**4. Discussion**

The formation of Gibbs free energy changes of the (ZrO2)*n* clusters and various ZrO2 nanoparticles are shown in Figure 8. Most of the Gibbs free energy change curves of the (ZrO2)*n* clusters are higher than those of the ZrO2 nanoparticles, which is in agreemen<sup>t</sup> with the law of the step-by-step change of the thermodynamic stability., from the perspective of the gradual decrease of the energy, this law indicates that the formation process of ZrO2 occurs from the atoms to clusters to nanoscale crystal particles to the macroscale crystal.

**Figure 8.** Formation Gibbs free energy changes of (ZrO2)*n* clusters and various ZrO2 nanoparticles.

The surfaces of nanoparticles usually contain two or three atomic layers [38,39], and the average length of the Zr–O bond is approximately 0.214 nm. Therefore, the surface atomic layer of zirconium oxide nanoparticles is approximately 0.428–0.642 nm thick. From Table 4, the sizes of the (ZrO2)*n* clusters are also within this range. In addition, from Table 5, the energy gap of the (ZrO2)2 cluster is the largest, indicating that this cluster is the most stable. Therefore, the (ZrO2)2 cluster may be the atomic layer on the surface of the ZrO2 nanoparticle. Based on the two-step nucleation theory, it is speculated that the nucleation pathway of ZrO2 is Zratom + Oatom → (ZrO2)*n* → (ZrO2)2 → core (ZrO2 particle)–shell ((ZrO2)2 cluster) nanoparticle → (ZrO2)bulk. The nucleation process of ZrO2 at 1873 K is shown in Figure 9.

**Figure 9.** Nucleation process of zirconium oxide at 1873 K.

## **5. Conclusions**

High-temperature deoxidation experiments and inclusion-extraction experiments have been performed. The nucleation process of ZrO2 inclusions in Zr deoxidized steel was investigated by classical nucleation theory and first-principles calculation. The main conclusions are as follows.

When the Zr content was 100 ppm, SEM-EDS showed that the main inclusions in the steel were ZrO2. μXRD analysis confirmed the existence of ZrO2, and monoclinic and tetragonal ZrO2 were simultaneously detected, which may be because tetragonal ZrO2 transformed to monoclinic ZrO2 during the rapid cooling process. The average size of the ZrO2 inclusions was 0.56 μm. Through classical nucleation theory, the relationships between the solute element activities and the nucleation radius and nucleation rate of ZrO2 were obtained. The theoretical value of the nucleation rate was compared with the values in the literature, and the experimental value of *I* was approximately 40 orders of magnitude larger than the theoretical value.

The thermodynamic properties of macroscale ZrO2 were calculated by first principles, and the results were consistent with the experimental values. (ZrO2)*n* (*n* = 1–6) cluster models were constructed, and the thermodynamic properties of the geometrically optimized cluster structures and nanoscale ZrO2 particles were calculated, which verified the rationality of the existence of the pre-nucleation phase in terms of the thermodynamics. Based on two-step nucleation theory, the nucleation pathway of ZrO2 is proposed to be Zratom + Oatom → (ZrO2)*n* → (ZrO2)2 → core (ZrO2 particle)–shell ((ZrO2)2 cluster) nanoparticle → (ZrO2)bulk.

**Author Contributions:** L.W., C.C. and S.Y. conceived the idea and designed the research. Y.L. performed the theoretical calculations. Y.L. and X.L. performed the μXRD analyses. X.L. performed the high-temperature experiments. Y.L. and L.W. wrote the original draft of the paper, and all authors contributed to the review and editing of the paper. L.W. acquired the funding and supervised the study. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Science Foundation of China (Grant Nos. 52064011 and 52274331) and the Science and Technology Planning Project of Guizhou (Grant Nos. Qian Ke He Ji Chu ZK [2021]258 and Qian Ke He Zhi Cheng [2021]342). This study was also supported by the Research Program for Talented Scholars of the Guizhou Institute of Technology (Grant No. XJGC20190962).

**Data Availability Statement:** Part of the data presented in this study are available on request from the corresponding author. The data are not publicly available due to intellectual property.

**Acknowledgments:** Thanks for the computing support of the State Key Laboratory of Public Big Data, Guizhou University.

**Conflicts of Interest:** The authors state that there are no conflict of interest to disclose.
