## 3 Results and Discussion
### Effect of Temperature
#### 3.4.3 Comparison with the Wormlike Chain Model  
Beyond the comparison with the scaling approach, it is important to mention that the front factor of the linear scaling law determined for the radius of the NaPSS stars is also similar to the contour lengths per repeat unit--an \"effective repeat distance\" which we note \(a_{\rm eff}\), obtained from the analyses of the measured form factors for fully sulfonated PSS chains through the wormlike chain model, estimated between 1.7 and 2.5 A [48; 49; 50]. This also means that both for linear chains and stars, the electrostatic blob size is not far from that of the repeat unit. Moreover, a comparison with a model of chain conformation is important for the description of the arm conformation. The scattering curves reported in this paper do not provide a direct measurement of the form factor. However, the radius extracted as discussed above gives at least a direct measurement of the end-to-end distance of the arms \(R\). In the frame of the wormlike chain description, it can be written:  
\[R^{2}\ =\ 2l_{\rm p}\times[L-l_{\rm p}(1-\exp(-L/l_{\rm p}))]\ =\ 2l_{\rm p}\times[Na_{\rm eff}-l_{\rm p}(1-\exp(-N(a_{\rm eff }/l_{\rm p}))] \tag{7}\]  
\(L=Na_{\rm eff}\) is the chain contour length; \(N\), the degree of polymerization; \(a_{\rm eff}\), the effective contour length per repeat unit; \(l_{\rm p}\), the persistence length. For \(l_{\rm p}/a_{\rm eff}\) very large (or very small \(L/l_{\rm p}\) values), Equation (7) reduces to \(R\sim N\cdot a_{\rm eff}\). We will see below that in our experiments, for each value of \(N\), the obtained value of \(l_{\rm p}\) is such that the contour length and the radius of the stars only represents a few persistence lengths \(l_{\rm p}\). To be more accurate, we consider the Zimm approximation of the Schulz distribution for the degree of polymerization:  
\[w(N)\ =\ (1/k!)\cdot(k/N_{N})^{k+1}\cdot N^{k}\cdot\exp(-k\cdot N/N_{N}) \tag{8}\]  
where \(N_{\rm N}\) is the number-average degree of polymerization and \(k\) is related to the polydispersity index \(I\) by the relation \(I=N_{\rm W}/N_{\rm N}=(k+1)/k\). The number-average of the square of the end-to-end distance is therefore:  
\[<R^{2}>_{\rm N}\ =2l_{\rm p}^{2}\left[\frac{a_{\rm eff}}{l_{\rm p}}N_{N}-1+ \left(\frac{k}{\left(\frac{a_{\rm eff}}{l_{\rm p}}N_{N}+k\right)}\right)^{k}\right] \tag{9}\]  
Thus, accounting for the arm polydispersity according to Equation (8) and using \(<\)\(N_{\rm a}\)\(>_{\rm N}\) (listed in Table 1), as well as a common average effective monomer size \(a_{\rm eff}\) = 2.1 A, we can find solutions for \(l_{\rm p}\): \(l_{\rm p}\) = 29 A, for \(<\)\(N_{\rm a}\)\(>_{\rm N}\) = 36 (\(R\) = 53 A); \(l_{\rm p}\) = 80 A, for \(<\)\(N_{\rm a}\)\(>_{\rm N}\) = 96 (\(R\) = 145 A); \(l_{\rm p}\) = 180 A, for \(<\)\(N_{\rm a}\)\(>_{\rm N}\) = 280 (\(R\) = 391 A). As announced above, within this approach, the radius of the stars \(R\) and the chain contour length (\(L=N_{\rm a}\). \(a_{\rm eff}\)) only involve a few persistence lengths (\(R\approx 2l_{\rm p}\), and \(L/l_{\rm p}\approx 3\)). This leads to a linear relationship between \(R\) and \(N_{\rm a}\). The observed variation of \(l_{\rm p}\) with the degree of polymerization can be surprising. However it may simply result from the variation of the average internal concentration in the stars \(c_{i}\), which depends on \(N_{\rm a}\) and which can be directly approximated to \(c^{\star}\) (where stars are in contact and fill the space). \[c_{i}\approx c*\ \propto\ <f>\cdot N_{\rm a}/R^{3} \tag{10}\]  
If we compare \(l_{\rm p}\) values with that of linear NaPSS chains (in the semidilute range) for a monomer concentration equal to \(c_{i}\), we find that the two variations are close (see Figure S1 in Supplementary Materials using data from Ref. [18] and others).