## 3 Results and Discussion
To investigate the effect of the latter, the fluorescence spectra of Py-PDMS in THF (\(\eta\) = 0.47 mPa.s) were compared to those acquired in dioxane (\(\eta\) = 1.37 mPa.s), a more viscous solvent with a chemical structure similar to that of THF [17]. As shown in Figure 3A,C, the spectra acquired in THF and dioxane are similar, implying similar PEF. Thus, the difference in PEF between the spectra obtained in THF and DMF should not be attributed to a difference in solvent viscosity. Rather, as will be demonstrated below from the analysis of the fluorescence decays, the lower PEF observed in DMF stems from the absence of pyrene aggregates generated by H-bonds in THF and dioxane between the amide linkers connecting the pyrene labels to the polysiloxane backbone. Here again, the analysis of the fluorescence spectra is complicated by the presence of an additional pyrene species, with the pyrene aggregates emitting as excimers upon direct excitation. The fluorescence decays of the Py-PDMS solutions were acquired and fitted according to the MFA. The MFA equations were derived by noting that any fluorescence decay can always be fitted with a sum of exponentials. This insight was applied to fit the fluorescence decay of the pyrene monomer with the sum of exponentials given in Equation (2), where one of the exponentials had a lifetime (\(\tau_{\text{M}}\)) matching that of isolated pyrenes (\(P_{\text{free}}\)*) and the other exponentials would represent pyrene excimer formation by diffusive encounters. The sum of exponentials describing excimer formation could be processed according to the standard kinetic scheme for excimer formation to yield a mathematical expression for the fluorescence decay of the excimer as shown with Equation (3). Global analysis of the monomer and excimer decays by optimizing both the pre-exponential factors and decay times ensured that all relevant parameters describing the kinetics of pyrene excimer formation could be retrieved with good accuracy. The fits were excellent, yielding \(\chi^{2}\) values that were always <1.30 and plots of the residuals and autocorrelation of the residuals that were randomly distributed around zero. An example of the fluorescence decay fits is shown in Figure 4. All parameters retrieved from the MFA of the fluorescence decays are listed in Tables S1-S3 in Supplementary Materials. Fitting the fluorescence decays yielded the average rate constant <\(k\)> for PEF calculated with Equation (4) and the molar fractions of all pyrene species present in solution. These included the pyrene labels that formed excimer by diffusion (\(Py_{\text{diff}}\)*), those that were aggregated and formed excimer by direct excitation of a dimer where the pyrene labels were well (\(E0\)*) or poorly (\(EL\)*) aligned, and those that were unable to form excimer (\(Py_{\text{free}}\)*) (either because they were unattached and free in solution (\(PAA\)*), located in a pyrene-deficient region of the polymer or attached to a short oligomer (\(Py\)*-ODMS)). These pyrene species had molar fractions \(f_{\text{diff}},f_{\text{E0}},f_{\text{EL}},f_{\text{free}},f_{\text{PAA}}\), and \(f_{\text{ODMS}}\), respectively. As shown in Equation (6), the MFA parameters can be combined to provide an absolute measure of the ratio of the fluorescence intensity from the excimer and monomer, \((I_{\text{E}}/I_{\text{M}})^{\text{TRF}}\). Figure 5 shows plots of \((I_{\text{E}}/I_{\text{M}})^{\text{TRF}}\) versus \((I_{\text{E}}/I_{\text{M}})^{\text{SSF}}\) (obtained from analyses of all fluorescence spectra) in THF, DMF, and dioxane. The linear relationship suggests that the parameters retrieved from the MFA of the TRF decays describe accurately the fluorescence behavior of the Py-PDMS samples. Furthermore, the point resulting from the ratio \((I_{\text{E}}/I_{\text{M}})^{\text{TRF}}\) and \((I_{\text{E}}/I_{\text{M}})^{\text{SSF}}\) for the fractionated sample, Py(3.5)-PDMS(f), also fell on the line of best fit for the unfractionated samples.