coefficient and R ext i is the curvature radius of the solid liquid interface of globulitic grains in class i. This curvature radius is assumed to be equal to the extended radius of grains in class i, i.e., R ext i, which is the radius that would have been reached if no mechanical blocking by neighboring grains had occurred. The portion of a spherical grain interface that is not mechanically blocked is considered ext to keep its curvature radius. The extended radius R i can be calculated from dr ext i ¼ w ldi for dendritic or ¼ w lsi for globulitic grains; ½16Š where at nucleation (t = t ni ), the initial grain radius is R i ext =2C/DT ni (critical radius for nucleation) if 2C/ DT ni 10 4 m, otherwise R i ext =10 4 m to avoid unrealistic large initial grain sizes. The extended radius of the spherical eutectic cells (R e ext ) is calculated from dr ext e ¼ w e ; ½17Š where at t = t ne, the radius is R ext e =10 6 m, an arbitrarily small value. 3. Grain growth velocity Primary solid grains of class i nucleate as solid spheres, i.e., with a globulitic morphology, and are forced to remain globulitic until R i ext is seven times the critical radius for nucleation (2C/DT) at the instantaneous undercooling of the external liquid, DT, according to the stability criterion proposed by Mullins and Sekerka. [20,21] When R i ext is larger than this limit, which decreases as the liquid cools down, grains of class i become dendritic if w ldi > w lsi or remain globulitic otherwise, as suggested by Wu and Ludwig. [12] However, dendritic grains might become globulitic again, as shown by Martorano et al., [22] if w ldi < w lsi and if, after solving Eqs. [2] and [3], e si > e gi. In this case, e gi is set equal to e si, since S ldi = 0, making Eqs. [2] and [3] identical. As adopted by several authors, [7,9] the average growth velocity of dendritic grain envelopes, w ldi, is assumed equal to the growth velocity of the tip of primary dendrite arms and is given below [9,23] w ldi ¼ D lm l ðk 1ÞC liq p 2 C I 1 V ðxþ 2 ½18Š I 1 X 1:195 V ðxþ ¼0:4567 ½19Š 1 X X ¼ C liq hc l C liq ð1 KÞ ; ½20Š where I 1 V is the inverse of the Ivantsov function and X is a dimensionless undercooling. Since X depends on C liq and hc l, which are uniform in the REV at a given time, then w ldi has the same value for the dendritic grains of all classes. An equation for the growth velocity of the globulitic grain envelopes, w lsi, was derived by applying Eq. [A7] from the Appendix to the solid liquid interface (A k- j = A lsi ), considering the curvature correction and the local thermodynamic equilibrium, and substituting Eqs. [A4] and [A5], giving 1 D s w lsi ¼ KC liq;gi hc si i si ð1 KÞC liq;gi þ D l d lsi C liq;gi hc l : ½21Š In this model, w lsi is affected by the solute diffusion fluxes into the solid (back-diffusion) anquid, consistently calculated by the effective diffusion layers (d lsi and defined in Section II C 4) adopted for the fluxes in the conservation Eqs. [2], [8], and [9]. Finally, the average growth velocity of eutectic grains, w e, is given by w e ¼ A e ðt e TÞ 2 ; ½22Š where the constant A e can be obtained from empirical or theoretical models of eutectic growth. 4. Effective diffusion lengths The effective diffusion lengths defined at the solid or liquid adjacent to interfaces are used to calculate the diffusion solute fluxes from one constituent to another. In the dendritic solid of grains in class i, the diffusion length at the solid si adjacent to the interface with the interdendritic liquid is given below assuming plate-like secondary dendrite arms [9] d sdi ¼ e si k i e gi 6 ; ½23Š where k i is the average spacing between secondary arms in grains of class i, which increases with time by coarsening during solidification according to the following equation [24] k i ¼ k i0 þ At ð t ni Þ a ; ½24Š where k i0 is the initial spacing; and A and a are constants depending on the alloy composition. Since in practical terms the value of k i0 significantly affects k i only shortly after nucleation, it was adopted as the critical radius of nucleation (k i0 =2C/DT ni ). For the globulitic grains in class i, a similar equation was derived for the diffusion layer in the solid, d lsi, after assuming that these spherical grains are located at the center of a unit cell of radius R fi. In this case, d lsi = R i / 5, where R i is the radius of the grain envelopes in class i. Because of grain impingement, R i is not equal to the extended radius, R ext i, calculated by Eq. [16]. As an approximation, R i is assumed to be the radius of a set of equal sized spheres occupying the total volume of grains in class i, resulting in METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 46A, JANUARY

8 R i ¼ 3e 1=3 gi ½25Š 4pn i The precise calculation of the thickness of the effective diffusion layer in the external liquid,, at the interface with the envelopes of dendritic or globulitic grains of class i, is complex. It should account for grains of different sizes, the formation of grain clusters after grain impingement, and the interaction between the solute fields of neighboring grains. As an approximation, the quasi-steady-state model proposed by Martorano et al., [22] which was developed for uniformly spaced, equal sized spheres was adjusted to consider the different grain sizes of the present model. This model has the advantage of reducing to the correct limits for relatively small or large envelope growth velocities and radii. To adjust the model, the unit cell size, R fi, is calculated considering that unit cells are spherical and occupy all the REV, which means their total volume fraction equals one. Also, the volume fraction of the unit cells of grains in class i is assumed to be equal to the fraction of grains in class i, relative to the total fraction of grains, i.e., e gi. PN e gi. Considering these assumptions, the cell size can be calculated from! 3 1=3 X N 1=3: R fi ¼ e gi, e gi ½26Š 4pn i When all grains are of the same size, N = 1 and this equation reduces to that used by Martorano et al. [22] The final equation to calculate the thickness of the effective diffusion layer becomes R i ¼ " R i R 3 fi R3 i R2 i Pe i þ R2 i þ Pe i R 3 fi R i R fi R i Pe i! Pe 2 R3 fi i R i e Pe i R fi R i 1 þ R2 i Pe 2 R 2 fi i R Iv fi Pei R i Pe i R fi R i e Pe i R fi R 1 i 13 Iv ð Pe iþ A5; Pe i ½27Š where Pe i ¼ R i i is the Peclet number for grains in class i and dr ext i / is calculated from Eq. [16]. For Pe i < 10 5, the following equation is more convenient [22] 0 ¼ 1 3 R 2 R i 2 R fi 1 R2 i A: ½28Š R 3 fi R3 i D l dr ext exact solution proposed by Aaron et al. [25] to calculate the growth velocity of a globulitic grain growing in an infinite medium of bulk concentration C l, in transient conditions. The diffusion length is calculated by d A li (C l C l )/( C l / r) atr = R i, where C l is the solution given by Aaron et al. [25] and r is the radial coordinate of a spherical reference system located at the grain center, resulting in d A li R i ¼ 1 p r ffiffiffiffiffiffiffi! 1=2erfc 2 Pe Pe i i expðpe i Þ ; ½29Š 2 where ercf is the complementary error function. Although this model is transient, the dimensionless diffusion length, d A li /R i, and Pe i are both constant during growth. To finally compare and d A li, Eq. [27] was adjusted for the case of an infinite medium, i.e., for the asymptotic limit of R fi fi, implying that R fi R i and dr R fi D ext, which gives /R i 1 Iv(Pe). This approximation is compared with the results from Eq. [29] in Figure 2, showing very good agreement for relatively low and large Pe i, and discrepancies of at most 20 pct in the intermediate values. Therefore, the diffusion length, used to calculated fluxes and growth velocities of globulitic grains, is consistent with the exact solution for spherical growth. Wu and Ludwig [12] assumed that this length is equal to the envelope radius ( /R i = 1), but Figure 2 shows that this is true only for relatively small growth velocities or envelope radii, i.e., for low Pe i. 5. Interfacial area concentration Grain envelopes are spherical before impingement and, during growth, might impinge on each other The growth velocity of globulitic grains, w lsi, given by Eq. [21], depends on this effective diffusion length. To examine the quality of the adopted model to calculate, a diffusion length equation was obtained from the Fig. 2 The dimensionless effective diffusion length by Martorano et al., [22] /R i, and by Aaron et al., [25] A /R i, as a function of the Peclet number (Pe i ). 382 VOLUME 46A, JANUARY 2015 METALLURGICAL AND MATERIALS TRANSACTIONS A

9 decreasing the specific interfacial area (S ldi and S lsi ) with the external liquid. To consider impingement, the specific area for isolated spherical envelopes (extended specific area) is modified by the Avrami correction, [26] being completely equivalent to the more frequent correction applied to the volume of grains, becoming S ldi ¼ e l n ext i 4p R ext 2 i and Slsi ¼ 0 for dendritic grains or S ldi ¼ 0 and S lsi ¼ e l n ext i 4p R ext 2 ; i for globulitic grains ½30Š where e l represents the Avrami correction. Analogously, the specific interfacial area between spherical eutectic cells 2. and the external liquid is given by S le ¼ e l n ext e 4p R ext e The interfacial area concentration between eutectic grains and the interdendritic liquid of all grains in all classes (used in Eq. [4]) can be calculated using the Avrami correction as P N S 2, die ¼ ð1 e s e l e e Þn ext e 4p R ext e where 1 e s e l e e is the total volume fraction of all interdendritic liquid, considering all grain classes. The concentration of interfacial area between the interdendritic liquid and the primary solid, S sdi, is calculated using the model proposed by Wang and Beckermann, [9] assuming that secondary dendrite arms are parallel plates, giving S sdi =2e gi /k i, whereas for globulitic grains S sdi =0. D. Numerical Solution The model equations were solved numerically for the time evolution of the main variables, namely T, e s, e si, e gi, e l, e e, e ee, hc si i si, and hc l. The differential equations were discretized using the explicit Euler method, implying that, on the right-hand side of the discretized equations, all variables were calculated at the beginning of the numerical time step, Dt. Nevertheless, the time derivatives dc liq /, de si /, and de l / are necessary on the right-hand side of Eqs. [2], [8], and [9], respectively, and these depend on C t+dt liq, e t+dt si, and e t+dt l, which are values at the end of the time step, t + Dt. By choosing a convenient order to solve the discretized equations, the values needed at t + Dt are available when necessary, eliminating iterations. During primary solidification, the strong coupling between Eqs. [1] and [2] can cause oscillations in the numerical solution. Instead of directly solving Eq. [1] to obtain T t+dt, the following combination of Eqs. [1] and [2] (added for all grain classes), [6], and [14] was discretized and solved 0 P N c p L e 1 gi e A dt m l ð1 KÞC liq ¼ XN D s X N S sdi ðs lsi w lsi Þþ KC liq hc si i si ð1 KÞC liq d sdi C liq hc l! X N S ldi þ D l A 0 Q ð1 KÞC liq d li L : ½31Š During one time step, Dt, the following general sequence of events is executed in the numerical algorithm: (a) If the undercooling for eutectic nucleation (DT ne ) is reached for the first time, primary solidification stops, all eutectic cells nucleate and grow during t+dt one time step Dt, with calculations of e e and e t+dt ee using Eqs. [4] and [5], respectively. The temperature T t+dt is calculated from the discretized form of Eq. [1]. (b) If eutectic cells have not yet nucleated and the undercooling of the external liquid, DT, reaches that for the nucleation of grains in class i, DT ni,all grains in this class, n i, nucleate with the size of the critical radius. (c) Primary solidification is then simulated for one time step Dt. To obtain T t+dt, dt/, and dc liq / = (1/m l )dt/, the discretized form of Eq. [31] is solved, but the summations in Eqs. [31] and [9] are calculated first using values at time t. To obtain S lsi, S sdi, and S ldi for the summations and to use Eq. [16] to obtain R ext,t+dt i, the morphology of the grains (dendritic or globulitic) must be known. If R ext;t i 14C=DT, grains are globulitic and w lsi and are calculated with iterations of Eqs. [21] and [27] or [28]. Grains are dendritic when R ext;t i >14C=DT and either e t t gi > e si or simultaneously e gi = e si and w ldi > w lsi ; otherwise, they are t t globulitic. (d) Still in the same time step, the following terms are calculated for all N grain classes: e t+dt si and de si / from Eq. [2]; hc si i si at t + Dt from Eq. [8]; and finally e t+dt gi from Eq. [3]. After obtaining these values for all classes, e s (Eq. [6]) and P N t+dt etþdt gi can be computed. (e) The fraction of external liquid, e l t+dt, is obtained from Eq. [7], allowing de l / to be calculated, which is used in Eq. [9] to obtain hc l at t + Dt. III. ANALYSIS OF THE MATHEMATICAL MODEL The present model was analyzed by comparing its results with those available in the literature. First, comparisons were carried out to show that, in particular conditions, the present model reduces to those previously proposed, representing an extension of these previous models. Next, comparisons with available experimental results were conducted to validate the model. Except for the simulation in the conditions adopted by Gandin et al., [11] the heat flux out of the REV was always calculated as Q ¼V ð 0 =A 0 Þc p _R, where _R is the cooling rate of the liquid immediately before solidification begins. The values adopted for A 0 and are irrelevant, because they cancel out in Eq. [1], and _R is given by the authors or estimated from their results. In this calculation of Q, it is implied that the extraction heat flux remains constant during all simulation. METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 46A, JANUARY

10 A. Comparisons with Previous Models The equiaxed solidification with instantaneous nucleation at the liquidus temperature was simulated after adopting N = 1 and DT ni=1 = 0.3. The results were compared with those presented by Wang and Beckermann [10] for an Al-5 pct Si alloy and by Martorano et al. [22] for an Al-3 pct Cu alloy. In Figure 3(a), good agreement is observed with the dimensionless cooling curves of Wang and Beckermann, [10] showing that recalescence occurs at higher temperatures as the final grain size (R fi ) decreases. After a different set of simulations, the evolution with time of the calculated solid fraction inside grain envelopes, e si /e gi (i = N = 1), is compared with the results of Martorano et al. [22] in Figure 3(b), also showing good agreement. For R fi = 1 mm, grains are initially globulitic (e si /e gi = 1), become dendritic (e si /e gi < 1), and change back to Fig. 3 Results of the present model for different grain sizes (R fi ) compared with (a) the dimensionless cooling curves obtained by Wang and Beckermann (W B) [10] and (b) the internal solid fraction (e si /e gi ) evolution with time calculated by Martorano et al. (MBG) [22]. Fig. 4 Present model results compared with those obtained by Quested and Greer [27] for Al-Ti alloys in three different conditions: (1) the standard condition (n T = m 3, _R = 3.5 K s 1, C 0 = pct Ti); (3) _R is increased to 7 K s 1 and (4) C 0 is increased to pct Ti. Curves of (a) undercooling as a function of time and (b) fraction of solid as a function of undercooling are shown. 384 VOLUME 46A, JANUARY 2015 METALLURGICAL AND MATERIALS TRANSACTIONS A

11 globulitic, whereas they are always globulitic for R fi = 0.1 mm. In these simulations, Martorano et al. [22] imposed constant cooling rate, which was approximated in the present model by an unrealistic low latent heat (L =1Jm 3 ). The relatively small discrepancies observed in Figures 3(a) and (b) are caused mainly by the Avrami correction for grain impingement, which was adopted in the present model, but not in the two previous models. [10,22] In Figure 4(a), undercooling curves are compared with those obtained by Quested and Greer [27] for globulitic grains in Al-Ti alloys. [4] Quested and Greer [27] assumed a log-normal distribution of sizes of particles in the melt, on which heterogeneous nucleation occurred. The nucleation undercooling was calculated using this distribution and the free-growth model. [4] In the present work, the distribution of undercoolings that results from this model was directly calculated from 2! 3 dn ddt ¼ n T p r / DT ffiffiffiffiffn 4C=DT exp ð Þln / pffiffiffi 5; ½32Š 2p 2 r/ where n T is the total number density of substrate particles for heterogeneous nucleation; / 0 and r / are the geometrical mean diameter and standard deviation, respectively, of the log-normal distribution of particle sizes. To define the grain classes (Section II C 1), DT nucl = 0.4 K and N max = 200. Larger N max values did not change simulation results significantly. Quested and Greer [27] adopted / 0 = lm andr / = and defined four cases, three of which are given in Figures 4(a) and (b) in the form of undercooling curves and fraction of solid as a function of undercooling, respectively. From case (1) to (3), the cooling rate was doubled, and from case (1) to (4), the solute content was doubled. Both changes caused a decrease in the minimum temperature of recalescence and consequently increased the number of nucleated grains, decreasing grain size. The agreement between present model calculations and these published results is good, indicating that Eq. [21], used to calculate the velocity of globulitic grains ( w lsi ) as a function of the effective diffusion layer ( ), is consistent with the growth velocity model used by Greer et al. [4] A detailed comparison is given in Table I for the final average grain size, size of recalescence in the cooling curves, solid fraction and time at the minimum temperature before recalescence, also indicating good agreement. In the model used by Quested and Greer, [27] grains were assumed always globulitic. The present model automatically predicted globulitic grains without imposing this restriction. A final comparison with the cooling curve calculated by Gandin et al. [11] is given in Figure 5, showing excellent agreement during both primary solidification (first recalescence) and eutectic solidification (second recalescence). In this case, the solidification of one grain (N = 1) of an Al-14 pct Cu alloy droplet was simulated using the and A 0 of a sphere of m radius (REV) and considering instantaneous nucleation with n ext e = n T = m 3. For the coarsening of secondary dendrite arms (Eq. [24]), A = m 1/3 and a = 1/3, which are in the range of recommended values for Al-Cu alloys. [28] The final calculated average (weighted by grain fractions) spacing between arms was 23.9 lm, in good agreement with the measured value of 25 lm. The total fraction of liquid predicted by the present model at the time of eutectic nucleation, which equals the eutectic fraction at the end of solidification, was 27.5 pct, in agreement with the Fig. 5 Cooling curve calculated with the present model and that presented by Gandin et al. [11] for the solidification of a levitated droplet of Al-14 pct Cu. Table I. Comparison Between Values Calculated with the Present Model (M) and Those Obtained by Quested and Greer [27] (QG) For: Grain Size, Recalescence, Solid Fraction and Time at the Minimum Temperature Before Recalescence (1) (3) (4) Case QG M QG M QG M Grain size, l (lm) Recalescence (K) e s Recalescence t Recalescence METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 46A, JANUARY

12 measured fraction of 29 pct. The eutectic growth constant in Eq. [22] was A e = ms 1 K 2. B. Cooling Rate and Distribution of Grain Sizes The average grain size calculated with the present model as a function of the total cooling rate agrees reasonably well with the experimental results of Eskin et al. [29] for an Al-2.12 pct Cu alloy (Figure 6). The total cooling rate reported by Eskin et al. [29] is the average cooling rate from the beginning to the end of solidification, which was found to be about one-fourth of _R, used to calculate the heat flux (Q). In these simulations, Fig. 6 Grain size ( l) calculated with the present model compared with that measured by Eskin et al. [29] as a function of the total cooling rate ( _R=4) for an Al-2.12 pct Cu alloy. the following normal distribution of nucleation undercoolings was adopted " dn ddt ¼ n T pffiffiffiffiffi exp DT DT 2 # N pffiffi ; ½33Š DT r 2p 2 DTr where DT r and DT N are, respectively, the standard deviation and the average nucleation undercooling. After several tests, the values n T = m 3, DT r = 0.8 K, and DT N = 4.5 K were chosen to give the agreement observed in Figure 6. These values are of the same order of magnitude as those adopted in previous models to simulate solidification without inoculant additions. [30,31] The remaining model parameters adopted in the simulations for the Al-Cu and Al-Si alloys presented in this and next sections are given in Table II. [11,32 36] In Figure 6, as the cooling rate increases to _R=4 10 K s 1, the calculated grain size decreases to the minimum possible value, i.e., l min = (0.5/n T ) 1/3. For all cooling rates, the model predicted that, at the minimum temperature before recalescence, grains were dendritic, in contrast to the globulitic grains assumed by Greer et al. [4] The simulation for _R=4 0:8 Ks 1 was examined in detail to reveal some of the present model features. The calculated cooling curve and time evolution of the number density of nucleated grains are given in Figure 7, showing that the average grain size is completely determined at the minimum of the recalescence in the cooling curve, because nucleation stops at this moment, as suggested by Maxwell and Hellawell. [3] The density of the distribution of grain envelope radii (R in Eq. [25]) was obtained from the numerical derivative of the accumulated distribution (calculated with the model) in relation to the grain radius, dn/dr. A distribution density, dn/dr ext, was also obtained for the extended envelope radius, R ext. These distribution densities are illustrated in Figure 8 for different times before the end of solidification, which occurs at t ~ 120 seconds. The distribution evolves to its final shape until the end of the nucleation period (t ~ 1.2 seconds) as new grains nucleate (Figrue 8(a)). Afterward, the distribution mainly Table II. Material Properties and Microstructural Parameters for the Simulations That Were Compared with Experimental Results Obtained in Al-Cu and Al-Si Alloys Property Al-Cu Al-Si D l (m 2 s 1 ) exp ( /8.314T) [32] D s (m 2 s 1 ) exp ( /8.314T) [33] L (J m 3 ) c pl (J m 3 K 1 ) [34] c ps (J m 3 K 1 ) [34] m l (K wt frac 1 ) K C (m K) T f (K) T e (K) A e (m s 1 K 2 ) [11] [35] A (m s a ) [36] a ( ) [36] DT nucl (K) 10* 10 N max ( ) 200* 200 *These parameters were used only in the simulations of the experiments by Eskin et al. [29]. 386 VOLUME 46A, JANUARY 2015 METALLURGICAL AND MATERIALS TRANSACTIONS A

13 Fig. 7 Calculated temperature and number density of grains (n) as a function of time for the solidification of an Al-2.12 pct Cu alloy at a total cooling rate of _R=4 = 0.8 K s 1. Fig. 8 Density of distribution of radii (dn/dr) and extended radii (dn/dr ext ) of envelopes during the solidification of an Al-2.12 pct Cu alloy at a total cooling rate of _R=4 = 0.8 K s 1 at different times: (a) 0< t 1.2 s and (b) 5s t 120 s. shifts to larger grain radii as a result of equal growth velocity for all (dendritic) grains, which grow at a uniform temperature. At early times (Figure 8(a)), the distribution density for the extended radii (dn/dr ext )is not shown, because it virtually coincides with the distribution for the actual envelope radii (dn/dr) as a result of negligible grain impingement. Nevertheless, at later times, grains are larger and begin to impinge on each other more frequently, increasing the difference between these distributions, as shown in Figure 8(b). One of the most important features of the present model is the calculation of the instantaneous distribution of grain sizes during solidification, as shown in Figure 8. To define the model equations, a continuous normal or log-normal distribution of nucleation undercoolings was assumed, which would give rise to a continuous distribution of grain sizes. To solve the model equations, however, this continuous distribution was discretized by adopting different grain classes for different ranges of grain sizes, considering that all grains in the class had the same size. Nevertheless, as the number of classes, N, increases, the numerical solution should converge to the solution of the continuous distribution problem and become independent of N. In Figure 9(a), the density of the distribution of grain radii calculated for several N values at the end of solidification is shown, indicating that changes are relatively small for N 200, which was the value adopted in all multigrain simulations in the present work. For N 50, the final distribution shape is significantly different from those obtained with N 100. The final average grain size given in Figure 9(b) as a function of N confirms this behavior by showing that grain size increases from its minimum possible value, obtained when only one grain class is adopted (instantaneous nucleation), and converges to ~ masn increases to 100, remaining approximately constant thereafter. As mentioned, these simulations were carried out for the solidification of an Al-2.12 pct Cu alloy cooling down to 800 K (t ~ 136 seconds), below the eutectic temperature, at the total cooling rate of _R=4 0:8 Ks 1. Simulations were run in a personal computer with a Intel Ò core i7 processor and took ~1 second for N = 1, increasing by ~1 second for each further grain class and resulting in METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 46A, JANUARY

14 Fig. 9 Effect of the number of grain classes (N) on the (a) final distribution density of grain radii and (b) final average grain size at the end of the solidification of an Al-2.12 pct Cu alloy at a total cooling rate of _R=4 = 0.8 K s 1. ~200 seconds for N = 200, which was adopted in most of the present work simulations. C. Amount of Eutectic The volume fraction of eutectic measured by Sarreal and Abbaschian [37] as a function of the cooling rate for an Al-4.9 pctcu alloy is compared with the present model results in Figure 10. Although columnar grains Fig. 10 Present model results compared with measurements by Sarreal and Abbaschian [37] for an Al-4.9 pct alloy in different cooling rates: (a) volume fraction of eutectic assuming either diffusion in the solid or without diffusion (D s = 0); (b) spacing between secondary arms. were observed in most of these experiments, the agreement is good after adopting the following model parameters: n T = m 3 (instantaneous nucleation), DT N = 0.05 K, n e ext = m 3, and DT ne = 0 K. From relatively low cooling rates to approximately 100 K s 1, the model correctly predicts 388 VOLUME 46A, JANUARY 2015 METALLURGICAL AND MATERIALS TRANSACTIONS A

15 an increasing eutectic fraction with an increasing cooling rate, caused by less time for solute diffusion in the solid (back-diffusion). When D s = 0 (dashene in Figure 10a), the eutectic fraction remains approximately constant in the range of lower cooling rates. For cooling rates larger than 100 K s 1, both model and experiments indicate a decrease in eutectic fraction. Sarreal and Abbaschian [37] attributed this effect to the depression of the dendrite tip temperature during columnar growth. Therefore, the first solid (tip) that forms has a larger concentration, decreasing microsegregation and the amount of eutectic. In the present model of equiaxed solidification, there is an analogous effect, because an increase in cooling rate decreases the minimum temperature before recalescence and the temperature plateau after recalescence. During this plateau, 60 pct or more of the solid forms. when the plateau temperature decreases, this solid should have a larger solute concentration and, therefore, less microsegregation and eutectic. The average dendrite arm spacing (weighted by grain fraction), calculated using typical values of A and a for dendrite arm coarsening in Al-Si alloys (Table II), is also compared with the measurements by Sarreal and Abbaschian, showing very good agreement (Figure 10(b)). D. Solute Content and Amount of Inoculant Predictions of the present model for the average grain size as a function of the Si concentration in Al-Si alloys are compared with the experimental measurements carried out by Johnsson [38] at a cooling rate of _R = 1 Ks 1 in Figure 11. Because inoculant was added to the melt, the log-normal undercooling distribution model was adopted and calculated using Eq. [32] for r / = 0.8 and / 0 = 0.6 lm or =0.2 lm. This undercooling distribution stems from the size distribution of inoculant particles illustrated in Figure 12. The latter / 0 value (0.2 lm) gave the best agreement observed in Figure 11, while the former (0.6 lm) closely approximates the distribution of inoculant particle sizes measured by Greer et al. [4] in a commercial grain refiner of the Al-Ti-B system. For pct Si ~3, both model and experimental results agree, showing a decrease in grain size for an increase in pct Si. This is usually attributed to a decrease in grain growth velocity (restriction to grain growth), increasing the maximum undercooling before recalescence, which increases the number of nucleated grains. [39,40] Nevertheless, for Si > 3 pct, calculated grain sizes still decrease as a result of the growth restriction, but measured grain sizes increase. Therefore, the mechanism responsible for this grain size increase, which is still unclear, is obviously not included in the present model equations. Possible mechanisms are the reduction of grain growth restriction and the poisoning of the surface of inoculant particles. [41] In these simulations, the model automatically predicted globulitic grains. An experimental verification was not possible, because Johnsson [38] did not report the grain microstructures. Simulations were also carried out forcing the dendritic growth law on the globulitic grains to verify whether the anomalous increase in grain size for pct Si > 3 could be explained by a change in growth morphology, but no significant change was seen (Figure 11). Fig. 11 Average grain size ( l) calculated with the present model and measured by Johnsson [38] for an increasing concentration of Si in Al- Si alloys. Calculations are carried out using two different undercooling distributions, defined by n T, / 0, and r /. Simulations in which the law for dendritic growth is imposed are also shown. Fig. 12 Normalized log-normal distribution of inoculant particle sizes for the standard deviation r / = 0.8 and two different geometrical mean diameters (/ 0 ). METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 46A, JANUARY