Fig. 1 Schematic phase diagram of an alloy with a miscibility gap in the liquid state

Fig. 2 Change in Gibbs free energy on alloy formation

Fig. 3 a Variation of the liquid free energy at 900 K calculated with ΔH0 = 20 kJ/mol, b phase diagram showing the miscibility gap and spinodal line in a regular solution system [21]

Fig. 4 Schematic diagram showing the L-L interface

Fig. 5 Schematic diagram showing: a the equilibrium contact between S1, L1 and L2 phases necessary for an aligned composite growth; b perfect wetting of S1 and L2 by L1 which leads to the formation of an irregular microstructure

Fig. 6 Variation of the interfacial energy with temperature [80]

Fig. 7 Undercooling values as a function of composition for Zn-Pb alloy system [26]

Fig. 8 Undercooling as a function of composition for Ga-Bi alloy system [141]

Fig. 9 Supersaturation of the matrix liquid and nucleation rate of the droplets versus time in Al-7 wt% Pb alloy melt cooled at the rate of 150 K/s [147]

Fig. 10 Number density of the droplets and average droplet radius versus time in Al-7 wt% Pb melt cooled at the rate of 150 K/s [147]

Fig. 11 Schematic of the phase diagram of an alloy with a miscibility gap in the liquid state along with the rapid unidirectional solidification process

Fig. 12 Temperature profile, upper consolute temperature (liquidus or binodal line) and nucleation rate in front of the S-L interface of Al-5 wt% Pb alloy solidified at the rate of 10 mm/s. Initial temperature of the melt is 850 °C [181]

Fig. 13 Number density and average radius of the droplets in front of the S-L interface of Al-5 wt% Pb alloy solidified at the rate of 10 mm/s. Initial temperature of the melt is 850 °C [181]

Fig. 14 Difference between the local and the initial concentration of the Pb in the melt in front of the S-L interface of Al-5 wt% Pb alloy. Initial temperature of the melt is 850 °C [181]

Fig. 15 Number density and average radius of the droplets in front of the S-L interface of Al-5 wt% Pb alloy solidified at the rate of 15 mm/s. The dotted lines are the results calculated by neglecting the effect of the collisions and coagulations between the droplets [182]

Fig. 16 Flow field of the melt in front of the solidification interface for the sample solidified at the rate of 5 mm/s. The arrows indicate the direction of the local velocity [183]

Fig. 17 Maximum nucleation rate of the MPDs in front of the solidification interface along the r direction for the sample solidified at the rate of 5 mm/s [183]

Fig. 18 Variation of the number density of the MPDs in front of the solidification interface for the sample solidified at the rate of 5 mm/s. The dashed lines are the results calculated by neglecting the convective effect. The thin lines are for the center (r = 0 mm), and the thick lines are for the position r = 3.4 mm of the sample. The melt flows upward at the center (r = 0 mm) and downward at r = 3.4 mm. The term nS/L is the droplet number density in the melt at the solidification interface. The term nL is the droplet number density in the melt at 4.16 mm above the solidification interface. The inset shows the convective effect [183]

Fig. 19 Volume fraction and average radius of the MPDs in front of the solidification interface for the sample solidified at the rate of 5 mm/s. The dashed lines are the results calculated by neglecting the convective effect. The thin lines are for the center (r = 0 mm), and the thick lines are for the position r = 3.4 mm of the sample. The melt flows upward at the center (r = 0 mm) and downward at r = 3.4 mm [183]

Fig. 20 Nucleation rate (I) and the average diameter (\( \left\langle D \right\rangle \)) of the MPDs, and driving force for the nucleation (ΔG) of the MPDs when the alloy solidifies continuously at the rate of 10 mm/s under the effect of the ECPs of different current densities. The frequency of the ECPs is 50 Hz. The duration of each electropulse is 6 μs [104]

Fig. 21 Convective velocities of the melt in front of the solidification interface along the central z-axes for the Al-5 wt% Pb alloy sample solidified in the magnetic field of different strengths [98]

Fig. 22 Microstructures of the Al-7 wt% Pb alloy sample solidified at the rate of 8 mm/s under the effect of the direct current of: a 0, b 170, c 438 A/cm2, respectively [102]

Fig. 23 Monotectic lamellae evolving with the steady-state growth behavior. The snapshot corresponds to the time: at = 0.0, b 2.4 × 10-2. The coloring illustrates the concentration field according to the legend embedded in the first snapshot, where the black and white regions represent the solid S1 and liquid L2 phases, respectively [190]

Fig. 24 Growth of a L2 droplet in front of a solid planar front shown at times: at = 0.0, b 4.0 × 10-3, c 1.2 × 10-2, d 2.0 × 10-2. The coloring illustrates the concentration field according to the legend embedded in the first snapshot, where the black and white regions represent solid and L2 phases, respectively [190]

Fig. 25 Coarsening mechanism of the two droplets. a, b Diffusion-controlled coagulation, the dimensionless times are a 0 and b 2.5 × 10-2, c-f flow-assisted coagulation, the dimensionless times are taken at time 1.25 × 10-4, 1 × 10-3, 2 × 10-3 and 5 × 10-3, respectively. The arrows indicate the velocity field [191]

Fig. 26 Liquid phase separation in Al-Bi as predicted by the phase field model without melt flow: a snapshot of the concentration field taken at time 0.051, b histogram showing respective droplet size distributions at three dimensionless times of 0.025 (red), 0.031 (white) and 0.051 (green) [191]

Fig. 27 Liquid phase separation in Al-Bi as predicted by the phase field model with melt flow: a snapshot of the concentration field taken at time 0.051, b histogram showing respective droplet size distributions at three dimensionless times of 0.025 (red), 0.031 (white) and 0.051 (green) [191]

Fig. 28 Zoom of the miscibility gap occurring in the liquid region of Bi-Zn system. Simulation pictures present the variations of the stable morphology at different temperatures and compositions [192]

Fig. 29 Phase field simulation of the microstructural evolution in the Cu-Fe system at 1680 K with the composition of: a 25 at.%, b 50 at.%, c 75 at.%. Yellow color the Cu-rich phase; blue color the Fe-rich phase [194]

Fig. 30 Snapshots of the phase separation process in a rapidly solidified ternary Fe-Sn-Si immiscible alloy droplet. The black part is the Sn-rich phase, and the white part is the Fe-rich phase. Snapshots at: aτ = 0, bτ = 10, cτ = 500, dτ = 1000, eτ = 2000, fτ = 3000, respectively [198]