## 3 Results and Discussion
### Interfacial Stability of NPs at the Oil-Water Interface  
To enhance interfacial stability, NPs must adhere strongly to the oil-water interface. Under such conditions, NP layers form around oil drops, preventing their close approach, and thus, prohibiting coalescence [36]. As particle size increases, the oil-water interface behavior improves because adhesion energy is enhanced. In principle, the adhesion energy is influenced by the IFT, as well as NP wettability and size. This relationship is demonstrated as follows: \(E=\pi R^{2}\gamma_{\textit{dw}}\) (\(1-cos\ \theta\))\({}^{2}\), where \(E\) is the adhesion energy of a particle, \(R\) is the particle radius, \(\gamma_{\textit{dw}}\) is the IFT, and \(\theta\) is the contact angle of the particle at the oil-water interface. Thus, small particles exhibit lower adhesion energy, compared with large particles. We found that all silica NPs remained in stable emulsions for ten days, and thus, readily adsorbed to the oil-water interface, this included 7 nm CSNPs (Figure 4).  
To evaluate oil droplet sizes in emulsions, we performed bright-field microscopy (Figure 5a). Regardless of particle size, all droplets ranged in size from 0.5 to 4 \(\upmu\)m and the mean diameter of all droplets (D drop.) ranged from 1.47 \(\upmu\)m (12 nm CSNP) to 1.76 \(\upmu\)m (12 nm FSNP) (Figure 5b). As shown in Table 1, the mean D drop. was used to calculate the mean droplet volume (V drop. = 1/6 \(\uppi\)\(\times\) D drop.3). All emulsions included 4.0 \(\times\) 10\({}^{12}\)\(\upmu\)m\({}^{3}\) of oil (V tot. oil vol.). Thus, we calculated the total numbers of oil droplets (N total drop. = V tot. oil vol./V drop.) and the surface area per droplet (S aver. drop. = \(\uppi\)\(\times\) D drop.2). The V drop. and S aver. drop. decreased as drop size (D drop.) decreased. Because V tot. oil vol. was identical in each instance, N total drop. increased as droplet size decreased. We found that the 12 nm CSNPs had the smallest droplet diameter, and thus, formed the largest number of droplets (Table 1).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **7 nm CSNP** & **12 nm CSNP** & **22 nm CSNP** & **12 nm FSNP** \\ \hline Mean droplet diameter (D drop.), \(\upmu\)m & 1.63 & 1.47 & 1.66 & 1.76 \\ \hline Mean surface area of droplet (Saver. drop. = \(\uppi\)\(\times\) D drop.2, \(\upmu\)m2) & 8.34 & 6.78 & 8.66 & 9.73 \\ \hline Mean droplet volume (V drop. = 1/6 \(\uppi\)\(\times\) D drop.3), \(\upmu\)m3 & 2.26 & 1.66 & 2.39 & 2.85 \\ \hline Total oil volume in emulsion (V tot. oil vol.), \(\upmu\)m3 & & & \(4.0\times 10^{12}\) & \\ \hline Total number of emulsion droplets (Ntot. drop. = V tot. oil vol./V drop.) & 1.76 \(\times\) 10\({}^{12}\) & 2.40 \(\times\) 10\({}^{12}\) & 1.67 \(\times\) 10\({}^{12}\) & 1.40 \(\times\) 10\({}^{12}\) \\ \hline Total surface area of emulsion droplets (Stot. drop. = Ntot. drop. Saver. drop.), \(\upmu\)m3 & 1.47 \(\times\) 10\({}^{13}\) & 1.63 \(\times\) 10\({}^{13}\) & 1.44 \(\times\) 10\({}^{13}\) & 1.36 \(\times\) 10\({}^{13}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mean surface area per droplet.  
Figure 4: Phase stability tests of emulsions stabilized by differently sized NPs (**a**) immediately upon emulsion formation and (**b**) after ten days.