## 3 Results and Discussion
### Results: Effects of Concentration \(c\) and Arm Degree of Polymerization \(N_{a}\)
In the same Figure 3b, we also plotted for comparison in dotted lines the variations of \(q^{\star}\) for different kinds of compact crystalline structure. The general agreement and accuracy are quite satisfying. We notice the slight departure from the \(c^{1/3}\) scaling law as the concentration increases. It may be ascribed to the fact that at large concentrations, \(q_{1}\)* enters the \(q\) region where the star form factor shows a steeper variation, producing a shift to lower \(q\). It could also be ascribed to some increase in packing compactness as concentration increases, which would be more sensitive for lower \(N_{\mathrm{a}}\). As a final remark on the variation of \(q_{1}\)*, we stress the fact that the \(c^{1/3}\) scaling law shows a continuous variation when the solution passes \(c^{\star}\). This classical behavior does not allow us, in principle, to differentiate between interpenetration of the stars and a progressive shrinking (compaction) of each star keeping them at the overlapping concentration. A better insight could come from the analysis of the shape of the scattering maximum, but modeling is beyond the scope of this paper. We discuss below (see Section 3.6) another fact in favor of interpenetration. Let us now turn towards the second type of maximum observed at \(q_{2}\)* \(>q_{1}\)*. Since this maximum never exists for neutral PS stars, it has an electrostatic origin. Its position dependence upon polymer concentration shows that it is not commensurate with the first one (Figure 3a). For low \(c\), \(q_{2}\)* is found to be constant. Actually, it only depends on the arm degree of polymerization \(N_{\mathrm{a}}\) for a fixed mean functionality \(<\)\(f\)\(>\). For a given \(N_{\mathrm{a}}\), we attribute this to the fact that the average internal polymer concentration \(c_{\mathrm{i}}\) inside each individual star is constant, as long as the stars do not overlap each other. This regime actually corresponds to the dilute regime, \(c<c^{\star}\). For \(c\) above a certain threshold, \(q_{2}\)* starts to increase. Our interpretation is that stars begin to interpenetrate each other; therefore their average inner concentration increases. In practice, this could be related to either an interdigitation or a compaction process. The only knowledge of the peak position (and unfortunately, its limited accuracy around \(c^{\star}\)) is indeed not enough to distinguish between them since both could in principle lead to a similar variation of the average correlation length with \(c\). Since the number of arms is rather limited, we consider hereafter the interdigitation process. Whatever the real situation is, the concentration at which \(q^{\star}\) passes from a plateau to an increasing line is therefore interpreted as the overlap threshold, \(c\)*. Note the interesting fact that in this interpenetrating range \(c>c\)*, the \(q_{2}\)* values are found to be unaffected by the arm degree of polymerization \(N_{\mathrm{a}}\). All \(q_{2}\)* values fall over the same dotted line, which corresponds correctly to the variation \(q^{\star}(\mathrm{\AA^{-1}})=(0.165\pm 0.002)\cdot c\) (mol\(\cdot\) L\({}^{-1}\))\({}^{1/2}\) found for linear fully sulfonated NaPSS [18; 19; 20; 21]. Therefore, when the stars studied here (with a point-like center) are interpenetrated, the average correlations behave as if the repeat unit's concentration was spatially homogeneous inside the star and equal to the nominal one, \(c\). The exponent of the scaling law \(q_{2}\)* \(\sim c^{\alpha}\), with \(\alpha=\frac{1}{2}\), for all these highly sulfonated stars, is at variance with the lower value of \(\alpha\) found for NaPSS stars of lower degree of sulfonation \(0.63<\tau_{\mathrm{S}}<0.78\) in Ref. [6]. The latter variation agreed with the one found from partially sulfonated linear PSS solutions (e.g., \(q^{\star}=0.130c^{0.465}\), for \(\tau_{\mathrm{S}}=0.69\), from SAXS measurements of [25; 26]).