Abstract

1. Introduction

During casting process of alloys a large variety of microstructures can be observed. The most commonly found ones are known as cell, dendrite and eutectic morphology. The formation of dendrites is a common mechanism of crystallization from high solidification speeds in the metals and alloys. A convenient and widely used technique to investigate the effects of solidification conditions on the resulting microstructures is based on the measurements of dendritic spacing, which is the distance between the dendritic primary, secondary or higher orders arms. On the other hand, dendritic arm spacing has strong influence on both microsegregation patterns and second phase formation in the interdendritic regions during solidification process. The formation of dendritic morphology during solidification and the microsegregation process have fascinated researchers for literally hundreds of years. The solidification thermal parameters have influence on the microstructure and microsegregation andthese, in turn, are key featuresin inducing non-uniformity on the as-cast material mechanical properties. For this reason, the microstructure and microsegregation has been extensively studied both theoretically¹1 Salvino IM, Ferreira LO, Ferreira AF. Simulation of Microsegregation in Multicomponent Alloys During Solidification. Steel Research International. 2012;83(8):723-732.

2 Salvino IM, Jácome PAD, Ferreira AF, Ferreira IL. An Analysis of the Physical Properties of Multicomponent Alloy on the Simulation Solidification by Phase-Field Model. Materials Science Forum. 2013;730-732:703-708.

3 Xavier CR, Delgado HG Junior, Castro JA, Ferreira AF. Numerical Predictions for the Thermal History, Microstructure and Hardness Distributions at the HAZ during Welding of Low Alloy Steels. Materials Research. 2016;19(3):520-533.

4 Ferreira AF, Castro JA, Ferreira LO. Predicting Secondary-Dendrite Arm Spacing of the Al-4.5wt%Cu Alloy During Unidirectional Solidification. Materials Research. 2017;20(1):68-75.

5 Sun Q, Zhang Y, Cui H, Wang C. Phase field modeling of multiple dendrite growth of Al-Si Binary alloy under isothermal solidification. China Foundry. 2008;5(4):265-267.

6 Wesner E, Chaoudhury A, August A, Berghoff M, Nestler B. A phase-field study of large-scale dendrite fragmentation in Al-Cu. Journal of Crystal Growth. 2012;359:107-121.

7 Trivedi R. Interdendritic Spacing: II. A Comparison of Theory and Experiments. Metallurgical and Materials Transactions A. 1984;15(6):977-982.

8 Kurz W, Fisher JD. Dendrite growth at the limit of stability: tip radius and spacing. Acta Metallurgica. 1981;29(1):11-20.

9 Burton JA, Prim RC, Slichter WP. The Distribution of Solute in Crystals Grown from the Melt. Part I. Theoretical. The Journal of Chemical Physics. 1953;21:1987-1990.

10 Voller VR, Beckermann C. A unified model of microsegregation and coarsening. Metallurgical and Materials Transactions A. 1999;30(8):2183-2189.^-1111 Voller VR. A semi-analytical model of microsegregation in a binary alloy. Journal of Crystal Growth. 1999;197(1-2):325-332. and experimentally¹²12 Spinelli JE, Cheung N, Goulart PR, Quaresma JMV, Garcia A. Design of Mechanical properties of Al-alloys chill castings based on the metal/mold interfacial heat transfer coefficient. International Journal of Thermal Sciences. 2012;51:145-154.

13 Freitas ES, Silva AP, Spinelli JE, Casteletti LC, Garcia A. Inter-relation of Microstructural Features and Dry Sliding Wear Behavior of Monotectic Al-Bi and Al-Pb Alloys. Tribology Letters. 2014;55(1):111-120.

14 Spinelli JE, Ferreira IL, Garcia A. Influence of melt convection on the columnar to equiaxed transition and microstructure of downward unsteady-state directionally solidified sn-pb alloys. Journal of Alloys and Compounds. 2004;384(1-2):217-226.

15 Goulart PR, Cruz KS, Spinelli JE, Ferreira IL, Cheung N, Garcia A. Cellular growth during transient directional solidification of hypoeutectic Al-Fe alloys. Journal of Alloys and Compounds. 2009;470(1-2):589-599.

16 Meza ES, Bertelli F, Goulart PR, Cheung N, Garcia A. The effect of the growth rate on microsegregation: Experimental investigation in hypoeutectic Al-Fe and Al-Cu alloys directionally solidified. Journal of Alloys and Compounds. 2013;561:193-200.

17 Medrano-Prieto HM, Garay-Reyes CG, Gómez-Esparza CD, Aguilar-Santillán J, Maldonado-Orozco MC, Martínez-Sánchez R. Evolution of Microstructure in Al-Si-Cu System Modified with a Transition Element Addition and its Effect on Hardness. Materials Research. 2016;19(Suppl 1):59-66.

18 Avazkonandeh-Gharavol MH, Haddad-Sabzevar M, Fredriksson H. Effect of partition coefficient on microsegregation during solidification of aluminium alloys. International Journal of Minerals, Metallurgy, and Materials. 2014;21(10):980-989.

19 Gündüz M, Kaya H, Çardili E, Marasli N, Keslioglu K, Saatçi B. Effect of solidification processing parameters on the cellular spacings in the Al-0.1 wt% Ti and Al-0.5 wt% Ti alloys. Journal of Alloys and Compounds. 2007;439(1-2):114-127.

20 Chen M, Kattamis TZ. Dendrite coarsening during directional solidification of Al-Cu-Mn alloys. Materials Science and Engineering: A. 1998;247(1-2):239-247.

21 Zhang C, Ma D, Wu KS, Cao HB, Cao GP, Kou S, et al. Microstructure and microsegregation in directionally solidified Mg-4Al alloy. Intermetallics. 2007;15(10):1395-1400.

22 Canté MV, Spinelli JE, Ferreira IL Cheung N, Garcia A. Microstructural Development in Al-Ni Alloys Directionally Solidified under Unsteady-State Conditions. Metallurgical and Materials Transactions A. 2008;39(7):1712-1726.

23 Silva AP, Garcia A, Spinelli JE. Microstructure morphologies during the transient solidification of hypomonotectic and monotectic Al-Pb alloys. Journal of Alloys and Compounds. 2011;509(41):10098-10104.

24 Zheng X, Luo AA, Zhang C, Dong J, Waldo RA. Directional Solidification and Microsegregation in a Magnesium-Aluminum-Calcium Alloy. Metallurgical and Materials Transactions A. 2012;34(9):3239-3248.^-2525 Ganesan M, Dye D, Lee PD. A technique for characterizing microsegregation in multicomponent alloys and its application to single-crystal superalloy castings. Metallurgical and Materials Transactions A. 2005;36(8):2191-2204. for the last decades. The Al-1.2wt%Pb and Al-3.2wt%Bi alloys have been chosen by Freitas et al.¹³13 Freitas ES, Silva AP, Spinelli JE, Casteletti LC, Garcia A. Inter-relation of Microstructural Features and Dry Sliding Wear Behavior of Monotectic Al-Bi and Al-Pb Alloys. Tribology Letters. 2014;55(1):111-120., to study the effect of microstructural parameters on the wear process. The solidification experiments were achieve under non-stead state conditions of heat flow with a large amplitude of cooling rates, which allows a variation on the dendritic morphology. The results obtained by Freitas et al.¹³13 Freitas ES, Silva AP, Spinelli JE, Casteletti LC, Garcia A. Inter-relation of Microstructural Features and Dry Sliding Wear Behavior of Monotectic Al-Bi and Al-Pb Alloys. Tribology Letters. 2014;55(1):111-120., show that interphase spacing and the morphology of the minority phase have a strong effect on the wear process. Spinelli et al.¹²12 Spinelli JE, Cheung N, Goulart PR, Quaresma JMV, Garcia A. Design of Mechanical properties of Al-alloys chill castings based on the metal/mold interfacial heat transfer coefficient. International Journal of Thermal Sciences. 2012;51:145-154. proposed a theoretical and experimental work to investigate the thermal parameters during solidification process of binary Sn-Pb alloys. The results obtained by Spinelli et al.¹²12 Spinelli JE, Cheung N, Goulart PR, Quaresma JMV, Garcia A. Design of Mechanical properties of Al-alloys chill castings based on the metal/mold interfacial heat transfer coefficient. International Journal of Thermal Sciences. 2012;51:145-154. indicate that CET (columnar to equiaxed transition) occurrence may have been influenced by convection process. In that paper, in downward solidification conditions, the convection process seems to favor the CET occurrence. Growth direction has effect on the primary dendritic arm spacing, in other words, decreasing the dendritic arm spacing for downward vertical solidification when compared with those grown vertical upwards. Goulart et al.¹⁵15 Goulart PR, Cruz KS, Spinelli JE, Ferreira IL, Cheung N, Garcia A. Cellular growth during transient directional solidification of hypoeutectic Al-Fe alloys. Journal of Alloys and Compounds. 2009;470(1-2):589-599. have investigated the resulting structure of solidification in hypoeutectic binary alloys (Al-Fe) with unsteady-state heat flow conditions, both solidification speed and cooling rate are quantitatively determined in that work. Goulart et al.¹⁵15 Goulart PR, Cruz KS, Spinelli JE, Ferreira IL, Cheung N, Garcia A. Cellular growth during transient directional solidification of hypoeutectic Al-Fe alloys. Journal of Alloys and Compounds. 2009;470(1-2):589-599. found that cellular microstructures are predominant in the entire castings for alloys investigated. Furthermore, cellular spacing as function of both solidification speed and cooling rate, it is not affected by concentration of iron solute, and metal/mold heat transfer coefficient can be determined via a power function of time. Meza et al.¹⁶16 Meza ES, Bertelli F, Goulart PR, Cheung N, Garcia A. The effect of the growth rate on microsegregation: Experimental investigation in hypoeutectic Al-Fe and Al-Cu alloys directionally solidified. Journal of Alloys and Compounds. 2013;561:193-200. have executed experiments of unidirectional solidification in aluminum-based systems (Al-Cu and Al-Fe). Composition from dendritic cores to interdendritic regions was numerically determined by Meza et al.¹⁶16 Meza ES, Bertelli F, Goulart PR, Cheung N, Garcia A. The effect of the growth rate on microsegregation: Experimental investigation in hypoeutectic Al-Fe and Al-Cu alloys directionally solidified. Journal of Alloys and Compounds. 2013;561:193-200.. The authors have adopted an experimental equation for solute concentration (Cu and Fe) which shows a good agreement to the experimental data. They have shown that solidification speed has significant effect on the resulting microsegregation curves. Several works¹1 Salvino IM, Ferreira LO, Ferreira AF. Simulation of Microsegregation in Multicomponent Alloys During Solidification. Steel Research International. 2012;83(8):723-732.^-52 Salvino IM, Jácome PAD, Ferreira AF, Ferreira IL. An Analysis of the Physical Properties of Multicomponent Alloy on the Simulation Solidification by Phase-Field Model. Materials Science Forum. 2013;730-732:703-708.^,1212 Spinelli JE, Cheung N, Goulart PR, Quaresma JMV, Garcia A. Design of Mechanical properties of Al-alloys chill castings based on the metal/mold interfacial heat transfer coefficient. International Journal of Thermal Sciences. 2012;51:145-154.

13 Freitas ES, Silva AP, Spinelli JE, Casteletti LC, Garcia A. Inter-relation of Microstructural Features and Dry Sliding Wear Behavior of Monotectic Al-Bi and Al-Pb Alloys. Tribology Letters. 2014;55(1):111-120.

14 Spinelli JE, Ferreira IL, Garcia A. Influence of melt convection on the columnar to equiaxed transition and microstructure of downward unsteady-state directionally solidified sn-pb alloys. Journal of Alloys and Compounds. 2004;384(1-2):217-226.

15 Goulart PR, Cruz KS, Spinelli JE, Ferreira IL, Cheung N, Garcia A. Cellular growth during transient directional solidification of hypoeutectic Al-Fe alloys. Journal of Alloys and Compounds. 2009;470(1-2):589-599.

16 Meza ES, Bertelli F, Goulart PR, Cheung N, Garcia A. The effect of the growth rate on microsegregation: Experimental investigation in hypoeutectic Al-Fe and Al-Cu alloys directionally solidified. Journal of Alloys and Compounds. 2013;561:193-200.

17 Medrano-Prieto HM, Garay-Reyes CG, Gómez-Esparza CD, Aguilar-Santillán J, Maldonado-Orozco MC, Martínez-Sánchez R. Evolution of Microstructure in Al-Si-Cu System Modified with a Transition Element Addition and its Effect on Hardness. Materials Research. 2016;19(Suppl 1):59-66.^-1818 Avazkonandeh-Gharavol MH, Haddad-Sabzevar M, Fredriksson H. Effect of partition coefficient on microsegregation during solidification of aluminium alloys. International Journal of Minerals, Metallurgy, and Materials. 2014;21(10):980-989. have been carried out which highlighted the importance of taking into account the thermal parameters effect on the microstructure and microsegregation during solidification process. Several experimental researches have been proposed for binary alloys of metallurgical interest. However, experimental works focused on the multicomponent alloys are still rare in the literature¹1 Salvino IM, Ferreira LO, Ferreira AF. Simulation of Microsegregation in Multicomponent Alloys During Solidification. Steel Research International. 2012;83(8):723-732.^,1717 Medrano-Prieto HM, Garay-Reyes CG, Gómez-Esparza CD, Aguilar-Santillán J, Maldonado-Orozco MC, Martínez-Sánchez R. Evolution of Microstructure in Al-Si-Cu System Modified with a Transition Element Addition and its Effect on Hardness. Materials Research. 2016;19(Suppl 1):59-66.. The present experimental paper is based on the research line of previous references, highlighting the correlation between the thermal parameters and tertiary dendrite arm spacing (λ₃) and microsegregation during solidification of a ternary Al-9.0wt%Si-4.0wt%Cu alloy under unsteady-state heat flow conditions. However, in the present work the numerical results for microsegregation are obtained by disregarding the equilibrium partition coefficient (K_eq) and, instead, adopting an effective partition coefficient (K_ef). The multielement interaction effect on the Si microsegregation in the ternary Al-9.0wt%Si-4.0wt%Cu alloy during solidification process is investigated. This approach constitutes the main innovative feature of the present work. A model for microsegregation analysis of the solidification process is referred as Scheil's equation¹⁸18 Avazkonandeh-Gharavol MH, Haddad-Sabzevar M, Fredriksson H. Effect of partition coefficient on microsegregation during solidification of aluminium alloys. International Journal of Minerals, Metallurgy, and Materials. 2014;21(10):980-989.^,2626 Chalmers B. Principles of Solidification. New York: John Wiley & Sons; 1964.. The assumptions considered in Scheil's equation (1) are described as follows: a) no diffusion in solid phase; b) infinitely fast diffusion occurs in the liquid and; c) equilibrium exists at the solid/liquid interface and so compositions from phase diagram can be considered.

where C_S is the concentration in the solid region, C₀ the initial overall composition and F_S the solid fraction. Although the influence of solidification speed (V_L) and cooling rate ($\dot{\mathit{T}}$) on the microstructural characteristics was the focus of many studies¹1 Salvino IM, Ferreira LO, Ferreira AF. Simulation of Microsegregation in Multicomponent Alloys During Solidification. Steel Research International. 2012;83(8):723-732.

2 Salvino IM, Jácome PAD, Ferreira AF, Ferreira IL. An Analysis of the Physical Properties of Multicomponent Alloy on the Simulation Solidification by Phase-Field Model. Materials Science Forum. 2013;730-732:703-708.

3 Xavier CR, Delgado HG Junior, Castro JA, Ferreira AF. Numerical Predictions for the Thermal History, Microstructure and Hardness Distributions at the HAZ during Welding of Low Alloy Steels. Materials Research. 2016;19(3):520-533.^-44 Ferreira AF, Castro JA, Ferreira LO. Predicting Secondary-Dendrite Arm Spacing of the Al-4.5wt%Cu Alloy During Unidirectional Solidification. Materials Research. 2017;20(1):68-75.^,1212 Spinelli JE, Cheung N, Goulart PR, Quaresma JMV, Garcia A. Design of Mechanical properties of Al-alloys chill castings based on the metal/mold interfacial heat transfer coefficient. International Journal of Thermal Sciences. 2012;51:145-154.

13 Freitas ES, Silva AP, Spinelli JE, Casteletti LC, Garcia A. Inter-relation of Microstructural Features and Dry Sliding Wear Behavior of Monotectic Al-Bi and Al-Pb Alloys. Tribology Letters. 2014;55(1):111-120.

14 Spinelli JE, Ferreira IL, Garcia A. Influence of melt convection on the columnar to equiaxed transition and microstructure of downward unsteady-state directionally solidified sn-pb alloys. Journal of Alloys and Compounds. 2004;384(1-2):217-226.

15 Goulart PR, Cruz KS, Spinelli JE, Ferreira IL, Cheung N, Garcia A. Cellular growth during transient directional solidification of hypoeutectic Al-Fe alloys. Journal of Alloys and Compounds. 2009;470(1-2):589-599.^-1616 Meza ES, Bertelli F, Goulart PR, Cheung N, Garcia A. The effect of the growth rate on microsegregation: Experimental investigation in hypoeutectic Al-Fe and Al-Cu alloys directionally solidified. Journal of Alloys and Compounds. 2013;561:193-200., literature on the influence of these parameters on the microsegregation profiles are still quite limited²¹21 Zhang C, Ma D, Wu KS, Cao HB, Cao GP, Kou S, et al. Microstructure and microsegregation in directionally solidified Mg-4Al alloy. Intermetallics. 2007;15(10):1395-1400.. Burton et al.⁹9 Burton JA, Prim RC, Slichter WP. The Distribution of Solute in Crystals Grown from the Melt. Part I. Theoretical. The Journal of Chemical Physics. 1953;21:1987-1990. proposed an equation for effective distribution coefficient (K_ef) as a function of the solidification speed (V_L). The equilibrium partition coefficient can be replaced by this effective distribution coefficient into the Scheil microsegregation model. Burton proposes the following equation for K_ef

where δ diffusion layer thickness of the segregated solute ahead the solid/liquid interface and D_L is the liquid solute diffusivity. Physical properties adopted for this work were the same those used by Qiang et al.⁵5 Sun Q, Zhang Y, Cui H, Wang C. Phase field modeling of multiple dendrite growth of Al-Si Binary alloy under isothermal solidification. China Foundry. 2008;5(4):265-267. and Wesner et al.⁶6 Wesner E, Chaoudhury A, August A, Berghoff M, Nestler B. A phase-field study of large-scale dendrite fragmentation in Al-Cu. Journal of Crystal Growth. 2012;359:107-121. and are shown in Table 1.

The δ depends on the solidification speed (V_L), the liquid viscosity and the agitation condition ahead the solid/liquid interface, and its value can vary from 10^-6 to 10^-3m, according to the Chalmers cited by Meza et al.¹⁶16 Meza ES, Bertelli F, Goulart PR, Cheung N, Garcia A. The effect of the growth rate on microsegregation: Experimental investigation in hypoeutectic Al-Fe and Al-Cu alloys directionally solidified. Journal of Alloys and Compounds. 2013;561:193-200..

2. Experimental Procedure

The experimental technique adopted was previously described in more details¹⁴14 Spinelli JE, Ferreira IL, Garcia A. Influence of melt convection on the columnar to equiaxed transition and microstructure of downward unsteady-state directionally solidified sn-pb alloys. Journal of Alloys and Compounds. 2004;384(1-2):217-226.

15 Goulart PR, Cruz KS, Spinelli JE, Ferreira IL, Cheung N, Garcia A. Cellular growth during transient directional solidification of hypoeutectic Al-Fe alloys. Journal of Alloys and Compounds. 2009;470(1-2):589-599.^-1616 Meza ES, Bertelli F, Goulart PR, Cheung N, Garcia A. The effect of the growth rate on microsegregation: Experimental investigation in hypoeutectic Al-Fe and Al-Cu alloys directionally solidified. Journal of Alloys and Compounds. 2013;561:193-200.. A ternary Al-9.0wt%Si-4.0wt%Cu alloy was prepared in a furnace to 800^oC, from the high-purity materials (99.9% Al, 99.7% Si and 99.9% Cu). The chemical analysis of the ternary alloy has been checked via conventional spark emission spectroscopy. The melt was then cast into a steel mouldof a special apparatus in orderto obtain a vertical upward transient directional solidification. A thin steel plate is used to close the base of the steel mold, which separates the melt from water-cooling system.

Solidification apparatus details are shown in Figure 1. The pouring temperature (T_P) was setup at 55^oC above its liquidus temperature (T_L). The surface of the cooling base of the mold was polished in order to allow a good contact with the melt and, therefore, a higher heat transfer process. This creates favorable conditions to obtain a higher range of cooling rates during the solidification process. The temperature data were measured by type K thermocouples. The thermocouples were positioned along of the casting 5, 10, 15, 20, 35, 45, 60 and 85mm from the mold bottom. The temperature data were sampled at 0.01s intervals by a data logger interface and stored in a personal computer. An Olympus Optical Microscope (Olympus Corporation, Japan) was used to produce digital images that were analyzed using Goitaca (https://sourceforge.net/projects/goitacaeq) image processing software in order to measure the tertiary dendritic spacing (λ₃). A scanning electron microscope JEOL (JEOL, Ltd., Japan) model JSM 5800LV with energy-dispersive spectrometer NORAN System 6, (Thermo Fisher Scientific), was used for the solutes concentration measurements. Although WDS could present a better detection limit than EDS, for the elements and composition range involved it has been considered that EDS would be cheaper and faster process. The concentration measurement initiates at the dendritic arm centre and ends at the centre of adjacent arm, as shown in Figure 2.

The concentration measurements were performed in samples taken from positions very close to the tip of the thermocouples. About 40 concentration measurements were performed for each position along the casting. The solidification speed for each position along the casting varied between 0.22-1.19mm/s.

3. Results and Discussion

The cooling curves recorded by the thermocouples of the ternary Al-9.0wt%Si-4.0wt%Cu alloy, from the onset of solidification experiment, are shown in Figure 3. The temperatures measured by the thermocouples along the casting allow determining the position of liquidus temperature during solidification process. Figure 4 correlates the position of liquidus isotherm (P) and solidification speed (V_L) with time during the solidification process for the ternary Al-9.0wt%Si-4.0wt%Cu alloy.

The cooling curves were used to found the position of liquidus temperature (P) in the solidification process. On the other hand, solidification speed (V_L) used in the present work, were derived from position of liquidus temperature, as shown in Figure 4. In that figure, the solidification speed (V_L) varies significantly with time immediately after the onset of solidification, followed by approximately constant values. From experimental equations (P and V_L) shown in Figure 4, it was possible to obtain an equation for the solidification speed as a function of position, Figure 5. The mould water-cooling system favors high solidification speed (V_L) at regions close to the cooling bottom. As we move away from mould bottom, the values decreasesas a result of increasing thermal resistance of the solidified material.

Metallographic samples taken from cross sections along the casting are shown in Figure 6. Those samples have been taken perpendicular to the solidification direction and, therefore, show basically the tertiary dendritic arms. In that figure, on the right side of each picture, it is presented information regarding the sample position in the casting (P), solidification speed (V_L), cooling rate ($\dot{\mathit{T}}$) and tertiary dendritic arm spacing (λ₃). Figure 6 illustrates the effect of thermal variables on the dendritic morphology during the solidification process, through those optical micrographs, one can see that dendritic arm spacing (λ₃) decreases dramatically at higher cooling rates ($\dot{\mathit{T}}$). Such an alteration in dendritic arm spacing at higher cooling rates is due to the increasing nucleation rates that lead to a microstructure refinement.

Figure 7 a-b shows the experimental Cu and Si composition profiles measurements. For clarity, the concentration profiles were considered for a range of solid fraction (0 to 60%), which is assumed to correspond to the solidification before eutectic and/or any intermetallic phase formation.

Figure 7 (a-b) reveals that both Cu and Si concentrations increase gradually as the solidification progresses. It can be seen from Figure 7 (a) that the Cu profile moves upward with the increase in solidification speed (V_L). This is due to microsegregation phenomenon at the solid/liquid interface, which is usually characterized by the equilibrium partition coefficient (K_eq), which is determined from the phase diagram. The significant deviations between Scheil's equation and experimental data shown in Figure 7 (a), is due to the vertical directional solidification process considered in present work that takes place under non-equilibrium conditions. In contrast to Figure 7 (a), one can see an inverse trend in Si microsegregation (Figure 7 (b)) that is profile of solute concentration moves downward with the increase in solidification speed (V_L). Closer examination of the experimental results shown in Figure 7 a-b, suggests that the Si concentration in the solid phase may have been influenced by multielement interaction in ternary Al-9.0wt%Si-4.0wt%Cu alloy. The growth of Cu-rich dendrites is accompanied by progressive enrichment of residual liquid in Si, i.e., large concentrations of Cu and correspondingly minute amounts of Si are recorded in the solid phase for same vertical positions (P) of the casting.

In order of improving the prediction capability of Scheil's equation for Cu concentration, the effective partition coefficient (K_{ef_Cu}) is considered in this present work. The eq. (2) has been used to create a plot of K_{ef_Cu} versus V_L. In order to found a experimental equation of the effective partition coefficient (K_{ef_Cu}) as a function of solidification speed (V_L), a curve fitting technique have been used on such points shown in Figure 8.

It is important to emphasize that the experimental equation shown at Figure 8 have been derived for a solidification speed range from 0.22 to 1.19mm/s obtained in the present work, and that for higher solidification rates a tendency to approach K_{ef_Cu} = 1 will exist, since Eq. (1) is operative between equilibrium partition coefficient (K_eq) and 1.

Due to the observed inverse trend in Si microsegregation, the Scheil's equation was adjusted preliminary to experimental data (Figure 9) and then the K_{ef_Si} as a function of solidification speed (V_L) can be determined empirically. The Figure 10 shows an empirical equation for effective partition coefficient (K_{ef_Si}) obtained from Scheil's equation adjusted to the experimental data, Figure 9. The solidification speed effect on the K_{ef_Si} can be incorporated for a range of solidification rates between 0.27 and 1.19mm/s.

The microsegregation of Cu and Si are shown in Figure 11 and 12, respectively. The profiles (Cu and Si) were obtained via Scheil's equation using the effective partition coefficients (K_{ef_Cu} and K_{ef_Si}). The results shown in Figs. 11 and 12 display good agreement between the numerical and experimental data. However, some discrepancies are observed at higher solid fractions (F_S>0.4).

The calculated results underestimate the experimental data for both solutes Cu and Si for F_S>0.4, because the solid back-diffusion was not taken into consideration in the Scheil's equation. In addition to solid back-diffusion, there are other micro-scale phenomena which can affect microsegregation, known as interdiffusion during cooling and coarsening¹⁰10 Voller VR, Beckermann C. A unified model of microsegregation and coarsening. Metallurgical and Materials Transactions A. 1999;30(8):2183-2189.^,1111 Voller VR. A semi-analytical model of microsegregation in a binary alloy. Journal of Crystal Growth. 1999;197(1-2):325-332.^,2525 Ganesan M, Dye D, Lee PD. A technique for characterizing microsegregation in multicomponent alloys and its application to single-crystal superalloy castings. Metallurgical and Materials Transactions A. 2005;36(8):2191-2204..

4. Conclusions

Solidification speed (V_L) varies significantly with time, immediately after the onset of solidification, followed by approximately constant values. The reason for this is due to increasing thermal resistance of solidified region, in the other words, the increasing of solid region exert a strong influence on the solidification kinetics. Tertiary arm spacing (λ₃) decreases with rates ($\dot{\mathit{T}}$), this occurs because there is a increasing of nucleation rate that lead to a microstructure refinement. The results reveal that concentrations of the solutes increase gradually with the solidification progress. These increasing concentrations are due to a solute distribution at the phase interface, which is characterized by the partition coefficient. Theprofile of Si concentration moves downward with increase in solidification speed, this trend suggests that the Si solute in the solid phase have been influenced by multielement interaction effects in ternary Al-9.0wt%Si-4.0wt%Cu alloy. The effect of the solidification speed (V_L) on the microsegregation has been examined by Scheil's equation through the incorporation of V_L into an effective partition coefficient (K_ef), which has been determined for the range of solidification speed (V_L) experimentally examined. The effective partition coefficient of Cu is calculated by Burton's equation, while an experimental equation is proposed for effective partition coefficient of Si, which is based on a best-fit curve to the experimental concentration profiles. The concentrations of solutes (Cu and Si) calculated by Scheil's equation using effective partition coefficient (K_{ef_Cu} and K_{ef_Si}) was shown to adjust well to the experimental data. The calculated results via Scheil's equation yielded deviations for a solid fraction above 0.4, such a deviations is due to solid back-diffusion was not taken into consideration in the said equation.

5. Acknowledgments

I would like to express my deep appreciation for the support provided by the Instituto Militar de Engenharia through its Material and Mechanical Engineering Section and PETROBRAS through its Research Center - CENPES, for the use of their Scanning Electron Microscopy and X-ray Microanalyses laboratories that allowed the realization of the present job. My special thanks to the staff of these institutions that have greatly helped in the activities.

6. References

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