# Particle Formation Mechanism of TiCl\({}_{4}\) Hydrolysis to Prepare Nano TiO\({}_{2}\)
## 2 TiCl\({}_{4}\) Hydrolysis Process and Mathematical Model
### Mathematical Model
#### 2.2.2 Nucleation Dynamics  
The hydrolysis process involves a change in the concentration of Ti\({}^{4+}\) over time. Referencing prior studies [20], the reaction rate for the formation of TiO\({}_{2}\) via hydrolysis of TiCl\({}_{4}\) takes the form of a power function:  
\[r_{Ti^{4+}}=-\frac{C_{t}}{dt}=k\epsilon_{t}^{\alpha} \tag{3}\]  
In this equation, \(r_{Ti^{4+}}\) signifies the reaction rate of \(Ti^{4+}\) hydrolysis, while \(t\) denotes the reaction time, \(C_{t}\) is the concentration of Ti\({}^{4+}\) at time \(t\), \(k_{1}\) is the reaction rate constant, and \(\alpha\) represents the reaction order. With the reaction sequence set at \(\alpha=1\), an experimental relationship can be drawn between Ti\({}^{4+}\) concentration and time:  
\[C_{Ti^{4+}}=\exp(-0.098t+2.708) \tag{4}\]  
Based on the principles of homogeneous nucleation along with the previous literature [21; 22], the initiation of primary homogeneous nucleation demands an energy barrier that surpasses the nucleation energy, \(\Omega\). Assuming a spherical shape for both the embryo and nucleus, we derive  
\[\Omega=\frac{16\pi\sigma^{3}V_{m}^{2}}{3(k_{B}RTlnS)^{2}} \tag{5}\]  
Combining the homogeneous nucleation theory with the Arrhenius equation, the nucleation rate is given as  
\[B_{1}=Z_{c}\exp(-\frac{\Omega}{RT}) \tag{6}\]  
By substituting Equation (4) into Equation (5), we arrive at  
\[B_{1}=Z_{c}\exp(-\frac{16\pi\sigma^{3}V_{m}^{2}}{3{k_{B}}^{2}(RT)^{3}(lnS)^{2 }}) \tag{7}\]  
In these equations,  
\[S=\frac{c}{c_{sat}} \tag{8}\]  
\[Z_{c}=\frac{\mathrm{k}(c-c_{sat})^{n}}{c_{sat}^{n^{\prime}}} \tag{9}\]  
Furthermore, the secondary nucleation of TiO\({}_{2}\) may occur due to the influence of crystalline slurry. Its rate can be represented by Formula (10). \[B_{2}=k_{2}\tau^{\alpha}G^{q}M_{T}^{p} \tag{10}\]  
The total nucleation rate can be obtained by substituting Formulas (7) and (10) into Formula (11). \[B=B_{1}\times B_{2} \tag{11}\]  
Formula (12) is obtained by further collation. \[B=\frac{k_{1}(c-c_{sat})^{n}}{c_{sat}^{n^{\prime}}}\tau^{\alpha}M_{T}^{p}Q^{ q}\exp(-\frac{k_{3}}{T^{3}(lnS)^{2}}) \tag{12}\]  
where \(k_{3}\) is a constant. \[C_{Ti^{4+}}=\exp(-0.098t+2.708) \tag{4}\]0  
In Equation (12), \(B\) denotes the assay nuclear rate (number/\(\upmu\)m\({}^{3}\cdot\)s), \(R\) stands for the gas constant (8.314 J/mol\(\cdot\)K), and \(k_{1}\) is the nucleation rate constant. The difference between\(c\) and \(c_{sat}\) refers to the supersaturation concentration, while \(c_{sat}\) signifies the saturation concentration (kg/m\({}^{3}\)). The supersaturation index, represented by \(n\), is considered to be 1, given that TiO\({}_{2}\) is fundamentally insoluble in water. The saturation index, denoted \(n^{\prime}\), is consistent with the supersaturation index. The impeller tip velocity (m/s) during the growth of the crystal core is represented by \(\tau\), with \(o\) being the agitation index, typically set to 1 based on empirical data. \(M_{T}\) denotes the suspension density (kg crystal/m\({}^{3}\) suspension), and \(p\) signifies the suspension density index. The formation of TiO\({}_{2}\) can lead to an agglomeration process, which impacts the suspension density. However, this influence is not incorporated into the calculation, suggesting that \(p\) is zero. \(Q\) symbolizes the growth rate (\(\upmu\)m/s), and \(q\) represents the growth rate index in the nucleation equation, typically set to 1. The molar volume of the crystal nucleus is represented by \(V_{m}\) (0.14364 nm\({}^{3}\)), while \(\sigma\) stands for the interfacial tension (0.05737 N/m). Lastly, \(k_{B}\) denotes the Boltzmann constant (1.380649 \(\times\) 10-23 J/K). Drawing on [23] and the diffusion theory [24], the reaction order \(\alpha\) defines the correction coefficient \(\eta\) with the formula \(\eta=\left(\frac{c_{g}-c_{sat}}{c-c_{sat}}\right)^{\alpha}\), where \(c\) signifies \(C_{Ti^{+}}\) and \(c_{s}\) represents the concentration on the solid surface.