## 3 Results and Discussion
### Synthesis of Randomly Grafted Arborescent Copolymers  
Structural variables such as the size and the branching density (porosity) of arborescent copolymers can be designed to meet requirements for specific applications. [PERSON] and [PERSON] previously reported the use of linear poly(ethylene oxide) with a molecular weight of 5000 (PEG\({}_{110}\)), grafted either randomly or at the end of the chains on arborescent PBG substrates to generate arborescent copolymer micelles [28]. Unfortunately, the randomly grafted arborescent G1 PBG-\(g\)-PEO\({}_{110}\) copolymers obtained by that approach yielded large aggregated species in phosphate-buffered saline (PBS), while G2 PBG-\(g\)-PEO\({}_{110}\) and G3 PBG-\(g\)-PEO\({}_{110}\) were both insoluble. The corresponding end-grafted samples PBG-\(e\)-PEO\({}_{110}\) displayed better dispersibility in PBS than the randomly grafted systems, albeit a small population of aggregates still existed in these samples. Therefore, we hypothesized that by randomly grafting longer hydrophilic PEO chains (M\({}_{n}\) = 10,000 g/mol), the solubility of the copolymers in PBS would be enhanced to yield stable unimolecular micelles with minimal aggregation. This would in turn affect the size and the properties of the micelles, and thereby provide control over the encapsulation and release behaviors of these unimolecular micelles. A detailed discussion of the synthesis of linear amine-terminated PEO, characterization results, and \({}^{19}\)F NMR analysis to determine the amine functionality level of PEO (\(f_{\rm NH2}\)) is provided in the SM. The synthesis of amphiphilic arborescent copolymers from amine-terminated PEO chains and the randomly carboxyl-functionalized PBG substrates was achieved under previously optimized reaction conditions [28]. The coupling of a G1 PBG substrate with amine-terminated PEO chains is illustrated in Scheme 2 as an example of the synthesis of an amphiphilic arborescent copolymer. The results obtained for grafting PBG substrates of generations G1 and G2 with PEO side chains are summarized in Table 2. The absolute molecular weight of the substrates and the arborescent copolymers was determined by SEC-MALLS analysis in DMSO. In all cases, the molecular weight of the copolymers increased relatively to the PBG substrates, while the dispersity remained low (D \(\leq 1.09\)). The grafting yield (G\({}_{\text{y}}\)), defined as the fraction of linear chain segments becoming attached to the substrate, was determined from the weight fraction of each component in the copolymers, along with the known amounts of substrate and side chains used in each grafting reaction. The calculation of the grafting yield for sample G1 PBG\({}_{15}\)-\(g\)-PEO is provided in Equation (S2) (Supplementary Materials) as an example. Several factors could have contributed to the relatively low (13-29%) grafting yields in the synthesis of the amphiphilic arborescent copolymers. It was assumed that partial deprotection of PBG would yield randomly distributed coupling sites within the substrates, but the deprotection of adjacent groups is also possible, which may lead to increased coupling site crowding. The grafting yield achieved for the G1 PBG-\(g\)-PEO copolymers was higher than for the G2 PBG-\(g\)-PEO copolymers, which is also expected due to the increased rigidity of PBG substrates of higher generations. The branching functionality \(f_{n}\), defined as the number of PEO side chains added, was determined by dividing the increase in molecular weight observed for the grafted copolymer by the molecular weight of the PEO side chains. A calculation for sample G1 PBG\({}_{15}\)-\(g\)-PEO is provided in Equation (S3) as an example. All the copolymers synthesized were purified by preparative SEC. The SEC traces for the purified samples, corresponding to the data summarized in Table 2, are compared in Figure 2. As expected, a decrease in elution volume was observed as the generation number increased.