## 3 Results and Discussion
Figure 4: TRF decays of the pyrene monomer (**left**, \(\lambda_{\text{em}}\) = 375 nm) and excimer (**right**, \(\lambda_{\text{em}}\) = 510 nm) for the Py(5.2)-PDMS sample in THF fitted according to the MFA. \(\lambda_{\text{ex}}\) = 344 nm. While the MFA parameters in Figure 5 support the attribution of the various pyrenyl species, the presence of Py-ODMS and residual PAA in the solution with pyrene labels that could not form excimer complicated the interpretation of \(<\)\(k\)\({}^{\text{MF}}\)\(>\)blob, which accounts for the change in pyrene content of the different Py-PDMS samples while assuming that all pyrene labels are covalently attached to PDMS. As shown in Equation (5), \(<\)\(k\)\({}^{\text{MF}}\)\(>\)blob depends on \(x\), the molar fraction of structural units in PDMS bearing a pyrene label and \(f_{\text{Mfree}}\), the molar fraction of pyrene labels bound to PDMS that do not form excimer and are probed in the monomer decay. Unfortunately, because the monomeric emissions from the \(Py_{\text{free}}\)*, \(PA\)*, and \(Py\)*-ODMS species exhibit a long lifetime of about 250 ns in THF, their individual contributions could not be isolated through MFA of the TRF decays; only the sum could be ascertained. The derivation of \(x\) was mentioned in the Experimental section and is described in details in the Supporting Information. The derivation of \(f_{\text{Mfree}}\) takes advantage of the fact that the MF and GPC analyses in Table 1 yield the sum of \(f_{\text{free}}\) +\(f_{\text{ODMS}}\) +\(f_{\text{ PAA}}\) and of \(f_{\text{ODMS}}\) +\(f_{\text{ PAA}}\), respectively. Subtracting the latter from the former yielded \(f_{\text{free}}\) which could then be utilized to determine the molar fractions, \(f_{\text{diff}}\)*, \(f_{\text{free}}\)*, and \(f_{\text{agg}}\)* (=\(f_{\text{E0}}\)* + \(f_{\text{EL}}\)*), the latter fractions representing all of the pyrene species attached to the polymeric PDMS sample. They correspond to the main component of the GPC traces shown in Figure 2. In turn, the fractions \(f_{\text{diff}}\)* and \(f_{\text{free}}\)* were employed to calculate \(f_{\text{Mfree}}\) (see Equation (S20)), which was then employed in Equation (5) to determine \(<\)\(k\)\({}^{\text{MF}}\)\(>\)blob. Thus, \(f_{\text{diff}}\)*,\(f_{\text{free}}\)*, and \(f_{\text{agg}}\)* and the average rate constant \(<\)\(k\)\({}^{\text{MF}}\)\(>\)blob were plotted in Figure 6 as a function of pyrene content for the Py-PDMS samples in THF, DMF, and dioxane. For comparison purposes, similar trends are plotted for the previously reported data in THF with Py-PIMA [11], a polymer whose backbone is expected to be much more rigid than that of PDMS. The profiles of the molar fractions display some common features. At low pyrene content, a large fraction of the lumophore groups (\(f_{\text{free}}\)*) did not form excimer, and the polymers showed very little pyrene aggregation (\(f_{\text{agg}}\)* \(\sim\)0). As reflected by the increases in \(f_{\text{diff}}\)* and \(f_{\text{agg}}\)* (except for Py-PDMS in DMF where \(f_{\text{agg}}\)* \(\sim\)0), more excimer was produced by diffusive encounters when the pyrene content was increased to intermediate values. At the highest pyrene contents explored, no isolated pyrene remained in solution (i.e., \(f_{\text{free}}\)* = 0). As found for the Py-PDMS samples in DMF, \(f_{\text{diff}}\)* can pass through a maximum or remain constant depending on the extent of pyrene aggregation. The absence of significant aggregation in DMF suggests that the amide linkers connecting 1-pyreneacetic acid to the 3-aminopropyl side chains of the PDMS substrate induces some H-bonding for Py-PDMS in THF and dioxane. Nevertheless, the level of aggregation remained moderate for the Py-PDMS samples even in these solvents;\(f_{\text{agg}}\) remained \(<\)0.35 for all solvents over the range of pyrene contents investigated. The pyrene content (\(x(\)_ffree_ = 0)), where _ffree\({}^{*}\)_ reaches zero in the plots shown in Figure 6, marks the point where all pyrene labels are able to form an excimer.