It is well-established that when a nonmagnetic substance is subjected to a magnetic field gradient, it will be acted upon by a magnetic force due to the interaction of the magnetization and the imposed magnetic field gradient. Depending on the magnetic property of the material, the direction of the induced magnetic force can be either along or opposite to the magnetic field gradient for paramagnetism or antimagnetism, respectively. In practical applications, a longitudinal gradient magnetic field is usually applied, which imposes a magnetic force on the material in the direction along or opposite to that of gravity, depending on its magnetic susceptibility. The z-axis component of the force can be expressed as 1 where V is the volume of the substance, μ₀ is the vacuum permeability, χ is the magnetic susceptibility per unit volume, and B is the magnetic flux density along the z direction.

When this simple description is extrapolated to a binary system consisting of a liquid matrix M and dispersed particles P of different magnetic susceptibilities, the resultant force acting on P, with consideration of the buoyancy effect, can be given by^[12] 2 where ρ is the density and g is the gravitational acceleration in the z direction. Supposing that the absolute susceptibility of P is larger than that of M, then if the value of is high enough, M tends to migrate in the matrix along the direction (a positive susceptibility) or opposite to the direction (a negative susceptibility) of the magnetic force.

In a magnetic field, if there is an electric current caused by the movement of an electrically conducting or partially electrically conducting fluid in the field, a Lorentz force will be produced by the interaction of the electric current and the applied magnetic field. The Lorentz force is given by 3 where σ is the conductivity of the fluid, is the velocity of the fluid, and is the effective electric current.

If there is a thermoelectric current at a liquid/solid interface caused by the Seebeck effect, then a TEMF will be produced by the interaction of the current and the applied magnetic field. For instance, during the growth of a dendrite in a magnetic field, an electric current can be induced by a temperature difference between the solid dendrite and the remaining liquid phase, on the basis of the Seebeck effect. The TEMF can be given by^[13] 4 where σ_L is the conductivity of the liquid, σ_S is the conductivity of the solid, ξ_L is the thermoelectric potential of the liquid, ξ_S is the thermoelectric potential of the solid, and f_S is the volume fraction of the solid.

When non-magnetic materials, such as paramagnetic and diamagnetic materials, are magnetized by a magnetic field, the material will be affected by the magnetic torque because the magnetization vector is not parallel to the magnetic field vector. The magnetic torque in the direction of the z axis can be expressed as^[14] 5 where χ₁ (along the x axis) and χ₂ (along the y axis) are the magnetic susceptibilities in the easy and difficult magnetization axes, respectively, and θ denotes the angle between the easy magnetization axis and the imposed magnetic field direction.

When magnetic particles are placed in a magnetic field, the particles become magnetic dipoles because they are magnetized. The dipole–dipole interaction acts as a mutual attraction between the particles in the plane parallel to the magnetic field, while the particles in the plane perpendicular to the magnetic field exhibit mutual repulsion. Assuming that the particles are spherical, the interaction energy between two neighboring magnetic dipoles, m₁ and m₂, can be expressed as^[15] 6 where and are the magnetic dipole moments of m₁ and m₂ magnetic dipoles, respectively, and denotes the distance between the centers of particles m₁ and m₂.

When the particles are paramagnetic, the induced magnetic dipole moment can be expressed by the following formula: 7 where r_p is the radius of the particle, χ_e is the effective magnetic susceptibility of the particles, and H_ex is the magnetic field intensity of the imposed magnetic field.

Migration of the solid phases in an alloy melt is strongly affected by natural convection.^[38] It has long been recognized that the Lorentz force can significantly suppress thermal convection in a melt,^[39–41] the sedimentation of phases in some alloys during solidification can therefore be obviously inhibited and more homogeneous microstructures can be obtained.^[42–45] The associated efficiency is mainly dependent on the physical properties of density and electrical conductivity of the alloying elements.^[46] Application of a gradient magnetic field provides another way to control migration of solid phases during solidification. This method is based on the magneto-Archimedes buoyant force, which is dependent on the magnetic susceptibility difference between the solid phases and liquid matrix.

In earlier examples, Al–Si alloys were chosen as model materials to demonstrate the controlling effects of a high-gradient magnetic field on the migration behavior of solid phases in an alloy melt. Wang et al.^[21] applied such fields during solidification of a hypereutectic Al–Si alloy and successfully controlled segregation of the primary Si particles. A similar phenomenon was observed by Jin et al.^[22] Following these studies, Lou et al.^[19] compared the controlling efficiency of a high uniform magnetic field with that of a high-gradient magnetic field. They found that by coupling the magnetic and Lorentz forces, the gradient magnetic field gave better suppression of segregation of TiAl₃ particles in Al–7 mass% Si–0.5 mass% Ti alloys.

Gravitational segregation of the alloying elements in some alloys was similarly reduced by using high-gradient magnetic fields. For example, Cu and Mg are often segregated at the lower part of an Al–Cu alloy and the upper part of an Al–Mg alloy, respectively. The application of negative high-gradient magnetic fields was found to suppress such segregations more or less (Fig. 4). Such suppression was, however, strongly dependent on the direction of the field gradient: if a positive gradient was applied, the gravitational segregation in both alloys increased. These effects were dependent on the physical properties of the elements, such as magnetic susceptibility, density, and electrical conductivity.^[47]

Figure 4.

	Figure Option ViewDownloadNew Window
	Figure 4. (color online) Changes of of solutes in Al–5 mass% Cu and Al–10 mass% Mg alloys with . indicates the segregation extent, where and are the average elemental concentrations at the dendrite cores in the upper and lower parts of the specimens, respectively.[47]

By controlling the migration behavior of particle-like phases in alloys during solidification, particle- or phase-gathered microstructures with volume fraction gradients have been fabricated in several alloy systems. These may be formed in situ during cooling of the melt,^[18] added in a preceding step,^[19] or transformed during annealing treatment.^[20] For example, during isothermal annealing in a high-gradient magnetic field, a hypoeutectic Mn–Sb alloy was first transformed to a semi-solid state to produce a mixture of a liquid matrix and paramagnetic MnSb particles, migration of the MnSb particles in the liquid matrix in the field gradient direction was then triggered, and, finally, their distribution was successfully controlled to yield a particle concentration gradient (Fig. 5). The volume fraction distribution of MnSb particles throughout the alloy from the lower surface, as shown in Fig. 6, suggested that the concentration gradient varied from 0 to about 20% MnSb along the magnetic field gradient direction. Such control was dependent on the direction of the gradient, the value of , and annealing time.

Figure 5.

	Figure Option ViewDownloadNew Window
	Figure 5. (color online) Micrographs of MnSb/Sb–MnSb gradients in Mn–89.7 mass% Sb alloys annealed in different gradient magnetic fields and for various holding time. (a) 0 T and 30 min; (b) and 30 min; (c) and 30 min; (d) and 90 min; (e) and 30 min.

Figure 6.

	Figure Option ViewDownloadNew Window
	Figure 6. (color online) Volume fraction distributions of MnSb particles along depth from the lower surface in Mn–89.7 mass% Sb alloys solidified at various magnetic field gradients for different annealing time.[20]

This approach has been extended to multiphase alloy systems, such as Al–12Si–11.8 Mg–6.5Ti, which has more than five different phases under equilibrium solidification conditions.^[48] Liu et al. solidified this alloy in high-gradient magnetic fields at a relatively low cooling rate. They found that, with decreasing temperature, the primary Ti₅Si₄ particles, the primary (Al,Si)₃Ti particles, the primary Mg₂Si, and the Al/Si eutectic precipitated in different temperature ranges and were driven by magnetic forces of different magnitudes and directions to move in the remaining melt for various times. Layered and graded distributions of these phases were formed in the final microstructure. Li et al.^[49] carried out a similar investigation of a solidified Bi–Mn alloy in a radial gradient magnetic field, which produced a radially distributed magnetic force. During the solidification process, the precipitated primary MnBi phases were driven by the magnetic force and separated from the liquid Bi matrix to form a ring-like structure in the plane perpendicular to the magnetic field direction.

Quite recently, a symmetrically distributed gradient magnetic field was introduced to the forming of particle-gathered microstructures.^[50] Such a field can produce symmetrically distributed magnetic forces. Choosing a hypoeutectic Mn–Sb alloy, a MnSb/Sb(eutectic)–MnSb(primary)–MnSb/Sb(eutectic) composite structure, with a symmetrically graded distribution in both morphology and composition, was directly fabricated in situ. Such composite structures were obtained during both solidification and isothermal annealing processes. Figure 7 shows the depth profile of the MnSb phase volume fraction measured from the top of the alloys solidified with and without a gradient magnetic field. Figure 8 shows a schematic diagram of the hypoeutectic Mn–Sb alloy isothermal annealing process and the curve of the stress state of the MnSb particles in the liquid melt in the symmetrically distributed high gradient magnetic field.

Figure 7.

	Figure Option ViewDownloadNew Window
	Figure 7. (color online) Volume-fraction distribution of primary MnSb particles in Mn–89.7 mass% Sb specimens isothermally annealed (a) without and (b) with a symmetrically graded magnetic field of 12 T; (c) distribution curve.[50]

Figure 8.

	Figure Option ViewDownloadNew Window
	Figure 8. (color online) (a) Stress state of Mn-enriched zones in liquid melt and (b)–(d) schematic diagram of Mn–89.7 mass% Sb alloy isothermal annealing process in symmetrically distributed high-gradient magnetic field (G is the gravity).[50]

Similar to that of the particle-like phases, if the solute migration both prior to or during a solidification process can be well-controlled, then the final microstructure of the alloys will be altered. This has support from some experimental results. As illustrated in Fig. 9, when a Mn–Sb melt with a eutectic composition is subjected to a gradient magnetic field, the magnetic force will drive the Mn-enriched zones to move along the gradient direction to form a Mn concentration gradient throughout the specimen, producing an extended alloy composition range in the binary Mn–Sb phase diagram, as indicated by the grid in Figs. 9(b) and 9(c). The solidification microstructure of the alloys will thus exhibit a continuous change in morphology due to the graded distribution of the solute concentration.^[51] According to this mechanism, layered microstructural gradients have been produced from eutectic^[51] and hypoeutectic^[18] microstructures by controlling the solute distribution in high-gradient magnetic fields (Fig. 10). Three layers were produced by solidification of Mn–Sb alloys in high-gradient magnetic fields. Both the primary MnSb and Sb phases exhibited a continuous change in volume fraction in a metallic MnSb/Sb matrix and their locations depended on the direction of the field gradients (Fig. 11). The evolution of the microstructure was strongly dependent on the alloy composition, specimen dimension, cooling rate, and the value.

Figure 9.

	Figure Option ViewDownloadNew Window
	Figure 9. (color online) Schematic diagrams of distributions of alloying elements in eutectic Mn–Sb melts prior to solidification in various gradient magnetic fields: (a) without magnetic field; (b) with a negative gradient field; (c) with a positive gradient field.[51]

Figure 10.

	Figure Option ViewDownloadNew Window
	Figure 10. (color online) Microstructures of eutectic Mn–Sb alloys solidified in gradient magnetic fields of (a) and (b) (B = 11.5 T). The alloy exhibited uniform eutectic microstructure in a high uniform magnetic field.[51]

Figure 11.

	Figure Option ViewDownloadNew Window
	Figure 11. (color online) Distributions of MnSb phase volume fraction along depth from the top surface of eutectic Mn–Sb alloys solidified in various gradient magnetic fields (B = 11.5 T).[51]

As an alternative to separately controlling the migration of solutes or particle-like phases, Liu et al.^[52] chose a multiphase alloy system—i.e., Bi–11.8 mass% Mn—as the experimental material and explored the possibility of simultaneously controlling the migration of solutes and particles during the solidification process. This alloy has four phases (primary Mn, primary MnBi, eutectic Bi, and eutectic MnBi) that successively precipitate during the solidification process. They solidified the alloy in various high-gradient magnetic fields at a low cooling rate. By controlling the migrations of both the Mn solute and primary MnBi phase during solidification, a Bi/BiMn(eutectic)–BiMn(primary)–Mn(primary) layered microstructure was produced. The location of each layer changed with the direction of the magnetic field gradient (Fig. 12). The critical values of for levitation of the Mn solute and BiMn phase were calculated using the model described in Ref. [21]. The distributions of the primary Mn and BiMn in the alloys in various gradient magnetic fields were predicted according to these calculated values (Fig. 13). The prediction confirmed that the migrations of solutes and particles can be simultaneously controlled.

Figure 12.

	Figure Option ViewDownloadNew Window
	Figure 12. (color online) Microstructures of Bi–11.8 mass% Mn alloys solidified in various high-gradient magnetic fields (B = 8.6 T): (a) B = 0 T; (b) ; (c) ; (d) .[52]

Figure 13.

Figure Option ViewDownloadNew Window

Figure 13. (color online) Schematic diagrams of distributions of primary Mn and BiMn in various high-gradient magnetic fields: (a) without magnetic field, both primary Mn and BiMn segregated in the upper part; (b) , primary Mn was homogeneously distributed; (c) , primary Mn segregated in the lower part; (d) , primary BiMn was homogeneously distributed; (e) , both primary Mn and BiMn strongly segregated in the lower part.[52]

Previous studies^[53–55] have indicated that, when subjected to a magnetic field, a crystal with an anisotropic magnetic susceptibility, driven by the magnetic torque originating from the magnetic anisotropy, will rotate to an angle to give a certain crystal orientation. By incorporating the magnetic force with magnetic torque, the change of solute and particle migrations induced by high-gradient magnetic fields has also been suggested as a means to modify the growth of crystals with magnetically anisotropic features. Gao et al.^[56] performed a series of solidification experiments on Tb–Dy–Fe alloys in various high-gradient magnetic fields to characterize their microstructure evolution. For the alloys solidified with and without a high uniform magnetic field, their saturation magnetization barely changed through their depth; however, for the alloys solidified in high-gradient magnetic fields, their saturation magnetization gradually increased from the top surface through the depth. Furthermore, the saturation magnetization increased in each section of the alloy more or less with increasing . In this alloy, (Tb, Dy)Fe₂ is the magnetic phase and the saturation magnetization of the alloy is strongly dependent on its proportion,^[57–59] so the change in saturation magnetization thus indicated a gradient distribution of this phase throughout the alloy. The orientation degree of the (Tb, Dy)Fe₂ phase along the direction, , was calculated from x-ray diffraction patterns measured from the transverse section of the solidified alloys using the model described in Ref. [60]. The calculation results showed that the applied high-gradient magnetic fields caused a graded increase of the orientation degree from the top surface through the depth. Furthermore, with increasing , the orientation degree increased for each part of the alloys. The orientation of the (Tb, Dy)Fe₂ phase was induced by the magnetic field on the basis of this rotation. The gradient distributions of the orientation degree and phase amount throughout the alloy were due to migration of the precipitated (Tb, Dy)Fe₂ phase, driven by the gradient magnetic force during solidification under the high-magnetic field gradient. Further investigation indicated that the gradient magnetic field effect is strongly dependent on the cooling rate.^[61]

Previous studies^[62,63] have also suggested that, when exposed to a uniaxial magnetic field, suspended particles in a liquid matrix tend to orient along the applied field because of the increase of the magnetic dipole moment to which they are subjected. The particles will migrate under the influence of the magnetic dipole–dipole interactions between neighboring particles, which are either attractive or repulsive depending on the directions of the dipoles.^[64,65] Wang et al.^[66] characterized the evolution of the solidification microstructure of a Mn–Bi alloy in high-gradient magnetic fields. The MnBi grains aggregated strongly at one side of the specimen with a varying distribution throughout, yielding a highly aligned and graded microstructure. The formation of such a special microstructure was realized by incorporating the magnetic force with the magnetic dipole–dipole interactions. Because the alloy underwent a paramagnetic to ferromagnetic transformation during the solidification process,^[67] the magnetic dipole–dipole interactions and the magnetic force caused a separation of the primary MnBi phase from the alloy melt and induced collision of particles, which thus resulted in the formation of the highly aligned and graded microstructure.

During the growth of alloys or mixed or doped crystals, especially alloys that grow in a directional manner, a magnetic field can produce a TEMF, which can induce thermoelectric magnetic convection (TEMC) and cause migration of solutes.^[68–71] Liquid motion due to TEMC can induce a Lorentz force that will, in turn, inhibit such motion to decrease the TEMC effect. Thus, there will be competition between the Lorentz force and TEMF in affecting convection, although this depends on the magnetic flux density.^[72,73] If we apply a high-gradient magnetic field to the directional solidification of alloys, the field will affect the solidification in three ways, i.e., via the Lorentz force, TEMF, and magnetic force. These forces can produce a coupling effect on solute and particle migration and may generate a more complicated solidification microstructure. This has been demonstrated in some recent experimental studies.

Wu et al.^[74] investigated the effects of a high-gradient magnetic field on the directional solidification of a hypereutectic Al–8 mass% Fe alloy. With the application of a gradient magnetic field, the primary Al₃Fe phase (that initially showed a directionally solidified morphology without application of a magnetic field) was twisted and fractured, some phases aggregated and distributed randomly in the samples to form a layered microstructure, and a eutectic area appeared at the top of the samples. The gradient magnetic field effect was dependent on the magnetic flux density (Fig. 14). The mechanism of this microstructural change was attributed to the magnetic force, which moved the Fe-enriched zones in the liquid matrix downward to the bottom of the samples during and even before directional solidification. This migration is due to the TEMF, which caused a cross-flow in the front of the solid/liquid interface, and due to the Lorentz force, which suppressed convection to maintain the movement of the Fe-enriched zones induced by the magnetic force and to counteract the TEMF effect (Fig. 15). The coupling effect of these three forces thus caused deformation and fracture of the primary Al₃Fe phase, morphological instability in the interface between the eutectic area and the primary Al₃Fe phase, and random distribution of the aggregated primary Al₃Fe phase.

Figure 14.

	Figure Option ViewDownloadNew Window
	Figure 14. (color online) Microstructure and volume fraction distribution in longitudinal section of directionally solidified Al–8 mass% Fe alloys grown in various magnetic fields: (a) 0 T; (b) 0.4 T; (c) 1 T; (d) 6 T. (e) Volume fraction distribution of the primary Al3Fe phase. V is the growth rate. Reproduced from Ref. [74], with the permission of AIP Publishing.

Figure 15.

Figure Option ViewDownloadNew Window

Figure 15. (color online) Schematics of alloys during directional solidification under an applied magnetic field: (a) distribution of Fe atoms and microstructure of Al3Fe phase as a liquid and solid under gradient magnetic field; (b) schematic of constitutional distribution; (c) distribution of magnetic flux density and its gradient along the axial direction; (d) schematic showing effect of magnetic force; (e) schematic showing the effect of TEMC. Reproduced from Ref. [74], with the permission of AIP Publishing.