## 4 Discussion
### Energy of Attachment of NPs at Interface vs. Size of the Microspheres  
Next, we attempt to understand in a different way how the curvature of the oil/water in emulsion droplet and subsequently in the resulting polymer microspheres can be influenced by the interfacial energy of the NPs and consequently by their detachment energy from water vs. oil. The free energy of adsorption and desorption of the NPs at the interfaces are related \(\Delta G_{adsorption}=-\Delta G_{desorption}\). The desorption energy or the detachment energy of an NPs can be calculated from the contact angle at the interfaces and is given by the following expression [35; 43]:  
\[-\Delta G_{detachment}=\pi R^{2}\gamma_{PW}(1\pm\cos\beta)=\Delta G_{adsorption} \tag{6}\]  
where \(\gamma_{PW}\) is the P/W interfacial energy. It is important to note that the sign in the parenthesis is taken negative when the NP detaches/desorbs from the oil/water interface into water and the sign is taken positive when the NP detaches from the oil/water interface into the oil. This distinction is important, because in the former case it gauges the affinity of the NPs with the emulsion droplets and in the latter case it gauges the affinity with the water phase. Using Equation (6) and plugging in the value for the contact angle \(\beta\), of the NP to the polymer microspheres, calculated the detachment energies of the NPs from the polymer microspheres, \(\Delta G_{\text{NP/P}}\), see Table S5, and from water, \(\Delta G_{\text{NP/W}}\), see Table S6. From the values in Tables S5 and S6, the attachment energies of NPs to water are greater in magnitude than those of NPs to polymer microspheres. Further, the greatest differences in the energies of detachment of the NPs with water vs. polymer are those for the NP-Gly, followed by NP-CN and NP-SH, for each corresponding polymer. In other words, the NP-Gly has the strongest affinity toward water phase. In fact, NP-Gly also produces the largest colloidosomes, and polymer microspheres after polymerization for all type of polymers, see Figures 5 and S12, also Figures S4-S6. To further emphasize this aspect, in Figure 7, we have generated a surface plot, from all NPs and monomers investigated, relating the population distribution function of the detachment energies \(\Delta G_{\text{NP/W}}\) (from water) and the microsphere size. From this plot, we observe an increase in the size of the microsphere with the energy of detachment, regardless of the monomer and NP type used for generating Pickering emulsions.  
These findings generate a new conjecture, namely if the curvature of the liquid menisci formed by the oil/water interface between the tightly packed NPs at the interface could determine the curvature of the entire emulsion droplet. For the case of NPs immersed less than half into the oil phase, the oil/water interfacial menisci will be convex on the water side and concave on the oil side. The NPs weakly attached to water (with small detachment energies from water), i.e., more immersed into the oil phase, will produce menisci with greater curvature than those NPs more strongly attached to water, i.e., less immersed into the oil phase, thus generating smaller emulsion droplets, with a larger curvature of the interface than the latter. This new conjecture needs a more in-depth analysis, whereas the scaling parameters must also be considered, and cannot be exhaustively treated with the available data in the current work.