Fig. 1 Microstructures of the Al-12 wt% Bi alloys solidified at the rate of 5 mm/s under the magnetic field intensities of a 0 T, b 1 T [16]

Fig. 2 Schematic diagram of the continuous solidification facility of an immiscible alloy in a transverse static magnetic field [17]

Fig. 3 Motion velocities of the average size droplets in front of the solidification interface along the central z-axes (axial position) for the Al-5 wt%Pb alloy continuously solidified at the rate of 5 mm/s. B is the magnetic strength, \({\mathbf{u}}_{\text{M}}^{z} \left( {\langle R\rangle } \right)\), \({\mathbf{u}}_{\text{S}} \left( {\langle R\rangle } \right)\) and \({\mathbf{u}}_{{{\text{M}}.\hbox{max} }}^{r} \left( {\langle R\rangle } \right)\) are the z vector of the Marangoni velocity, the Stokes velocity and the maximums of the r-component (radial position) of the Marangoni velocity of the droplets with average size, respectively [17]

Fig. 4 Convective flow velocities of the melt along the central z axes for the Al-5 wt%Pb samples continuously solidified at the rate of 5 mm/s in magnetic fields of different strengths [17]

Fig. 5 Average radius of the MPDs in front of the solidification interface along the central z axes for the Al-5 wt%Pb alloy sample continuously solidified at the rate of 5 mm/s. The insignificant difference between the solid line and the stars indicates that the applied magnetic field does not affect the solidification process through changing the Marangoni and Stokes moving velocities of the MPDs. The coincidence between the dotted line and the circles demonstrates that the magnetic field affects the solidification process and microstructure mainly through restraining the convective flow of the melt. \({\mathbf{V}}_{\text{C}}\), \({\mathbf{u}}_{\text{S}}^{{}}\) and \({\mathbf{u}}_{\text{M}}\) are the convective velocity of the matrix liquid, the Stokes motion and Marangoni migration velocities of the droplet, respectively [17]

Fig. 6 Maximum nucleation rates of the MPDs along the radial direction for the Al-5 wt%Pb samples continuously solidified at the rate of 5 mm/s under the effect of static magnetic fields [17]

Fig. 7 Micro-X-ray tomography images of the Al-10 at.%In alloys directionally solidified at the rate of 2.7 μm/s under the effect of a magnetic field: a 3D image of the In-rich phase in a specimen solidified without a magnetic field; b the slice images of a; c 3D image of a specimen solidified in a 10 T magnetic field; d the slice images of c [19]

Fig. 8 Comparison of droplet size distribution in the electromagnetically levitated Cu50-Co50 alloy samples solidified in magnetic fields of different strengths [20]

Fig. 9 Microstructures of the regions close to the surface of the Al-7 wt%Pb samples continuously solidified at a given rate of 8 mm/s under the effect of different DC densities: a 0 A/cm2, b 170 A/cm2, c 195 A/cm2, d 438 A/cm2. There is a layer poor in the Pb-rich particles on the surface of the sample solidified without a DC. The sample solidified with a DC density of 170 A/cm2 shows a microstructure with a uniform distribution of the Pb-rich particles across the whole section of the sample. The sample shows a layer rich in the Pb-rich particles on the surface when the DC density is 195 A/cm2. A continuous shell of Pb-rich phase forms on the surface of the sample solidified under the effect of a DC density of 438 A/cm2 [25]

Fig. 10 Volume fraction of the MPDs along the radial direction of the Al-7 wt%Pb alloys solidified at the rate of 8 mm/s under the effect of a direct current. j is the current density. \({\mathbf{u}}_{E}\) is the moving velocity of the MPDs due to the electric current. The circles are the volume fraction of the minority phase calculated by assuming that at j?=?438 A/cm2; the DC does not affect the motions of the MPDs. The negligible differences between the circles and the dot lines demonstrate that a DC affects the phase segregation along the radial direction of the sample mainly by changing the spatial motions of the MPDs [24]

Fig. 11 Microstructures of Bi-10 wt%Cu-10 wt%Sn alloy solidified continuously at the rate of 10 mm/s; a without ECPs, b-d with ECPs, the frequency and duration of each ECP are 50 Hz and 6 μs, and the peak values of the pulse current density are b 1?×?108 A/m2, c 2?×?108 A/m2, d 3?×?108 A/m2 [3]

Fig. 12 Microstructures of Cu-25 wt%Bi-25 wt%Sn alloy solidified continuously at the rate of 10 mm/s; a without ECPs, b with ECPs, the frequency and duration of each ECP are 50 Hz and 6 μs, and the peak value of the pulse current density is 3?×?108 A/m2 [31]

Fig. 13 Nucleation rate (I), average diameter (〈D〉) of the MPDs and driving force for the nucleation (ΔG) of droplets in front of the solidification interface along the central axis for the a Bi-10 wt%Cu-10 wt%Sn, b Cu-25 wt%Bi-25 wt%Sn samples continuously solidified at the rate of 10 mm/s under the effect of the ECPs. The frequency of the ECPs is 50 Hz, and the duration of each ECP is 6 μs. Jp is the peak values of pulse current density [3]

Fig. 14 Schematic diagram showing the solidification process of an immiscible alloy under the effect of an orthogonal electric-magnetic field. W is the gravity. F is the resultant force of the buoyancy and the electromagnetic force [34]

Fig. 15 Structures of Zn-30 wt%Bi alloys solidified under different electric magnetic body force magnitudes: a 0 N/cm3; b 51 N/cm3; c 57 N/cm3; d 85 N/cm3; e 91 N/cm3; f 102 N/cm3. The frequency of the electric magnetic body force and the cooling rate of the samples are 50 Hz and 25 K/min, respectively [36]