Investigation of thermoelectric magnetic convection and its effect on solidification structure during directional solidification under a low axial magnetic field

Investigation of thermoelectric magnetic convection and its effect on solidification structure during directional solidification under a low axial magnetic field

Available online at www.sciencedirect.com Acta Materialia 57 (2009) 2180–2197 www.elsevier.com/locate/actamat Investigation of thermoelectric magnet...

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Available online at www.sciencedirect.com

Acta Materialia 57 (2009) 2180–2197 www.elsevier.com/locate/actamat

Investigation of thermoelectric magnetic convection and its effect on solidification structure during directional solidification under a low axial magnetic field Xi Li a,b,*, Annie Gagnoud b, Zhongming Ren a, Yves Fautrelle b, Rene Moreau b a

Department of Material Science and Engineering, Shanghai University, Shanghai 200072, China SIMAP-EPM-Madylam/CNRS, ENSHMG, BP 95, 38402 St. Martin d’Heres Cedex, France

b

Received 13 September 2008; received in revised form 17 January 2009; accepted 22 January 2009 Available online 23 February 2009

Abstract Thermoelectric magnetic convection (TEMC) at the scale of both the sample (L = 3 mm) and the cell/dendrite (L = 100 lm) was numerically and experimentally examined during the directional solidification of Al–Cu alloy under an axial magnetic field (B 6 1T). Numerical results show that TEMC on the sample scale increases to a maximum when B is of the order of 0.1 T, and then decreases as B increases further. However, at the cellular/dendritic scale, TEMC continues to increase with increasing magnetic field intensity up to a field of 1 T. Experimental results show that application of the magnetic field caused changes in the macroscopic interface shape and the cellular/dendritic morphology (i.e. formation of a protruding interface, decrease in the cellular spacing, and a cellular– dendritic transition). Changes in the macroscopic interface shape and the cellular/dendritic morphology under the magnetic field are in good agreement with the computed velocities of TEMC at the scales of the macroscopic interface and cell/dendrite, respectively. This means that changes in the interface shape and the cellular morphology under a lower magnetic field should be attributed respectively to TEMC on the sample scale and the cell/dendrite scale. Further, by investigating the effect of TEMC on the cellular morphology, it has been proved experimentally that the convection will reduce the cellular spacing and cause a cellular–dendritic transition. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Thermoelectric magnetic convection; Macroscopic interface; Cellular and dendritic morphology

1. Introduction Most models of the formation of microstructure neglect the influence of convection and only consider diffusional transport of either heat or mass away from the growing crystal. However, at low growth rates, convection and its influence on microstructure formation and development can become dominant [1,2] and should not be neglected. Indeed, Chakraborty et al. [3–7] have investigated numerically the effect of convection on the solidification structure during electromagnetohydrodynamic stirring processing, and have shown that convection will affect the solidifica*

Corresponding author. E-mail address: [email protected] (X. Li).

tion structure significantly. The effect of a permanent magnetic field on convection in directional solidification has been discussed extensively [8]. It has been determined that the application of a magnetic field during the directional solidification of materials can significantly reduce the thermosolutal buoyant flow [9–11]. However, no influence on the microstructure or longitudinal macrosegregation was detected upon application of either a transverse or an axial magnetic field (0.1 T) during the directional solidification of the near eutectic Pb–57 wt.% Sn alloy, which contained only a small volume fraction of primary dendrites [12]. To investigate this further, Tewari et al. [13] studied the effect of a higher transverse magnetic field (0.45 T) on cellular microstructures in directionally solidified Pb–17.7 wt.% Sn alloy, and showed experimentally that no influence on

1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.01.016

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macrosegregation was detected. However, at very low pulling rates (less than 1 lm s1), the cellular array was found to be severely distorted. This was due to the damping braking of one of the components of the natural thermosolutal convection by the magnetic field. Alboussiere and Moreau [14] and Laskar [15] investigated the effect of a magnetic field on Bi–60 wt.% Sn and Cu–45 wt.% Ag alloys, solidified vertically under solutally and thermally stabilizing conditions with a 0.6 T transverse or a 1.5 T axial magnetic field and found that large ‘‘freckles” appeared in this case, showing that a new movement had been created. Alboussiere et al. [14] suggested that this new flow was induced by the interaction between the magnetic field and thermoelectric effects, and Lehmann [8] subsequently offered some experimental evidence for thermoelectromagnetic convection (TEMC). Recently, Yesilyurt et al. [16] presented numerical solutions for buoyant convection and TEMC in an actual (single) crystal growth process with an axial magnetic field. Their results indicated that thermoelectric currents at the growth interface during directional solidification promoted convection when a low-intensity magnetic field was applied. For high magnetic field strengths, meridional flow typically slows down due to the Lorentz force. According to Khine and Walker [17], the magnitude of TEMC is maximum when the Hartmann number, Ha, is about 10. Nevertheless, the existence of TEMC and its effect on the solidification structure has not been proven. Moreover, the real effect of convection on the dendritic network is also still far from being completely understood and quantified. When some convection occurs in the liquid bulk and in the mushy zone, nonlinear coupling between the fluid flow, the mass and heat transport and the solidification process makes theoretical predictions difficult. Therefore, it is essential to investigate TEMC and to deepen our understanding of the mechanism of the effect of convection on the solidification microstructure. The aim of this work is twofold: on one hand, TEMC and its effect on solidification structure have been investigated numerically and experimentally; on another hand, by studying the influence of TEMC on the solidification structure, the understanding on the effect of the convection on the solidification structure may be extended and deepened. Our previous work [18] described thermoelectric magnetic force in solids and TEMC in liquids. This paper extends the work in Ref. [18] and investigates numerically and experimentally TEMC on the scale of the sample and the cell/dendrite and its influence on the morphology of the interface and the cell/dendrite in detail. Numerical results have indicated that on the sample scale, the value of TEMC increases to a maximum when B is of the order of 0.1 T and then decreases as B increases further. However, on the cellular scale, the value of TEMC increases with increasing magnetic field intensity up to 1 T. Experimental results have shown that an application of a magnetic field causes the interface to protrude, reduces the cellular spacing and results in an cellular–dendritic transition in directionally solidified Al–Cu alloy. Further, analy-

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sis on the macro- and microsegregation of the solute Cu shows that the concentration of the solute Cu increases at the boundary of the samples under the magnetic field. The formation of the protruding interface and the reduction in the cellular spacing are discussed in terms of TEMC on the sample and cell scales, respectively. 2. Numerical modeling and experimental details 2.1. Numerical modeling The calculation was performed using the finite-element code FLUX-EXPERT, available in our laboratory (SIMAP-EPM-Madylam, CNRS). The study configuration is two-dimensional and the liquid and solid regions are considered. In this model, electrical and hydrodynamic phenomena are modeled during the imposition of a temperature gradient and a magnetic field. Here, only the perpendicular component of the velocity is considered and the unknowns of the problem are the electrical potential and the velocity perpendicular to the study plane. The basic equations of this problem are: ! divð J Þ ¼ 0 ð1Þ ! ! ð2Þ lDw þ ð J ^ B Þz ¼ 0 ! where J is the electric current density which contains two components (Jx, Jy), w is the third component of the veloc! ity, l is the dynamic viscosity, and B is the magnetic field which has two components (Bx, By). Eq. (1) is written in the two regions and Eq. (2) is written only in the liquid region. In order to take the thermoelectricity into account, complementary terms have to be added to the classical forms of Ohm’s law, which becomes: ! ! ! ! ! J ¼ rr u  rS r T þ rð V ^ B Þ ð3Þ ! where V is the fluid velocity which has one component (0, 0, w), and u, T, S respectively denote the electric potential, the temperature and the absolute thermoelectric power. The Galerkin’s projective method was used to model Eqs. (1) and (2) and they are projected on the ai functions: Z Z Z ! ai r  ð J Þdv ¼ 0 ð4Þ X Z Z Z Z Z Z ! ! ai lDwdv þ ai ð J ^ B Þz dv ¼ 0 ð5Þ X

X

The integration by parts and the application of the Green–Ostograski theorem of these two projections lead to the following relations: Z Z Z Z Z ! ! r ðai Þ  J dv ¼ ai J  ! n ds ð6Þ X C Z Z Z Z Z ! ! ! lr ðai Þ  r ðwÞdv  ai rr ðwÞ  ! n ds C Z XZ Z ! ! ai ð J ^ B Þz dv ¼ 0 ð7Þ  X

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where ! n is the external normal of the region. The normal component of the current density is conserved between two media and the conditions on the external boundaries are the zero normal component of the current density or imposed electrical potential; hence the following weak form is obtained: Z Z Z ! ! r ðai Þ  J dv ¼ 0 ð8Þ X

For the velocity equation, since the boundary condi! tions are unconstrained, r ðwÞ  ! n ¼ 0, no velocity exists on the boundary; therefore, the following weak form is obtained: Z Z Z Z Z Z ! ! ! ! lr ðai Þ  r ðwÞdv  ai rð J ^ B Þz dv ¼ 0 X

X

ð9Þ In these two weak forms, we introduce the expression of the current density; thus, the following two relations will be produced:

Z Z Z ! ! ! ! ! rr ðai Þ  r udv  rr ai  ð V ^ B Þdv X X Z Z Z ! ! rS r ðai Þ  r Tdv ð10Þ ¼ X Z Z Z Z Z Z ! ! ! ! lr ðai Þ  r ðwÞdv þ ai rðr u ^ B Þz dv X X Z Z Z ! ! ! ai r½ð V ^ B Þ ^ B z dv  Z Z xZ ! ! ¼ ai rSðr T ^ B Þz dv ð11Þ

Z Z Z

X

With these formulations the variation of the absolute thermoelectric power between two media is taken into account and the circulation of electric current around the interface is modeled. Eqs. (10) and (11) are described using the equations generator of the Flux-Expert software. The strong coupling between electrical and hydrodynamic phenomena is taken into account. One numerical system is written and the unknowns are the electrical potential and

Fig. 1. Schematic illustration of the convections at the macro-interface and the tip of cell/dendrite: (a) Schematic illustration of the TEMC on the scale of the protruding-interface or the tip of cell/dendrite; (b) flows on the interface and the tip of cell/dendrite: 1.TEMC at the interface; 2. recirculation loops caused by TEMC on the interface; 3. TEMC at the interface at the cell scale; (c) TEMC amplitude (U) as a function of the external magnetic field (B) at the scales of the interface and the cell/dendrite.

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the velocity. The coupling terms are in the matrix of the system. The obtained system is nonlinear and an iterative method is used to solve it. 2.2. Experimental To examine the above numerical results and investigate the relationship between convection and the solidification structure (the macroscopic interface shape and the cellular morphology), two Al–Cu hypoeutectic alloys, Al–0.85 wt.% Cu and Al–4.5 wt.% Cu, were used to investigate the influence of the lower magnetic field on the macroscopic interface shape and the cellular/dendritic morphology. The alloys used in this study were prepared from high-purity Al (99.99%) and Cu (99.99%) in an induction furnace and then put in a 10 cm diameter high-purity graphite crucible and heated to 900 °C, magnetically stirred for half an hour and poured into a graphite mold to cast samples with a diameter of 3 mm and length of 200 mm. The cast sample was enveloped in a high-purity corundum tube with an inner diameter of 3 mm and a length of 200 mm. The samples were directionally solidified in a Bridgman apparatus at various growth speeds. The experimental apparatus and many experimental details can be found in Ref. [19]. The temperature gradient and the growth speed were adjusted to form cellular and dendritic regions. Microstructures were examined in the etched condition by optical microscopy. Electron probe microanalysis (EPMA) was used to measure the distribution of the solute Cu with an error of 3–5%.

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field (B) at the scales of the sample and the cell/dendrite are different, as shown in Fig. 1c. Therefore, here TEMC on the scale of the sample (L = 3 mm) and the primary arm (L = 100 lm) has been numerically simulated and the effect of TEMC on the interface shape and celluar/dendritic morphology has been investigated. 3.1. Numerical simulation results Fig. 2 indicates the meshes for a protruding interface (L = 3 mm) and a cell (L = 100 lm) during the numerical simulation. Table 1 shows the physical properties of the Al–Cu alloy and the initial condition used during the numerical simulation process. Fig. 3 shows the typical distribution of electric density and velocity around a protruding interface under a 0.1 T magnetic field. Fig. 3a–c show the TE current, the TE magnetic force and TEMC, respectively. It can be seen that a current swirl appears in the vicinity of the interface and that there is rotation of the liquid around the protruding interface, due to the radial component of the current density. It should be noticed that this swirl is remarkably localized in the vicinity of the inter-

3. Numerical simulation and experimental results It is well known that in any material a temperature gra~ produces a Seebeck electromotive force S rT ~ , dient rT where S is the thermoelectric power of the material [20]. If the gradients of S and T are not parallel, then a thermoelectric (TE) current is generated in the system. Thus, the interaction between the TE current and the magnetic field will produce a thermoelectric magnetic force and a convection (TEMC) will develop in the vicinity of the interface on the microscopic scale. For the protruding interface and the cell/dendrite, it is easy to understand that the temperature at the tips may be significantly higher than that at their base. Thus, a non-isothermal interface will form and a TE current will be produced. Fig. 1a describes TEMC in the case of an external magnetic field parallel to the solidification direction, and it can be observed that a rotatory TEMC around the protruding interface and the tip of the primary arm (cell and dendrite) forms. Fig. 1b shows a three-dimensional diagram of the convection at the sample and the cell/dendrite scales. Convection should consist mainly of three flows: TEMC on the sample scale, the recirculation loops caused by TEMC on the interface and TEMC on the cell/dendrite scale. Moreover, owing to the difference in the TEMC amplitude (U) for different scales, TEMC amplitudes as a function of the external magnetic

Fig. 2. Mesh of the interface and the cell for the numerical simulation.

Table 1 Physical properties of Al–Cu alloy and initial condition used during the numerical simulation process. Properties

Magnitude

Electrical conductivity of solid (rs, X1m1) Electrical conductivity of liquid (rL , X1m1) Thermoelectric power of solid (SS , VK1) Thermoelectric power of liquid (SL , VK1) Temperture gadient (G, K/cm)

13.7  106 3.8  106 1.1  106 1.0  107 60

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Fig. 3. Distribution of the electric current density, force and velocity around the modeled interface at the scale of the sample (L = 3 mm) under a 0.1T magnetic field: (a) Current streamlines near the interface (in A/m); (b) force in the liquid and solid near the interface (in N/m3); (c) computed velocity (in m/s).

face and does not extend into the bulk of the liquid. Further, TEMC under various magnetic field intensities has been investigated and modeled, as shown in Fig. 4. It can be seen that with the increase in the magnetic field intensity, the velocity of TEMC increases and reaches a maximum value for a field of 0.1 T; if the magnetic field intensity increases further, the value of TEMC decreases. This means that there is an optimum magnetic field that maximizes the fluid velocity. Moreover, it has been found that with the increase in the magnetic field intensity, the convection extends into the bulk of the liquid. Fig. 5 shows typical distributions of the thermoelectric density at the tip, in the middle and at the bottom of the cell. Fig. 5a–c and d show the current streamlines on the cell and the thermoelectric magnetic force density, respectively. It should be noticed that except in regions very close to the tip or to the bottom, where the curvature is pronounced, the current density is nearly uniform in each domain and almost parallel to the interface. Furthermore, TEMC under various magnetic field intensities has been numerically simulated, and Fig. 6 shows the computed velocity on the scale of the cell/dendrite under various magnetic fields. It can be observed that as the magnetic field intensity increases, TEMC increases and extends into the bulk liquid and the mushy zone. TEMC exhibits two main characteristics which are noteworthy: one is its extension in the vertical

direction as the increase in the applied magnetic field becomes larger. The other is the increasing variation in the flow rate (or velocity) with increasing magnetic field intensity up to 1 T. Fig. 7 shows the computed velocity on the scale of cell/dendrite for various dendrite/cell tip radii under magnetic field intensities of 0.1 and 1 T. It can be seen that with the increase in the dendrite/cell tip radius, TEMC extends in the vertical direction and its value decreases. The above numerical results show that an application of an axial magnetic field will produce a rotary TEMC on the protruding interface and at the tip of the primary arm, and that for different scales, the amplitude of TEMC under a given magnetic field intensity is different. Under a lower magnetic field, B 61 T, TEMC on the scale of the sample (L = 3 mm) reaches a maximum under a magnetic field of the order of 0.1 T. However, on the scale of the primary arm (L = 100 lm), the amplitude of TEMC increases with increasing magnetic field intensity up to 1 T. 3.2. Experimental results Al–Cu hypoeutectic alloys (Al–0.85 wt.% Cu and Al–4.5 wt.% Cu) were used to investigate the influence of the magnetic field on the macroscopic interface shape and the cellular or dendritic morphology.

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Fig. 4. Computed velocity on the scale of the sample (L = 3 mm) under the various magnetic field intensities.

Fig. 8 shows longitudinal structures near the liquid– solid interface in directionally solidified Al–4.5 wt.% Cu alloy at a growth speed of 2 lm s1 and a temperature gradient of 62.8 K cm1 under various magnetic field intensities. It can be observed that under the 0.05 T magnetic field (Fig. 8a), the interface becomes more regular and protruding compared with the interface without the magnetic field. At the same time, the cell has deviated from the solidification direction. When the magnetic field intensity increases to 0.1 T, the interface becomes irregular and a large vortex

has appeared in the mushy zone. If the magnetic field intensity is increased further to 0.3 T, it can be observed that the interface becomes regularly protruding again and its shape is very similar to that under the 0.05 T magnetic field. However, it should be emphasized that compared with the cellular morphology under a magnetic field of 0.05 T (Fig. 8e), the cell under the magnetic field of 0.3 T becomes smaller (i.e. the cellular spacing decreases). The above results mean that on the sample scale, the effect of the magnetic field on the interface shape increases as the magnetic

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Fig. 5. Distributions of the thermoelectric current density and force on the scale of the cell under a 0.1T magnetic field: (a) (b) and (c) the current streamlines on the tip, the middle and bottom of the cell/dendrite, respectively. (d) Distribution of the thermoelectric magnetic force density on the cell.

Fig. 6. Computed velocity on the scale of the cell/dendrite under the various magnetic field intensities: (a) 0.1T; (b) 0.5T; (c) 1T.

field intensity increases, and reaches a maximum value under a magnetic field of 0.1 T; if the magnetic field intensity increases further, the effect decreases. However, for the cell/dendrite scale, the effect of the magnetic field on the cellular and dendritic morphology increases with increasing magnetic field intensity up to 1 T. Fig. 9 shows longitudinal structures near the liquid–solid interface and the transverse structure at 3 mm in directionally solidified Al–4.5 wt.% Cu alloy at a growth speed of 10 lm s1 and a temperature gradient of 62.8 K cm1 under 0, 0.1 and 0.5 T magnetic

fields. It can be observed that the solid–liquid interface of the samples without a magnetic field exhibits a macroscopic planar interface and typical cellular microstructure, as shown in Fig. 9a. In comparatison, for the samples grown under a 0.1 T magnetic field, the microstructure becomes dendritic, the interface becomes irregular and a vortex appears on the transverse section. Under a 0.5 T magnetic field, the interface becomes a regular protrusion and only on the boundary of the sample do the dendrites become broken.

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Fig. 7. Computed velocity at the scale of cell/dendrite with various dendrite/cell tip radiuses under the 0.1T and 1T magnetic field intensities.

Moreover, it is well known that the Seebeck effect is proportional to the difference (P = SS-SL) between the thermoelectric power of the solid phase SS and the liquid phase SL, and that the thermoelectric power varies as the solute concentration of the alloy changes. Therefore, the Al–0.85 wt.% Cu alloy was also used to investigate the effect of the magnetic field on the solidification microstructure. Fig. 10 shows the longitudinal structures near the liquid–solid interface in directionally solidified Al– 0.85 wt.% Cu alloy at a growth speed of 5 lm s1 and a temperature gradient of 62.8 K cm1 under various magnetic field intensities. It can be observed that com-

pared with the interface shape in the absence of a magnetic field (Fig. 10a), application of a 0.5 T magnetic field causes the interface to become more protruding and that the amplitude of this protrusion decreases under a magnetic field of 1 T. Fig. 11 shows the longitudinal structures near the liquid–solid interface in directionally solidified Al–0.85 wt.% Cu alloy at 10 lm s1 under various magnetic field intensities. It can be observed that with the increase in the magnetic field intensity, the interface becomes gradually protruding and the cellular spacing and the cellular tip radius decrease. The protruding amplitude of the interface reaches a maximum when B

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Fig. 8. Longitudinal structures near the liquid-solid interface in directionally solidified the Al–4.5wt.% Cu alloy at a growth speed of 2 lm/s and a temperature gradient of 62.8 K/cm under various magnetic field intensities: (a) 0T; (b) 0.05T; (c and e) 0.1T; (d and f) 0.3T.

is 0.3 T and then decreases as B increases further. In addition, the cellular spacing under various magnetic fields has been measured (Fig. 12). It can be observed that as the magnetic field intensity increases, the cellular spacing decreases for magnetic fields of up to 1 T. These phenomena are similar to those of the Al–4.5 wt.% Cu

alloy; however, the effect of the magnetic field on the microstructure in the Al–0.85 wt.% Cu alloy is smaller than that of the Al–4.5 wt.% Cu alloy and the magnetic field intensity that is needed to maximize the effect on the interface shape is larger than the one in the Al–4.5 wt.% Cu alloy.

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Fig. 9. Longitudinal microstructures near the liquid-solid interface and transverse microstructures at the position of 3 mm from the interface in directionally solidified the Al–4.5wt.% Cu alloy at a growth speed of 10 lm/s and a temperature gradient of 62.8 K/cm under various magnetic field intensities: (a) 0T; (b) 0.1T; (c) 0.5T.

The above experimental results show that the magnetic field tends to enhance the cellular–dendritic transition. In order to prove this, the effect of the magnetic field on the microstructure near the cellular–dendritic transition has been investigated by observing the structure of the Al–4.5 wt.% Cu alloy solidified directionally at a growth speed of 20 lm s1 and a temperature gradient of 62.8 K cm1. Fig. 13 shows the longitudinal structures near the liquid– solid interface and the transverse microstructure 3 mm from the interface solidified directionally without and with a magnetic field. It can be observed that the microstructure in the absence of a magnetic field is near the cellular–dendritic transition. After the application of a magnetic field, the microstructure developes dendrites. This means that application of a magnetic field has indeed enhanced the cellular–dendritic transition and caused unstable cell growth. It is well known that convection will affect large-scale compositional inhomogeneities, the solute layer, and mass flux around the cellular and dendritic tip. Therefore, to

prove the existence of TEMC and investigate its influence on distribution and segregation, the distribution of the solute Cu on the scales of the sample and the cell/dendrite has been measured by EPMA. Figs. 14–16 show the radial distribution of Cu content in the mushy zone at 300 lm from the interface for samples of Al–0.85 wt.% Cu alloy directionally solidified at 10 lm s1 under 0, 0.1 and 0.5 T magnetic fields. Comparison of the distribution of Cu content without and with the magnetic field shows that the concentration of the solute Cu increases at the boundary of the sample solidified under the magnetic field. Moreover, the distribution of Cu content at the scale of the cell (Fig. 16) shows that the distribution of the solute Cu is periodic, and that the distribution period becomes short and irregular under a magnetic field. Further, the radial distribution of the Cu content on the transverse section at 2, 3 and 5 mm from the interface of the sample in Al–0.85 wt.% Cu alloy directionally solidified at 10 lm s1 under a 0.3 T magnetic field has been investigated.

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Fig. 10. Interface shape and cellular morphology in directionally solidified the Al–0.85 wt.%Cu alloy at a growth speed of 5 lm/s and a temperature gradient of 63 K/cm and the corresponding schematic diagram of the interface shape: (a) 0T; (b) 0.5T; (c) 1T.

The results (Fig. 17) show that the radial distribution of the Cu content is periodic and this is consistent with the ring-like structure in Ref. [18]. Moreover, it has been found that with the increase in the depth of the mushy zone, the concentration of the solute Cu increases in the center and at the boundary of the sample. This means that the macrosegregation under the magnetic field forms gradually. 4. Discussion Two effects of a permanent magnetic field on convection can be distinguished during directional solidification. At the sample scale, convection is damped by magnetohydrodynamic (MHD) effects. The motion of the conducting liquid in the presence of a magnetic field B creates a local current density jMHD given by Ohm’s law. The induced Lorentz force jMHD*B then breaks the motion. The efficiency of the braking is characterized by the Hartmann number, which is the ratio of the electromagnetic damping forces to the viscous forces [21]:

rffiffiffi r BL Ha ¼ l

ð12Þ

where r is the electrical conductivity, l is the dynamic viscosity of the fluid and L is a typical flow length. At the scale of the liquid bulk (L > 102 m), this number is about 150 for a field around 1 T and this implies natural convection can be braked, as experimentally evidenced in Ref. [22]. However, at the scale of interdendritic flow, where the typical size is around 100 lm, this number is much too low to yield any significant effect [15]. The appearance of TEMC during the directional solidification for a non-isothermal interface and the cell/dendrite interface [8,23] will modify the MHD effects. Thus, thermoelectric magnetohydrodynamic (TEMHD) effects will affect the solidification structure. The relationship between TEMC and the interface shape and cell/ dendrite morphology has been discussed as follows. 4.1. Effect of TEMC on the interface shape The numerical results show that TEMC on the sample scale increases with the magnetic field intensity to a maxi-

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Fig. 11. Interface shape and cellular morphology in directionally solidified the Al–0.85 wt.%Cu alloy at a growth speed of 10 lm/s and a temperature gradient of 62.8 K/cm and the corresponding schematic diagram of the interface shape under various magnetic field intensities: (a) 0T; (b) 0.1T; (c) 0.3T; (d) 1T.

mum when B is of the order of 0.1 T and then decreases as B increases further. It has been found experimentally that the application of a magnetic field causes this interface to become more protruding and the amplitude of this protrusion reaches a maximum under a magnetic field of 0.1–0.5 T. The change in the interface shape is in good agreement with the change in TEMC at the sample scale. This means that the formation of a protruding interface under a magnetic field may be attributed to TEMC on the sample scale and the secondary convection caused by it. From the measurement results (Figs. 14–17), it has been found that for the protruding interface, the concentration of the solute

Cu at the boundary of the sample increases. Since the solidification temperature decreases as the local concentration of the heavier species Cu increases, the solidification temperature for the Cu-rich melt at the ampoule wall is lower than that for the Al-rich melt at the centerline, leading to protrusion of the interface. Therefore, the formation of a protruding interface under a magnetic field should be attributed to the heavier species Cu flowing down to the bottom of protruding surfaces due to TEMC and secondary convection (Fig. 1b). Furthermore, the convection on the interface scale will cause the primary arm to deviate from the solidification direction. Moreover, it is easy to

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understand that the transferring speed of the solute Cu depends on the convection on the sample scale, and that with the increase in convection, the transferring rate of the solute Cu towards the boundary increases. Therefore, the change in the protruding amplitude of the interface is consistent with TEMC on the sample scale under a magnetic field.

180

Cell spacing (µm)

160 140 120 100 80 60

4.2. Effect of TEMC on the cellular and dendritic morphology

40 20 0 -0.2

0

0.2 0.4 0.6 0.8 Magnetic field intensity (T)

1

1.2

Fig. 12. Effect of the magnetic field on the cellular spacing for samples in directionally solidified the Al–0.85 wt.%Cu alloy at the growth speed of 10 lm/s.

The above results indicate that the application of a magnetic field has changed the morphology of cell and dendrite. Indeed, the magnetic field has enhanced the cellular–dendritic transition and reduced both the cellular spacing and the cellular tip radius. From the above experimental results, it can be seen that the change in cellular/

Fig. 13. Longitudinal microstructure near the liquid-solid interface and transverse microstructure at the position of 3 mm from the interface in directionally solidified the Al–4.5wt.% Cu alloy at 20 lm/s under various magnetic field intensities: (a) 0T; (b) 0.1T; (c) 0.2T; (d) 0.5T.

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a

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20

Concentration Cu (wt%)

18

0T

16 14 12 10 8 6 4 2 0 -1500

-1000

-500

0

500

1000

1500

Position from the centre of the sample (µm)

b

20

Concentration Cu (wt%)

18

0 .1 T

16 14 12 10 8 6 4 2 0

-1500

c

-1000

-500

0

500

1000

20 18

Concentration Cu (wt%)

1500

Position from the centre of the sample (µm) 0 .5 T

16 14 12 10 8 6 4 2 0

-1500

-1000

-500

0

500

1000

1500

Position from the centre of the sample (µm) Fig. 14. Radial distribution of the Cu content in the mushy zone at 300 lm from the interface in directionally solidified the Al–0.85 wt.%Cu alloy at 10 lm/s under various magnetic fields: (a) 0T; (b) 0.1T; (c) 0.5T. Blue pane indicates the rich of the solute Cu near the boundary of the samples. (For interpretation of color mentioned in this figure the reader is referred to the web version of the article.)

dendritic morphology under a magnetic field is in good agreement with the change in TEMC on the cell/dendrite scale. This means that the change of cell and dendrite under the magnetic field may be attributed to TEMC on the cellular/dendritic scale. The effect of convection on the cellular/dendritic morphology has been investigated widely. Natural convection has been shown to affect the dendritic tip morphology

(tip radius and secondary branching) during free dendritic growth at low rates in transparent systems [24]. Dupouy et al. [2] found that interdendritic convection strongly affected the dendritic array, in a way dependent on the convective configuration and the local direction of the flow. Therefore, the change in the cellular and dendritic morphology should be attributed to TEMC on the cell/dendrite scale. Convection will affect the primary spacing, and Leh-

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a

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1200

1400

1200

1400

1200

1400

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0

200

400

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c

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Concentration Cu (wt%)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0

200

400

600

800

1000

Position from the centre of the sample (µm) Fig. 15. Radial distribution of the Cu content in the mushy zone at 300 lm from the interface of samples in directionally solidified the Al–0.85 wt.%Cu alloy at 10 lm/s under various magnetic fields: (a) 0T; (b) 0.1T; (c) 0.5T.

mann et al. [25] have proposed relating the primary arm spacing to flow velocity U, which is parallel to the primary arm, as follows: k0 k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ðU =RÞ

ð13Þ

where k0 is the primary spacing without convection, U is the velocity and R is the growth speed. This is based on the hypothesis that constitutional undercooling of interdenritic

liquid cannot become higher than dendritic tip undercooling, since a tertiary arm would then grow and reduce the spacing. This expression shows that convection reduces the primary arm spacing. Moreover, investigations on dendritic spacing and morphology under microgravity have indicated that the primary spacing is much larger in space than on Earth, and this was attributed to the absence of natural convection in space [26]. Therefore, it is reasonable to attribute the decrease in the cellular spacing to TEMC at the

X. Li et al. / Acta Materialia 57 (2009) 2180–2197

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0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 200

b

205

300

305

400

300

305

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305

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0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 200

205

Distance, µm

c

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Concentration Cu (wt%)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 200

205

300

Distance, µm Fig. 16. Segregation of Cu contend on the cell/dendrite at 300 lm from the solid/liquid interface under various magnetic field intensities for the samples in directionally solidified the Al–0.85 wt.%Cu alloy at 10 lm/s under various magnetic fields: (a) 0T; (b) 0.1T; (c) 0.5T.

cell/dendrite scale. The above experimental results have shown that a magnetic field caused a decrease in the cell tip radius. Cantor et al. [27] have investigated the growth of dendrites under convection and found that convection decreased the cellular tip radius. Therefore, the decrease in the cellular tip radius should be attributed to TEMC. Moreover, as discussed in Ref. [28], a key parameter for side-branching instability is the relative width K, defined as: D¼



2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4k=ðprÞ

ð14Þ

where k and r are the primary spacing and the cell tip radius, respectively. This parameter characterizes the openness of the pattern or, in other words, the degree of occupancy of the ‘‘spacing” by the cell. It follows that side-branching can develop locally in places where the nearest-neighbor distances are larger than a critical value. Although TEMC reduces both the cellular spacing and the cell tip radius simultaneously, when the decrease in the amplitude of the cellular tip radius caused by TEMC is larger than the decrease in the cellular spacing, the value

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a 10

5. Conclusion 2mm

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6

4

2

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6

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2

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2.4

5mm

TEMC at the scales of the sample and the cell/dendrite has been numerically and experimentally examined during directional solidification under an axial magnetic field ðB 6 1TÞ. Numerical results show that for the different scales, the amplitude of TEMC is different under a given magnetic field. At the sample scale, the value of TEMC increases to a maximum when B is of the order of 0.1 T, and then decreases as B increases further. However, for the cellular scale, the value of TEMC increases with increasing magnetic field intensity. The effect of TEMC on the solidification structure has been investigated by investigating the influence of the lower magnetic field ðB 6 1TÞ on the macroscopic interface shape and cellular/dendritic morphology. It has been found that application of a magnetic field causes the interface to be more protruding, reduces both the cellular spacing and the cellular tip radius, and results in the occurrence of the cellular– dendritic transition. Moreover, the distribution of the solute Cu in the mushy zone has been measured by EPMA. It was found that the concentration of the solute Cu increases at the boundary of the samples under a magnetic field and the distribution of the solute Cu in the cellular and dendritic scale becomes irregular. The change in the interface shape and cellular/dendritic morphology under a magnetic field is in good agreement with the change in the TEMC computed velocity at the sample and cell/dendrite scales, respectively. The formation of the protruding interface under the magnetic field is attributed to the segregation of the solute Cu caused by TEMC, and secondary convection caused by TEMC at the sample scale and the change in the cellular/dendritic morphology under the magnetic field is attributed to TEMC on the cell scale. Acknowledgments

2.2

Concentration Cu (wt%)

2.0

This work is supported partly by the European Space Agency through the IMPRESS project, Natural Science Foundation of China (No. 50801045) and the Changjiang Scholars and Innovative Research Team in University (No. IRT0739). The authors are indebted to Prof. Thierry Duffar, EPM/CNRS, Grenoble, for helpful and fruitful discussions. Stimulating discussions with Professors Michel Rappaz and Ge´rard Lesoult are gratefully acknowledged.

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 -200

References 0

200

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Distance, µm Fig. 17. Radial distribution of the Cu content at the various positions from the interface for the Al–0.85 wt%Cu alloy directionally solidified at 10 lm/s under a magnetic field of 0.3T: (a) 2 mm; (b) 3 mm; (c) 5 mm.

of K will increase. As a consequence, TEMC will enhance the cellular–dendritic transition.

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