Micro-Segregation along the Monovariant Line in a Ternary Eutectic Alloy System

Transcription

1 Materials Transactions, Vol. 44, No. 5 (23) pp. 8 to 88 Special Issue on Solidification Science Processing for Advanced Materials #23 The Japan Institute of Metals Micro-Segregation along the Monovariant Line in a Ternary utectic Alloy System Toshiaki Himemiya Wakkanai Hokusei ollege, Wakkanai 97-3, Japan To investigate the solidification after a single-phase dendritic solidification of a ternary alloy or to develop the process for making in-situ composite of a ternary alloy, a micro-segregation model along the monovariant line has been introduced. The solidification mode accompanied with this micro-segregation model is assumed as cellular or dendritic eutectic solidification. This model assumes a partial diffusion in the solid; the diffusion of the first solute element in the first solid phase works completely or finitely but no other diffusions work in either solid. This is also a model of the solidification of an iron-carbon-metallic ternary alloy. Two kinds of formulations are made; the first is the general formulation between the solid compositions or the liquid compositions the volume fractions of the two solids the second is with a simplified phase diagram. omparison of the results between Scheil-type solidification, complete diffusion of the first solute in the first phase finite diffusion of the first solute in the first phase has been made for three cases of simplified phase diagrams. The affect of the solutal transition in the phase diagram has been demonstrated the affect of diffusivities in the solid on the micro-segregation along the monovariant line has been illustrated. (Received November 8, 22; Accepted February 7, 23) Keywords: complete diffusion, finite diffusion, solid, complete mixing, liquid, ternary alloy, monovariant line, iron-carbon-metallic alloy, diffusivity, cellular-eutectic solidification, dendritic-eutectic solidification. Introduction When a liquid solidifies with a single-phase dendritic solidification mode, the residual liquid composition within the inter-dendritic valley sometimes reaches a eutectic line. After this the liquid solidifies along the monovariant line. ) Along the monovariant line as shown in Fig., the liquid solidifies producing a mixture of two solid phase (-phase -phase); it lowers its liquidus temperature advancing the three-phase eutectic point. Fig. Solidification path in a ternary eutectic alloy system. The micro-segregation problem is trivial when the solidification proceeds along the monovariant line in the case of equilibrium solidification. It is also clear that the composition of the liquid reaches the three-phase eutectic point in the case of Scheil-type solidification (no diffusion in the solid). ut, in this case, prediction of the volume fraction of the three-phase eutectic is important. In cases other than equilibrium or Scheil-type, the prediction of the final composition of the solidification is meaningful. specially, we are interested in the case where the diffusion of the first solute in the -phase is complete or finite while the diffusion of the second solute in the -phase does not work, the diffusion in the -phase is negligible, such as Fe X (carbon is the first solute X is a second metallic solute) system, which has phase as austenite or ferrite as an intermetallic compound. hen et al. 2) analyzed the micro-segregation of an Al Li u system along the monovariant line proposed the general formulation for the Scheil-type solidification. ut, they did not consider the affect of the diffusion in the solids. In looking at this micro-segregation problem from another point of view, a complement of the growth theory of a cellular or dendritic eutectic 3 5) can be made. For an industrial purpose, we want to not only produce an in-situ composite with high performance but also to control the solidification process to the final stage, that is, to the intergranular solidification. Growth theories usually aim to predict the first crystallized phase from the liquid. On the contrary, the micro-segregation model treats the change of the composition of the residual liquid. These two kinds of theories could be coupled. In this work, some assumptions based on a simplified phase diagram will be introduced to illustrate the affect of the diffusion in the solid. Formulations are made for ) equilibrium solidification, 2) Scheil-type solidification, 3) complete diffusion of the first solute ( element) no diffusion of the second solute ( element) in the -phase no diffusion in the -phase, 4) finite diffusion of element in

2 82 T. Himemiya other conditions same as 3). Not only the affect of the diffusion in the solid but also the affect of the solutal transition in the phase diagram are considered. Finally, the relationship of the prediction of the micro-segregation to the growth theory of the cellular or dendritic eutectic will be discussed. 2. Simplification of Phase Diagram To illustrate the affect of the diffusion in the solid, it is convenient to define a simplified phase diagram of a ternary alloy. Figure 2 shows the assumptions of the phase diagram. In this phase diagram, it is assumed that the liquidus line the solidus lines are approximated as straight lines. The liquidus starts at ð ; ; ; Þ ends at ð ; Þ. The solidus line of the phase starts at ð ; ; ; Þ ends at ð ; ; ; Þ. The solidus of the phase starts at ð ; ; ; Þ ends at ð ; ; ; Þ. The start of the eutectic reaction (expressed with the tie triangle ) may be a pure binary eutectic system or a part of a peritectoeutectic isotherm. The end triangle of the eutectic valley (expressed with ) may be a part of a three-phase eutectic isotherm or a part of a peritecto-eutectic isotherm. In the latter case, after the liquid reaches the end point, a new phase may crystallize instead of the phases, phases might appear. In this paper, to simplify our consideration, it can be assumed that the monovariant line ends at a three-phase eutectic point. We will assume that the tie triangle, L can be expressed with a parameter s! L ¼ ; ; þ s ; ðþ L ; ; ¼ ; ; ; þ s ; ; ;! ¼ ; ; ; þ s ; ; ; ; ð2þ : ð3þ diagram. The parameter s may be interpreted as a normalized content along the monovariant line in a ternary alloy system. 3. Assumptions about Volume lement, Initial onditions Structures To consider the micro-segregation problem, a volume element must be introduced. In the solidification along a monovariant line, a volume element perpendicular to the growth direction of a cellular or dendritic eutectic ought to be considered (see Fig. 3). The hatched rectangle is the volume element. This means that the liquid is entrapped within the arms of the cells or the dendrites of the eutectic. The liquid in the volume element is assumed to be completely mixed. If the spacings of the phase ( e ) are much smaller than the width of volume element, that is, e, each solid phase ( or ) can be treated as a continuous element within the considered element such as in Fig. 4(b), in which diffusion works nearly one-directionally. In the transverse direction of the volume element, there are various sizes of mixture of phase phase, although the mean size is fixed by the growing condition. If e, then phases can be treated as combined volumes the liquid can be assumed to be completely mixed. This condition might be valid for a In the above equations, a linear progress of the tie triangle is assumed in addition to the linear morphology of the phase Fig. 3 Volume element to be considered for micro-segregation along a monovariant line. (a), s=.5,, s=,, A,,,,, s= Fig. 2 Simplified phase diagram at several values of a normalized content, s. (b) Dendrite ellualr or Dendritic utectec -phase -phase diffusion Liquid Liquid Fig. 4 (a) ontinuous structure of the solidified phases. (b) Affect of the intensities of the diffusions in the solid on the ratio between the volume fractions of phases.

3 Micro-Segregation along the Monovariant Line in a Ternary utectic Alloy System 83 dendritic-eutectic but might not be valid for a cellulareutectic as discussed later. In this micro segregation model, we will treat the fractions of the phase (f ) the phase (f ) as variants dependent on the liquid composition ( L or s). As shown in Fig. 4(b), it could be predicted that the solid phase where the diffusion in the solid works more intensely will grow more than the other phase. Therefore it is reasonable to illustrate the relation of s to f f. As was said previously, the solidification with cellular or dendritic eutectec solidification can be started from the initial composition near the monovariant line. ut single-phase dendritic solidification might be followed by solidification along the eutectic monovariant line due to the microsegregation of the dendritic growth. In the latter case, the initial condition of the segregation problem along the monovariant line is s ¼ s at f ¼ f ; f ¼ ð4þ where s 6¼ f 6¼. In this situation, the meaning of the assumption of no diffusion in the solid or complete diffusion in the solid is clear, the formulation of the Scheil-type model or the case of complete diffusion of the atom no diffusion of the atom in the phase no diffusion of any atom in the phase can be extended to this situation. ut for the finite diffusion case, extension of the formulation to this situation is quite difficult, because across the single-phase dendrite to two-phase dendritic-eutectic mixture, the sizes of phases change suddenly. Across the boundary layer, we must consider the diffusion with finite diffusivity. Therefore, the extension of the formulation of finite diffusion case might be approximate needs other formulations. To take into account the affect of the diffusivity in the solid, the morphology of the cellular or dendritic eutectic must be considered. In this work, we will assume that both phases are continuous, the sizes do not change suddenly (see Fig. 4(a)). If these assumptions hold, the diffusivity problem approximately becomes one-directional. (The situation of Fig. 4(a) can be simplified as Fig. 4(b).) 4. Formulation of Segregation along the Monovariant Line In the formulations of micro-segregation along the monovariant line of a ternary eutectic alloy, we will consider the problem in two steps for each case. First, we will consider the problem as a general condition, not restricted by the simplified phase diagram assumption. In this treatment, we will find the general formulations between the amounts of both phases, f f, the tie triangle L ; that is, the compositions of the liquid both solids. Next, we will introduce the simplified phase diagram assumption. In these formulations, the derivatives =ds =ds are expressed as functions of s, f f, respectively. We will consider four cases: ) the equilibrium solidification although this case is trivial, we must come back to this case when we discuss the relation of the solidification progress to the diffusion in the solid; 2) the Scheil-type case; 3) the complete diffusion of the atom no diffusion of the atom in the phase no diffusion of any atom in the phase;, 4) finite diffusion of the atom no diffusion of the atom in phase no diffusion of any atom in phase. As shown later, it is obvious that the second third cases are special instances of the fourth case. A more general case can be considered for the diffusion in the solid in both phases. The case with finite diffusion of the the atom in the the phase can be formulated. ut to consider the solute diffusion in a solid phase on the solidification phenomena is practical only for Fe X system. In a Fe X system, since the phase is ferrite or austenite, where the diffusion coefficent of carbon is about 9 m 2 s (in the case of austenite), the diffusion coefficient of a metallic solute ( = X) is about 3 m 2 s more than 3 orders different from each other, we can neglect the diffusion of element compared to the (carbon) solute. The phase is almost an intermetallic compound for a Fe X system, where the diffusion coefficient of carbon solute may be estimated as 5 m 2 s, the diffusion coefficient of a metallic solute may be no greater than that of carbon. Thus we can neglect the diffusion of solutes in phase compared to the diffusion of solute in phase. For most non-iron-based alloys or for stainless steel, the Scheil-type approximation can be used. 4. ase : quilibrium solidification From the mass balance conditions, we can easily obtain ð L Þ þð L Þ ¼ð f f Þd L þ f d þ f d ð L Þ þð L Þ ¼ð f f Þd L þ f d þ f d ð5þ ð6þ where the superscripts of, that is, L, denote the kinds of the phase, the subscripts of, that is, denote the solute element. f or f means the volume fraction of the -phase or -phase. These equations can be expressed with s, f f in the case of the simplified phase diagram. 4.2 ase 2: Scheil-type solidification As has been shown by hen et al., 2) the Scheil-type solidification can be formulated easily. In this type of solidification, since the diffusions in the solid phase do not work, we can consider only the solid-liquid interface. From the conservation condition of the solutes, the following equations are obtained:

4 84 T. Himemiya ð L Þ þð L Þ ¼ð f f Þd L ð7þ ð L Þ þð L Þ ¼ð f f Þd L : ð8þ Here the symbol means that the the contents in the solid are at the solid-liquid interface. With the use of the simplified phase diagram, we can obtain ða þ sbþ þðcþsdþ ¼ b ð f f Þds ð9þ ðg þ shþ þðpþsqþ ¼ c ð f f Þds ðþ where b ¼ ; ; a ¼ ; ; ; b ¼ð ; Þ a; c ¼ ; ; ; d ¼ð ; Þ c; c ¼ ; ; g ¼ ; ; ; h ¼ð ; Þ g; p ¼ ; ; ; q ¼ð ; Þ p ðþ ð2þ ð3þ ð4þ ð5þ ð6þ ð7þ ð8þ ð9þ ð2þ are constants calculated from the phase diagram. The above differential equations can be expressed for =ds =ds respectively as ds ¼ fð bp c cþþð b q c dþsgð f f Þ ðap cgþþðaq þ bp ch dgþs þðbq dhþs 2 ð2þ ds ¼ fð ca b gþþð c b b hþsgð f f Þ ðap cgþþðaq þ bp ch dgþs þðbq dhþs : 2 ð22þ In addition, these pairs of differential equations can be solved analytically; for instance, for f ¼ =ds, a differential equation is derived with the form: ds ¼ þ A s þ A 2 2 s þ 3 A 3 s 2 þ A 4 s þ A 5 f ð23þ where A, A 2, A 3, A 4, A 5,, 2 3 are constants calculated from a to q, b c. This differential equation is a type of separation of variables can be integrated, but f ¼ =ds cannot be expressed as an elementary function of s. Therefore, we would solve eqs. (2) (22) witha numerical method (the uler method). 4.3 ase 3: omplete diffusion of atom no diffusion of atom in phase no diffusion of any atom in phase In this case, from the conservation condition of the element over the volume element, the following equation can be formulated as dð f Þþ þ ð f f Þ L g¼: ð24þ Omitting the second order term of the differences of the content or the volume fraction, we obtain ð L Þ þð L Þ ¼ð f f Þd L þ f d ð25þ with the equation for the element as ð L Þ þð L Þ ¼ð f f Þd L : y introducing the simplified phase diagram, from eq. (24) we have ða þ sbþ þðcþsdþ ¼ b ð f f Þds þ f dsð ; ; Þ; ð26þ the same equation for the element as eq. (). Thus the following equations are obtained ds ¼ fð bð f f Þþf ð ; ; Þgðp þ sqþ c ð f f Þðc þ sdþ ð27þ ðap cgþþðaq þ bp ch dgþs þðbq dhþs 2 ds ¼ cð f f Þða þ sbþ fð b ð f f Þþf ð ; ; Þgðg þ shþ : ð28þ ðap cgþþðaq þ bp ch dgþs þðbq dhþs 2 The term f ð ; ; Þ is added to both numerators. 4.4 ase 4: Finite diffusion of element no diffusion of element in phase no diffusion of any elements in phase The basic assumption of the mass balance of the elements is: ðsolute rejected in the solid -liquid interfaceþ ¼ ðsolute increase in the liquidþ ð29þ þðsolute tranported by the diffusion in the -phaseþ:

5 Micro-Segregation along the Monovariant Line in a Ternary utectic Alloy System 85 The right-h side of the above equation the first term of the left-h side can be estimated in the same way as eq. (24). The second term of the left-h side would be estimated as f d by virtue of Wolczynski s model.6,7) His model is a kind of micro-segregation model of rody Flemings 8) except for the estimation of solute redistribution in the solid. Thus we obtain the equation for the element as ð L Þ þð L Þ ¼ð f f Þd L þ f d : ð3þ Here is a non-dimensional parameter expressing the intensity of the diffusion in the solid which can be interpreted by analogy as D S t f =L 2 where D S is the diffusion coefficient of solute in the phase, t f is the local solidification time of a cellular or dendritic eutectic, L is the half of the spacing of the cellular or dendritic eutectic needles (¼ ð=2þ ). For the element we have ð L Þ þð L Þ ¼ð f f Þd L : These equations reduce to the Scheil-type equation (eqs. (7) (8)) (case 2) if ¼, reduce to the case of the complete diffusion of in the phase, no diffusion of in the phase no diffusion of in the phase (case 3) if ¼. Therefore, the physical validity of these equations at the limit has been proven. With the simplified phase diagram, we have ds ¼ fð bð f f Þþf ð ; ; Þgðp þ sqþ c ð f f Þðc þ sdþ ðap cgþþðaq þ bp ch dgþs þðbq dhþs 2 ds ¼ cð f f Þða þ sbþ fð b ð f f Þþf ð ; ; Þgðg þ shþ ðap cgþþðaq þ bp ch dgþs þðbq dhþs 2 : ð32þ ð3þ 5. Results of the alculation Let s examine the affect of the diffusivity of the element () on the volume fractions of the solids with use of the simplified phase diagram. In these calculations, from s ¼ to s ¼,,, total nodes were made, with over, nodes the results changed little. Therefore the results shown below are with, nodes. First, calculations are made for the phase diagram shown in Fig. 5. The phase diagram elements are: ¼ð:2; :Þ, ¼ð:85; :Þ, ¼ð:5; :Þ, ¼ð:5; :2Þ, ¼ ð:9; :Þ, ¼ð:4; :3Þ. In the brackets, the numerals are mole fractions. Solidification starts at s ¼ : with f ¼ f ¼. The calculated relations between s f, f for ¼ ¼ are shown in Fig. 6. The liquid composition L, the solid compsition of at the solid-liquid interface the solid composition of at the solidliquid interface can be easily estimated from s with eqs. () to(3). At the end of solidification (at s ¼ ) the volume A,, Fig. 5 Phase diagram where ; is less than ;. Fractions of Solid Phases Fraction of phase Fraction of phase case 3, ξ= of case 4 case 3, ξ= of case Normalized ontent, s Fig. 6 Normalized content (solidification progress) volume fraction of solid phases for Fig. 5. fraction of with ¼ (case 3) is a little smaller than that with ¼ (case 2). This is because ( ; ; ) is negative in this case. Roughly speaking, the difference between case 2 3 is small. Second, the phase diagram shown in Fig. 7 is considered. In this phase diagram, ¼ð:25; :2Þ,,,, are same as in the previous assumption, where ( ; ; ) is positive. The calculated results are shown in Fig. 8. In this case, the volume fraction of with ¼ (case 3) is a little larger than that with ¼ (case 2). In the third phase diagram, we will consider the case shown in Fig. 9. In this diagram, ¼ð:; :2Þ, ¼ð:; :85Þ, ¼ð:; :5Þ, ¼ð:2; :5Þ, ¼ ð:; :9Þ, ¼ð:3; :4Þ. In this case, the affect of the diffusion in the solid () was apparent. The results of calculation with ¼ (case 2), ¼ :5 (case 4) ¼ (case 3) are shown in Fig.. With ¼ :5, the solidification ends before the liquid reaches the three-phase eutectic point (). ¼ :5 is a possible value of the diffusivity of carbon in iron. alculations can be made without f ¼ f ¼ as the

6 86 T. Himemiya Fractions of Solid Phases case 3, ξ= of case 4 case 3, ξ=.5 of case 4 Fraction of phase ξ= of case 4 case 2, Fraction of phase case 2, ξ= of case 4 ξ=.5 of case 4 ξ= of case 4 A,, Normalized ontent, s Fig. Normalized content (solidification progress) volume fraction of solid phases for Fig. 9. Fractions of Solid Phases A Fig. 9 Fig. 7 Phase diagram where ; is greater than ; case 3, ξ= of case 4 Fraction of phase case 3, ξ= of case 4 Fraction of phase starting point. As a example, the result of the calculation starting at s ¼ : with f ¼ :3 f ¼ is shown in Fig.. This is the case where the -dendrite solidifies first the liquid reaches the monovariant line when the volume Normalized ontent, s Fig. 8 Normalized content (solidification progress) volume fraction of solid phases for Fig. 7. Phase diagram where the monovariant line starts at the A line. Fractions of Solid Phases case 3, ξ= of case 4 Fraction of phase case 3, ξ= of case 4 Fraction of phase Normalized ontent, s Fig. Normalized content (solidification progress) volume fraction of solid phases for Fig. 9 with start of f ¼ :3. Ratio of -phase to -phase ξ = (Scheil-type) (case 2) ξ = (case 3) ξ =.5 of case 4 quilibrium (case ) Normalized ontent, s Fig. 2 The affect of the intensities of the diffusions in the solid () on the ratio of the volume fractions of -phase to -phase for Fig. 9. fraction becomes.3. ompared with Fig., the solidification ends faster with ¼. Last, the ratios between the volume fraction of phases are shown. affects the ratio as predicted in Fig.. As increases, the degree of the symmetry of the diffusion in the both solids decreases, the the growth of the -phase will exceed the growth of the -phase as shown in Fig. 2.

7 Micro-Segregation along the Monovariant Line in a Ternary utectic Alloy System Discussion From Figs. 6, 8, it can be pointed out that the affects of the unsymmetrical properties of the diffusivities in the solids () on the micro-segregation behavior along the monovariant line is influenced not only by the value itself but also by the solutal transition in the phase diagram. In a single-phase solidification of a multi-component alloy, the solidification path of a dendritic growth is affected by the phase diagram (solute distribution coefficients) the diffusivities in the solid. 9) Along the monovariant line, although the path is fixed, the ratio between both phases the composition of the final stage are affected by the phase diagram by the diffusivities in the solids. Solidification along the monovariant line with << after single-phase dendritic solidification can also be considered. In Fig., we have shown only the case with ¼ ¼. With the case of <<, the results would exist between ¼ ¼, but a sudden change of structure occurs from the single-phase dendrite to a two-phase mixture, the approximation of the one-directional diffusion is difficult to hold. Next, we will discuss the micro-segregation problem related to the growth theory of the cellular or dendritic eutectic. First, the restriction of the volume element must be pointed out. The assumption that the liquid contents along the width of the volume element are uniform requires e. This means the present model is for a dendritic-eutectic but not for a cellular-eutectic. To construct a micro-segregation model for a cellular-eutectic is more difficult. This is because for a two-phase dendritic-eutectic, e, but for a twophase cellular-eutectic, e is at best one order less than as shown by Himemiya. 4) Furthermore, the liquid in the valley of the cellular-eutectic needles are not entrapped in a volume element (within the secondary arms). Solutes of the liquid within cellular-eutectic needles diffuse along the valley. A method analogous to Sharp Flemings ) might be useful for the case of a cellular-eutectic. Second, the effect of microsegregation on the ratio of f =f has been demonstrated. In the previous work of the growth model for a cellular or dendritic eutectic, 4,5) it was assumed that the ratio is constant. Although the growth model only treats the top of the needle of the cellular or dendritic eutectic, the possibility to join the micro-segregation problem with the growth model is suggested. 7. Summary A micro-segregation model along the monovariant line in a ternary alloy has been introduced. This model assumes a partial diffusion in the solid; the diffusion of the first solute element in the first solid phase works completely or finitely but no other diffusions work in both the solids. After describing the general formulations, a simplified phase diagram has been introduced comparison between ) the Scheil-type solidification, 2) complete diffusion of the first solute in the first phase 3) finite diffusion of the first solute in the first phase has been made. The affect of the shape of the phase diagram diffusivities in the solid have been demonstrated. When the difference of the contents of the first element between the start of the monovariant line the end of the line is small, the affect of the diffusivity in the solid is small. ut, when the difference is large, the affect becomes clear. In the latter case, the solidification ends before the residual liquid reaches the three-phase eutectic point with the intensity of the diffusion ¼ :5. The calculation of the solidification along the monovariant line after a single-phase dendritic solidification can also be made. A comparison of the relative ratio of both solid phases with ) equilibrium, 2) Scheil-type, 3) finite diffusion of the first solute in the first solid 4) complete diffusion of the first solute in the first solid has been made, it is found that when the the degree of asymmetrical properties of the diffusivities in the solids increases more, the relative ratio deviates more from unity. RFRNS ) R. M. Sharp M.. Flemings: Metall. Trans. 5 (974) ) S.-W. hen, Y.-Y. huang, Y. A. hang M. G. hu: Metall. Trans. 22A (99) ) D. G. Mcartney, J. D. Hunt R. M. Jordan: Metall. Trans. A (98) ) T. Himemiya: Mater. Trans., JIM 4 (999) ) T. Himemiya: Mater. Trans., JIM 4 (2) ) W. Wolczynski: Modelling of Transport Phenomena in rystal Growth, ed. by J. S. Szmyd K. Suzuki, (WIT Press, Southampton-oston, 2) pp ) W. Wolczynski, W. Krajewski, R. bner J. Kloch: alphad 25 (2) ) H. D. rody M.. Flemings: Trans. Metall. Soc. AIM 236 (966) ) T. Himemiya W. Wolczynski: Mater. Trans. 43 (22) Appendix: Nomenclatures Symbol Meaning Definition Unit A,, A 5 constant calculated from phase diagram the first solute element of the ternary alloy, 2, 3 constant calculated from phase diagram the second solute element of the trenary alloy the initial solute content in the liquid mole fraction L solute content in the liquid mole fraction solute content in the -phase mole fraction solute content in the -phase at the solid- mole fraction liquid interface

8 88 T. Himemiya solute content in the -phase mole fraction solute content in the -phase at the solid- mole fraction liquid interface D S diffusion coefficient of solute in - m 2 s phase end of the monovariant line the content of solute at mole fraction the content of solute at mole fraction b content difference see eq. () mole fraction c content difference see eq. (6) mole fraction start of the monovariant line ; the content of the solute at mole fraction ; the content of the solute at mole fraction L half of the spacing of the cellular or dendritic ð=2þ m eutectic needles a content difference see eq. (2) mole fraction b content difference see eq. (3) mole fraction c content difference see eq. (4) mole fraction d content difference see eq. (5) mole fraction f the fractions of the -phase f initital value of f f the fraction of the -phase g content difference see eq. (7) mole fraction h content difference see eq. (8) mole fraction p content difference see eq. (9) mole fraction q content difference see eq. (2) mole fraction t f local solidification time of a cellular or s dendritic eutectic s normalized content along the monovariant see eqs. () (3) line s initial value of s the first phase of the solids end of solidus,, corresponding to ; the content of solute at mole fraction ; the content of solute at mole fraction start of the solidus,, corresponding to ; the content of the solute at mole fraction ; the content of the solute at mole fraction the second phase of the solids end of solidus,, corresponding to ; the content of solute at mole fraction ; the content of solute at mole fraction start of the solidus,, corresponding to ; the content of the solute at mole fraction ; the content of the solute at mole fraction e the spacings of the phase see Fig. 3 m spacing of the needles of the cellular or see Fig. 3 m dendritic eutectic nondimensional intensity of diffusion of D S t f =L 2 element in -phase