## 3 Results and Discussion
### Randomly CuAAC-Grafted Arborescent Copolymers from G1 and G2 Substrates
Both the PGly and PEO shell components are water-soluble and required no further modification. First- and second-order analysis of the correlation function, \(\mid\)g\({}_{1}\)(\(\tau\)) \(\mid\) and \(\mid\)g\({}_{2}\)(\(\tau\)) \(\mid\), respectively, provide information on the size dispersity of the system. Monodispersed samples are expected to yield identical results for their first- and second-order analysis, since the correlation function can be represented by a single exponential decay under these conditions [23]. Therefore, as the size distribution of a sample broadens or becomes bimodal, the difference between the first- and second-order analysis results increases. The first- and second-order hydrodynamic diameters (\(d_{\rm h1}\) and \(d_{\rm h2}\), respectively) obtained for the copolymers and their respective PBG cores in DMF are compared in Table 4. The uncertainties reported are either the standard deviation for a series of 5 measurements or 1 nm, whichever was larger. This approach was used in an earlier investigation [16], together with more detailed CONTIN analysis of the DLS data of obtain size distributions, and it was shown that both methods led to similar conclusions. For the sake of completeness, CONTIN analysis results were also included as Supporting Information (Figures S1 and S2) in the current investigation, and reference is made to these results in support of the conclusions from the simpler analysis based on \(d_{\rm h1}\) and \(d_{\rm h2}\) as required. The variability of CONTIN analysis, due to its sensitivity to baseline noise [24] and difficulties encountered in resolving multimodal distributions [25], should be kept in mind when attempting to compare size distributions derived from DLS. Consequently, such comparisons should be considered semi-quantitative at best. Figure 6: SEC traces in DMF with 0.1% LiCl for (top to bottom) G2 PBG-\(g\)-P(HEA-TMS) crude, G2 PBG-\(g\)-PGly, G2 PBG-\(g\)-PEO, G1 PBG-\(g\)-P(HEA-TMS), G1 PBG-\(g\)-PGly, G1 PBG-\(g\)-PEO, G2 PBG, and G1 PBG. The relatively narrow molecular weight distributions observed in the SEC measurements (\(M_{\text{w}}/M_{\text{n}}\leq 1.14\)) suggest that the first- and second-order hydrodynamic diameters of the arborescent copolymers should be in close agreement. This is indeed the case for most samples characterized in DMF, since this solvent is relatively good for both the core and the shell components of the molecules. One notable exception is the \(d_{\text{h1}}\) and \(d_{\text{h2}}\) values for G1 PBG-\(g\)-PHEA in DMF, which are much larger than for the G1 PBG core in DMF, suggesting that aggregated species were present in that sample even in DMF. The \(d_{\text{h1}}\) and \(d_{\text{h2}}\) values also vary by 14 nm, which is likewise consistent with aggregation. A similar trend is observed for G1 PBG-\(g\)-PGly, with significantly larger \(d_{\text{h1}}\) and \(d_{\text{h2}}\) values and a 13 nm difference. The size distributions for both randomly-grafted copolymers, generated by CONTIN analysis of the DLS data and provided as Supporting Information (Figures S1 and S2), are consistent with these results. Sample G1 PBG-\(g\)-PHEA has the broadest size distribution among all the samples, with sizes spanning a range of over one order of magnitude and an average diameter ca. five-fold larger than the G1 PBG substrate (Figure S1). While the size distribution for G1 PBG-\(g\)-PGly is narrower, the distribution is centered at diameters ca. 10-fold larger than the substrate. Aggregation is not expected for arborescent PBG in DMF, but once it is partly deprotected (through loss of the benzyl ester group) its solubility in DMF may be decreased. The aggregation of the G1 PBG copolymers in DMF is, therefore, attributed to the fact that they have a rather open structure relatively to the G2 PBG copolymers, and the presence of free carboxyl groups, both of which favor the aggregation of the PBG cores in DMF.