*4.2. Precipitation and Growth of Spinel Crystals*

The precipitation of spinel crystals includes nucleation and crystal growth. The slag system satisfies the basic requirements of supersaturation and supercooling for crystal nucleation and the essence of crystal nucleus growth is the transfer of atoms and other particles in liquid to the surface of the crystal nucleus. The composition of spinel crystals is obviously different from the chemical composition of slag and its growth rate depends on the long-range diffusion of solute atoms, which is essential to ensure the continuous growth of crystal [19,20]. For a slag system under constant temperature, the chemical reaction merely occurs between the free oxides such as (FeO), (Cr2O3), (MgO), (Al2O3), (CaO) based on slag molecular theory. The ability of molecules in slag to carry out chemical reactions is closely related to their activity, and the concentration of free oxides is the activity. Therefore, the reaction rate of Equations (1)–(5) are related to the concentrations of (FeO), (Cr2O3), (MgO), (Al2O3) and (CaO). As shown in Figure 6, various spinel crystals are precipitated when the slag with the concentration of solute atom *C*<sup>0</sup> is cooled to the temperature T. Furthermore, the concentration of solute atoms in the parent phase and precipitated phase at the phase interface is *CM* and *CN*, respectively. During the *dt* time, the phase boundary pushes the distance of *dr* toward the parent phase, and the quantity of solute atoms needed for the volume increase of the precipitated phase is (*CN* − *CM*) *dr*. Moreover, this part of the solute is provided by the diffusion of solute atoms in the parent phase.

**Figure 6.** Concentration distribution of the solute atoms in slag during spinel crystal growth. *C*<sup>0</sup> represents the average concentration of solute atoms in the matrix, *CM* represents the concentration of solute atoms in the matrix side at the crystal interface, *CN* represents the concentration of solute atoms in the crystal side at the crystal interface, and *u* represents the growth rate of the crystal.

The growth rate of the spinel crystals can be reflected by Equations (6) and (7) [21]:

$$
\mu = \frac{dr}{dt} = \frac{D}{c\_N - c\_M} (\frac{\partial c}{\partial r})\_{r=R} \tag{6}
$$

$$D = \frac{k\_B T}{3\pi d \mu} \tag{7}$$

where *d*, μ, *T*, and *kB* refer to the particle diameter, melt viscosity, absolute temperature, and Boltzmann constant, respectively. ( <sup>∂</sup>*<sup>c</sup>* ∂*r*) *<sup>r</sup>*=*<sup>R</sup>* refers to the concentration gradient of the particle near the phase interface.

It can be seen from the above equation that the growth rate of the precipitated phase crystal nucleus is directly proportional to the diffusion coefficient of the solute in the parent phase and the concentration gradient of the solute atom near the phase interface, while being inversely proportional to the difference in the equilibrium concentration between the two phases at the interface. For a system with a constant temperature, the diffusion coefficient D (cm2/s, the order of magnitude is 10−5–10−<sup>7</sup> generally) is constant and is closely related to the viscosity in slag. In this way, the growth rate of spinel crystals mainly depends on the particle concentration difference (*CN* − *CM*) and concentration gradient (( <sup>∂</sup>*<sup>c</sup>* ∂*r*) *<sup>r</sup>*=*R*) at the interface of the spinel crystal. After the particle reaches the interface, it is rapidly consumed, meaning that the concentration difference of components near the phase interface is basically unchanged. For convenience of discussion, the (*CN* − *CM*) of components near the phase interface was considered approximately as a constant. Therefore, the effect of heating time on the growth rate of the spinel crystals mainly depends on the concentration gradient of the particles in the melt.

As shown in Figure 7, the theory of the spinel transition fraction X (X = process/final precipitation amount × 100) reached 88.37%, 92.25%, and 96.29% at 1450 ◦C, 1400 ◦C and 1350 ◦C, respectively. This reveals that the process of spinel crystallization is close to the terminal at the experimental temperature. In addition, during the solidification of the CaO-SiO2-MgO-Al2O3-Cr2O3-FeO system, the contents of chromium in the residual liquid-slag and spinel crystals are as shown in Figure 8. At 1450 ◦C, the chromium content in the liquid phase was only 0.49%, which was far less than that in the spinel crystals. Obviously, the concentration gradient of particles was small due to the completion of the precipitation region of the spinel crystals and the limited content of chromium in the liquid phase. Moreover, the content of the residual liquid phase decreased with the completion of spinel crystal crystallization. The diffusion condition of the solute atoms becomes worse, which causes the diffusion coefficient *D* to become evidently low. Slow particle diffusion results in the prolongation of the heating time having little effect on the increase in the size of the spinel crystals. If the holding point is carried out at a higher temperature, the higher content of chromium in the liquid phase and the higher concentration gradient of the solute will increase, leading to increases in the growth rate of the spinel crystals and the size of spinels.

**Figure 7.** Theoretical transition fraction of a spinel crystal.

**Figure 8.** Content of chromium in the spinel crystal and liquid phase during slag solidification.
