## 2 Experimental  
### Materials  
Polystyrene (PS) stars were prepared according to a so-called arm first method. PS arms, with a narrow molecular weight distribution, were synthesized by anionic polymerization and, in monocarbanionic chain form, initiate the polymerization of a small amount of divinylbenzene (DVB). Star-shaped PS with a minute core of cross-linked poly-(DVB) is obtained [32]. The smallest possible DVB/(PS living end) ratio is chosen in order to restrict the size of the star core while a reasonable functionality, \(f\), is reached. The DVB content of the star \(\tau_{\rm DVB}\) is defined as the ratio of the number of DVB monomers to the total number of monomers. To remove residual PS linear chains and reduce the width of the \(f\)-distribution, some PS stars were fractionated by further size exclusion chromatography (SEC). The same technique was also applied to obtain the molar mass distributions of both arms and stars, respectively. Such coupled measurements lead to mean functionalities <\(f\)> and variances \(\sigma^{2}\) of \(f\)-distributions [7; 33]. PS stars were then sulfonated according to the [PERSON] _et al._ procedure [34; 35]. After neutralization with sodium hydroxide (NaOH), NaPSS stars were purified by extended dialysis against pure water (conductivity of the order of 1 \(\mu\)S\(\cdot\) m\({}^{-1}\)) followed by freeze-drying. Their characterization was carried out by elemental analysis. In this way, the degree of sulfonation \(\tau_{\rm S}\), defined as the ratio of the number of sulfonated units to the total number of units (DVB core excluded), and the weight fraction of water content \(\tau_{\rm w}\), were determined for each NaPSS star. In addition, static light scattering experiments were performed on NaPSS stars in dilute aqueous solutions with added salt (NaCl) in order to verify that the sulfonation process did not disturb the star shape of the initial PS samples. A PS star is referred to as S\(x\), in order to identify a star (S) architecture and a rough estimate of the arm degree of polymerization (\(x\approx\) <N\({}_{\rm a}\)>\({}_{\rm N}\)). For the related NaPSS star, \(-\)Sy is added to the initial PS star reference in order to specify the sulfonated character (S) and the arm degree of sulfonation (\(y=\tau_{\rm S}\) (%)). Table 1 lists the characteristics of the different samples used in this study. We note that the high degrees of sulfonation used in this study (\(\tau_{\rm S}\) = 91% and 94%) were achieved through a second sulfonation according to the same procedure. \begin{table} \begin{tabular}{l c c c c c c c c c c c c c c} \hline \hline
**Sample** & \(\mathbf{X_{1}\%}\) & \(\mathbf{\frac{<M_{2}\times N}{g\cdot mol^{-1}}}\) & \(\mathbf{\textit{I}_{a}}\) & \(\mathbf{<N_{2}\times N}\) & \(\mathbf{\frac{M_{N}}{g\cdot mol^{-1}}}\) & \(\mathbf{\textit{I}}\) & \(\mathbf{\tau_{\text{type}}\%}\) & \(\mathbf{\textit{<f>}}\) & \(\mathbf{\textit{<g>}}\) & \(\mathbf{\textit{<f>}}\) & \(\mathbf{\textit{<g>}}\) & \(\mathbf{\textit{<g>}}\) \\  & & & & & & & & & & & & & \(\mathbf{\textit{<g>}}\) & \(\mathbf{\textit{<g>}}\) & \(\mathbf{\textit{<g>}}\) & \(\mathbf{\textit{<g>}}\) & \(\mathbf{\textit{<g>}}\) & \(\mathbf{\textit{<g>}}\) \\ \hline
540 & 3 & 3.8 k & 1.05 & 36 & 40 k & 1.13 & 3.70 & 10.0 & 0.35 & 0 & 105.1 & 96.68 & 0 & 1.924 & 18.60 \\
540-885 & 3 & – & 1.05 & 36 & – & 1.13 & 3.70 & 10.0 & 0.35 & \(\mathbf{\textit{<g>}}\) & 188.6 & 109.90 & 10.7 & 0.515 & 5.66 \\ S100 & \(<\)2 & 10 k & 1.09 & 96 & 126 k & 1.06 & 1.86 & 10.8 & 0.32 & 0 & 104.6 & 96.54 & 0 & 0.993 & 9.59 \\ S100-891 & \(<\)2 & – & 1.09 & 96 & – & 1.06 & 1.86 & 10.8 & 0.32 & \(\mathbf{\textit{91}}\) \(\mathbf{\textit{<g>}}\) & 195.8 & 111.22 & 10 & 0.075 & 0.83 \\ S300 & \(<\)2 & 29 k & 1.10 & 280 & 364 k & 1.07 & 0.64 & 12.4 & 0.25 & 0 & 104.3 & 96.45 & 0 & – & – \\ S300-894 & \(<\)2 & – & 1.10 & 280 & – & 1.07 & 0.64 & 12.4 & 0.25 & \(\mathbf{\textit{94}}\) \(\mathbf{\textit{<g>}}\) & 199.6 & 111.93 & 11 & 0.01 & 0.11 \\ \hline \hline \end{tabular}