## 3 Results and Discussion  
### Results: Effects of Concentration \(c\) and Arm Degree of Polymerization \(N_{a}\)  
The scattering functions of the parent PS stars (non sulfonated, organic solvent) are not shown in this paper (see Ref. [6]). We mainly focus on the scattering functions \(g\)(\(q\),\(c\)), or \(g\)(\(q\),\(\phi\)) of the related fully sulfonated NaPSS stars. Their evolutions with the unit concentration \(c\), or the corresponding volume fraction \(\phi\), as well as the number average arm degree of polymerization \(<\)\(N_{\text{a}}\)\(>\)\(N_{\text{N}}\) (noted \(N_{\text{a}}\) hereafter) are shown in Figures 1 and 2. First Discussions: Using the Variations of the Maxima Positions \(q_{1}{}^{*}\) and \(q_{2}{}^{*}\) with Concentration \(c\) to Attribute Them to Defined Distances (\"Indexation\")  
We consider now Figure 3a which contains most of the information enclosed in this paper, through the simple survey of the main features of the c-dependence of the position of each kind of maxima. It allows us to lay stress on the difference in nature of both maxima. It turns out that it enables their indexation. Indeed, both show power laws for \(q^{*}\) according to \(c\) (seen as straight lines in log-log representation), but with different exponents (slopes in log-log representation). First, \(q_{1}{}^{*}\)(open circles and open squares) almost scales as \(c^{1/3}\) on the whole explored concentration range, with different front factors for the two stars S100-S91 and S300-S94, _i.e._, for the two distinct \(N_{\rm a}\) values (the two lines are shifted in Figure 3a). This c-dependence is different from the one found for the polyelectrolyte peak of linear NaPSS polyelectrolytes in the semi-dilute regime, which is close to \(c^{1/2}\) (dotted line in Figure 3a). The exponent 1/3 suggests that the first maximum corresponds to a position order in three dimensions. \(q_{1}{}^{*}\) is then related to a mean distance between objects in a 3d liquid- or crystalline-like structure (scaling laws for simple cubic, bcc, and fcc lattices are presented in  
Figure 3: Indexation of the main maxima of the PS and NaPSS star scattering functions. **(a)** Scattering vectors \(q^{*}\) (PS) [1], \(q_{1}{}^{*}\) and \(q_{2}{}^{*}\) (NaPSS) _versus_ statistical unit concentration \(c\). The scaling law associated with NaPSS linear polyelectrolytes of sulfonation degree close to one is also plotted (\(q^{*}\) = (0.165 \(\pm\) 0.002)\(c^{1/2}\)[18; 20]); **(b)**\(q^{*}\) (PS) [1] \(q_{1}{}^{*}\) (NaPSS) _versus_ star concentration \(c_{\rm star}\) (not to be confused with \(c^{*}\)). The expected scaling laws for a simple cubic lattice (..), a body-centered cubic (bcc)-lattice (- - -) and a face-centered cubic (fcc)-lattice (\(\times\)\(\times\)\(\times\)). Figure 3b). From this point of view, it is noticeable that scattering functions of parent PS stars in THF, a good solvent, show similar maxima with identical positions \(q^{\star}\) for the same concentration (Figure 3a; \(q^{\star}\) for S100, full circles, is identical to \(q_{1}\)* for S100-S91, open circles, at the same concentration). This similarity between neutral and charged stars gives further support to a direct relation between \(q_{1}\)* and the mean distance between star cores, \(2\pi/q_{1}\)*. By the way, it also confirms nicely that no significant degradation of the star-branched architecture occurs under the sulfonation process. Finally, all the \(q_{1}\)* values can be roughly superimposed into a single curve if the star concentration \(c_{\mathrm{star}}\) (not to be mistaken with \(c^{\star}\)) is used as variable instead of the polymer concentration \(c\) (Figure 3b). Taking into account polydispersity, \(c_{\mathrm{star}}\) is defined as:  
\[c_{\mathrm{star}}\;=\;(<m>/M_{\mathrm{N}})c \tag{4}\]  
\(M_{\mathrm{N}}\) is the number average molecular weight of the star; \(<\)\(m\)\(>\), the molar mass of its effective units (Table 1); \(c\), the statistical unit concentration.