## 3 Results and Discussion
### Effect of NP Number Density on the Oil-Water IFT  
The NP spatial density at the oil-water interface is important in terms of interfacial stability. An increased NP number density reduces the interfacial area covered by each NP. To explore the effect of NP number density on IFT, we calculated the interface areas covered by one NP, assuming that all NPs were located at the oil-water interface. First, we calculated the total number of NPs present in the emulsion (N \({}_{\text{tot. NP}}\)) considering the total mass of NPs (M \({}_{\text{tot. NP}}\)) and the density of each bare NP (\(\rho\) NP) used for emulsion formation. As listed in Table 2, M \({}_{\text{tot. NP}}\) for each silica NP was 0.0199 g, thus, 0.5 wt\(\%\) in the aqueous phase. \(\rho\) NP ranged from approximately 0.5 \(\times\) 10\({}^{-21}\) (12 nm FSNP) to 2.5 \(\times\) 10\({}^{-21}\) g/nm\({}^{3}\) (7 nm CSNP). Thus, the total volume of NPs in emulsion (V \({}_{\text{tot. NP}}\) = M \({}_{\text{tot. NP}}\)/\(\rho\) NP) ranged from approximately 7.93 \(\times\) 10\({}^{18}\) nm\({}^{3}\) (7 nm CSNP) to 3.98 \(\times\) 10\({}^{19}\) nm\({}^{3}\) (12 nm FSNP), and the total number of NPs (N \({}_{\text{tot. NP}}\) = V \({}_{\text{tot. NP}}\)/V\({}_{\text{NP}}\)) ranged from approximately 1.46 \(\times\) 10\({}^{15}\) (22 nm CSNP) to 4.42 \(\times\) 10\({}^{16}\) (7 nm CSNP). The interfacial area covered by each NP (A \({}_{\text{inter. NP}}\)) was the total surface area of an emulsion droplet (S \({}_{\text{tot. drop. }}\)) divided by N \({}_{\text{tot. NP}}\) A \({}_{\text{inter. NP}}\), thus ranged from 309.99 nm\({}^{2}\) (12 nm FSNP) to 9836.12 nm\({}^{2}\) (22 nm CSNP). The FSNP dispersion has a low density; many particles attain the interface when NPs of the same mass are injected for emulsion generation. Therefore, the interfacial area per particle is low for the FSNP dispersion. Figure 5: (**a**) Droplets of different sizes in NP-stabilized emulsions and (**b**) their size distributions. We measured the changes in the interfacial properties of oil-water mixtures according to the variation in interfacial area per NP. Assuming that NP-NP interactions can be neglected, the IFT (\(\gamma_{c}\)) is \(\gamma_{c}=\gamma_{o}-\frac{N_{P}E_{d}}{A}\), where \(\gamma_{o}\) is the IFT of the bare interface, \(E_{d}\) is the desorption energy required to remove one NP from the interface, and \(N_{P}\) is the number of NPs in a given interfacial area A [13,37]. \(\gamma_{c}\) is the IFT between the aqueous phase containing silica NPs and the oil phase, and \(\gamma_{o}\) is 14 mN/m, yielding the IFT between oil and the aqueous phase in the absence of silica NPs (the aqueous phase contains PSS-\(co\)-MA and the zwitterionic TPS surfactant) (Figure 6). Thus, any increase in the IFT-difference (\(\gamma_{o}-\gamma_{c}\)) indicates that NPs reduce the IFT (\(\gamma_{c}\)). We found that as the interfacial area per NP decreased, the IFT-difference increased (Figure 6a), because the NP-NP spacing became smaller, thereby the interface area per NP decreased to approximately 1/30. We calculated the desorption energy required to remove one NP (\(E_{d}\)), this comprised the IFT-difference (\(\gamma_{o}-\gamma_{c}\)) as a function of \(E_{d}\) (Figure 6b). A low \(E_{d}\) indicates that an NP is easily adsorbed to and detached from the interface, therefore, more effectively reducing the IFT at low \(E_{d}\). Although the \(E_{d}\) was indeed low, we confirmed that all CSNP dispersions and the FSNP dispersion exhibited adequate emulsion stability (i.e., no phase separation) (Figure 4). The 12 nm FSNP dispersion adsorbed readily to the oil-water interface at low values of \(E_{d}\) and exhibited stable interface behavior, thereby increasing the IFT-difference (\(\gamma_{o}-\gamma_{c}\)).