**3. Results**

#### *3.1. Inclusion Characterization*

To obtain the size distribution of the ZrO2 inclusions, 100 SEM images were continuously taken at 5000× magnification, and the total observed area was 0.11 mm2. The average size of the ZrO2 inclusions in the sample was 0.56 μm, and the size distribution of ZrO2 inclusions approximately approached a normal distribution (Figure 1).

**Figure 1.** Size distribution of ZrO2 in Zr deoxidized steel.

The inclusions extracted by electrolysis were characterized by μXRD, and the morphology and composition were analyzed by SEM-EDS (Figure 2). From the mapping image, the inclusions in the Zr deoxidized steel were zirconium oxide. A few Al inclusions were also detected because of the trace amount of Al in the raw materials. The results of μXRD suggested that the zirconium oxide was ZrO2. In addition, both monoclinic and tetragonal ZrO2 were detected. This may be because of the transformation of tetragonal ZrO2 to monoclinic ZrO2 during rapid cooling.

**Figure 2.** Morphology, composition, and X-ray diffraction pattern of the inclusions in Zr deoxidized steel.

#### *3.2. Classical Nucleation Calculation*

According to classical nucleation theory, the relationship between the critical nucleation radius of ZrO2 and the activities of the solute elements at 1873 K is shown in Figure 3a. When the Zr activity is in the range 0.0001–1, and the oxygen activity is in the range 0.0001–0.1, the critical nucleation radius of ZrO2 is 0.3–1.2 nm. The relationship between the nucleation rate and solute element activities is shown in Figure 3b. When the Zr activity is in the range 0.0001–1, and the oxygen activity is in the range 0.001–0.1, the critical nucleation rate of ZrO2 is in the range 100–560 cm<sup>−</sup>3·s<sup>−</sup>1. The points in Figure 3 are the experimentally measured Zr and oxygen activities.

**Figure 3.** Relationships between the activities of the solute elements and the (**a**) critical nucleation radius and (**b**) nucleation rate of ZrO2.

To obtain the activities of zirconium and oxygen, the composition of the zirconium deoxidized steel and corresponding thermodynamic data were substituted into

$$a\_i = f\_i[\text{mass\%} \text{\'s}] \tag{7}$$

$$\log f\_i = \sum \varepsilon\_i^j [\text{mass\%} \ i] \tag{8}$$

where *ai*, *fi*, and [mass% *i*] are the activity, activity coefficient, and concentration of element *i*, respectively, and *eji*is the first-order interaction coefficient (Table 4).

**Table 4.** Interaction coefficients of O and Zr at 1873 K [26].


The experimentally estimated value of ln *I* can be calculated by Equations (5) and (6). The experimental value of ln *I* was 57 cm<sup>−</sup>3·s<sup>−</sup>1. From Figure 3, the theoretical value of ln *I* is approximately −40 cm<sup>−</sup>3·s<sup>−</sup>1. Therefore, the experimental value of the nucleation rate *I* was approximately 40 orders of magnitude larger than the theoretical value.

#### *3.3. First-Principles Calculations*

According to the two-step nucleation mechanism, the nucleation process of inclusions in liquid steel should include the multiphase deoxidation reaction, which can be expressed by the following two steps [32–35]. In the first step, the deoxidized elements in the molten steel melt and dissolve, and the deoxidized element atoms react with the dissolved oxygen in the molten steel to form oxide clusters. In the second step, the clusters combine to form cluster aggregates. The cluster aggregates then form critical crystal nuclei.

#### 3.3.1. Structures of (ZrO2)*n* Clusters

With increasing *n* value, the average bond length of the (ZrO2)*n* cluster initially slightly increases, and it finally fluctuates at approximately 2.0 Å (Table 5). The nucleon binding ability in the nucleus is stronger and more stable for larger average binding energy (*Ebin*). The average binding energies of the (ZrO2)*n* (*n* = 1–6) clusters are negative (Table 5), indicating that the binding between nuclei is relatively stable.


**Table 5.** Structures, average bond lengths, sizes, and average binding energies of (ZrO2)*n* (*n* = 1–6) clusters.

With an increasing number of atoms (*n* = 1–6), the average binding energy of the (ZrO2)*n* cluster decreases, especially between the (ZrO2)1 and (ZrO2)2 clusters. This may be because of the energy error caused by the different numbers of atoms in different clusters. Therefore, the energy gap was used to compare the stabilities of the clusters. The energy gap is the difference between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO). The energy gap reflects the ability of electrons to transition from an occupied orbital to an empty orbital, and it represents the ability of molecules to participate in chemical reactions. The system is more stable for a larger energy gap [36].

The energy gaps of the (ZrO2)*n* clusters are given in Table 6. The (ZrO2)2 cluster has the largest energy gap, indicating that the (ZrO2)2 cluster is the most stable of the (ZrO2)*n* clusters (*n* = 1–6). The LUMOs and HOMOs of the (ZrO2)*n* clusters are shown in Figure 4. The blue and yellow area denote the orbitals of electron cloud, where the color is to distinguish the plus or minus of orbital wave function.

**Table 6.** LUMO–HUMO energy gaps of the (ZrO2)*n* (*n* = 1–6) clusters.


**Figure 4.** HOMOs and LUMOs of the (ZrO2)*n* clusters.

3.3.2. Thermodynamic Properties of the (ZrO2)*n* Clusters

The thermodynamic properties of ZrO2 are shown in Figure 5. Where *S* is the entropy, and *Cp* is the heat capacity. In this study, the ZrO2 crystal structure was tetragonal. The lines and points represent the calculated thermodynamic properties and values in the literature, respectively [37]. In the temperature range of 0–1000 K, there is a certain degree of deviation between the calculated and the values in the literature, but the variation trend of the thermodynamic properties with temperature is consistent. To the best of our knowledge, the reason that caused deviation is mainly the machine error derived from the thermochemical software package, and it was not possible to meet strict consistency criteria when combining data from various sources to form a data set for a substance. Another reason is that phase transformation occurred between monoclinic and tetragonal ZrO2. In general, the calculated value is in good agreemen<sup>t</sup> with the value from the literature.

**Figure 5.** Thermodynamics properties of ZrO2.

#### 3.3.3. Gibbs Free Energy Changes of (ZrO2)*n* Clusters and Nanoparticles

The formation Gibbs free energy (Δ*G*) curves of the (ZrO2)*n* clusters and ZrO2(s) are shown in Figure 6a,b, respectively. The formation Gibbs free energies of (ZrO2)*n* (*n* = 1–6) are negative, suggesting the (ZrO2)*n* (*n* = 1–6) clusters form. However, the formation Gibbs free energy change from (ZrO2)1 to ZrO2(s) is positive, so the reaction from (ZrO2)1 to ZrO2(s) does not occur when the temperature is greater than 1000 K.

**Figure 6.** (**a**) Formation Gibbs free energy curves of (ZrO2)*<sup>n</sup>*. (**b**) Formation Gibbs free energy curve of ZrO2(s).

The Gibbs free energy changes from nanoscale ZrO2 to ZrO2(bulk) and from the Zr and O atoms to nanoscale ZrO2 are shown in Figure 7a,b, respectively. The Gibbs free energy of the macroscale ZrO2 crystal is less than zero, indicating that nanoscale ZrO2 will spontaneously transform to the macroscale ZrO2 crystal.

**Figure 7.** (**a**) Gibbs free energy change from nanoscale ZrO2 to ZrO2(bulk). (**b**) Gibbs free energy changes from the Zr and O atoms to nanoscale ZrO2.
