## 4 Discussion  
To further investigate the modification mechanism of nano-particles, we discuss the behavior of nano-particles in liquid mediums. To date, the polarization of nano-particles has been fully proved under the applied electric field. The charges will distort the electric field and accumulate on the surface of nano-particles, which form a \"barrier\" around the nano-particles [37, 38]. Due to the mobility of electrons largely outweighing the ions, our primary concern here is the movement of electrons. While nano-particles can capture electrons and reassemble into negatively charged nano-particles, the breakdown process is suppressed, which can explain why the breakdown strength of nano-liquids is improved after nano-modification in our experiment. The modification effect is closely associated with the trapped electrons process by the polarized nano-particles, which in turn, is significantly affected by the interface between nano-particles and base liquids. Researchers have found a positive correlation between the modification effect and interface, which is greatly affected by the volume fractions [43, 46]. Consequently, we discuss the interface between nano-particles and base liquids in different volume fractions. As a special colloid, the stability of nano-liquids can be improved by lowering the potential energy of nano-particles. According to the DLVO (Derjaguin, Landau, Verwey and Overbeek) theory, we can \begin{table}
\begin{tabular}{c c c c c} \hline \hline _Volume Fractions_ & \(BS_{TiO_{3}}(\mathrm{kV/cm})\) & \(BS_{Al_{2}O_{3}}\) (\(\mathrm{kV/cm}\)) & \(ME_{TiO_{2}}\) & \(ME_{Al_{2}O_{3}}\) \\ \hline
0\% & 670.6 & 670.6 & 0\% & 0\% \\
0.2\% & 686.2 & 704.6 & 2\% & 5\% \\
0.4\% & 741.9 & 729.9 & 11\% & 9\% \\
0.6\% & 791.5 & 790.3 & 18\% & 18\% \\
0.8\% & 761.7 & 770.7 & 14\% & 15\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The average breakdown strength and modification effect of TiO\({}_{2}\) and Al\({}_{2}\)O\({}_{3}\) nano-liquids. \(BS\) represents the average breakdown strength, which is expressed in units of (\(\mathrm{kV/cm}\)), and \(ME\) represents the modification effect. Figure 6: Weibull distribution of pure transformer oil and nano-liquids’ breakdown strength with volume fractions of added TiO\({}_{2}\) nano-particles and Al\({}_{2}\)O\({}_{3}\) nano-particles ranged from 0.2% to 0.8%. obtain the attractive potential energy \(V_{A}\) and repulsive potential energy \(V_{R}\) by analyzing the van der Waals force and electric double-layer forces [47; 48]  
\[V_{A}=-\frac{H}{12}(\frac{1}{x^{2}+2x}+\frac{1}{x^{2}+2x+1}+2 ln\frac{x^{2}+2x}{x ^{2}+2x+1}) \tag{3}\]  
\[V_{R}=\frac{64\pi rcST\gamma^{2}}{k^{2}}e^{-ka} \tag{4}\]  
where \(x=\frac{a-2r}{2r}\), \(\gamma=\frac{2\pi\varphi_{0}}{4\pi T}\), \(k=(\frac{N_{A}cz^{2}a^{2}}{cST})^{\frac{1}{2}}\), \(H\) is the Hamaker constant, \(r\) is the particle radius, \(a\) is the centroid distance, \(T\) is the temperature of nano-liquids, \(z\) is the charge number in base liquids, \(q\) is the unit charge, \(\varphi_{0}\) is the surface potential of nano-particles, \(\varepsilon\) is the dielectric constant of base liquids, and others are the constant related to the liquid state. The potential energy \(V\) of nano-particles is equal to the sum of the attractive and repulsive potential energy  
\[V=V_{A}+V_{R} \tag{5}\]  
The potential energy \(V\) of nano-particles is related to many factors, such as particle radius, centroid distance and temperature. The stability of nano-liquids is directly affected by potential energy. The lower the potential energy, the more stable the system. Because of the small particle size (about 1-100 nm), there is a high surface energy and specific surface area in nano-particles. In order to analyze the interaction forces between the suspended nano-particles, we assumed that the nano-particles are spherical particles of the same size and evenly dispersed.