## 3 Results and Discussion
### Counterions Condensed and/or Trapped Inside the Stars: Information from the Second Maximum  
#### 3.5.1 Estimating the Inner Counterions Concentration from \(q_{2}\)*  
The second maximum in the scattering functions of PE stars represents the electrostatic repulsion between arms. In the dilute regime, it is exclusively concerned with the repulsion between arms belonging to the same star. Indeed, it is an intramolecular characteristic, as proved by the superimposition of the scattering functions \(g(q,c)\) associated with distinct values of \(c<c^{\star}\), in the corresponding \(q\)-range (Figure 1a).  
It has been often assumed that the correlation length associated to the position of this second maximum \(q_{2}\)* is connected to an average internal screening length \(K_{\rm i}{}^{-1}\), through the scaling law \(q_{2}\)* \(\propto K_{\rm i}\)[6]. \(K_{\rm i}{}^{-1}\) is an average Debye screening length, which can be linked to the ionic strength and therefore, to an average inner concentration of counterions \(c_{\rm ion}\) through:  
\[K_{\rm i}{}^{2}\ =\ 4\pi l_{\rm B}c_{\rm ion} \tag{11}\]  
Counterions can be condensed around the chains, or simply trapped between the arms. If we consider that condensed ions do not participate to the screening and that only trapped ions (of concentration \(c_{\rm trapped}\)) are active  
\[K_{\rm i}{}^{2}\ =\ 4\pi l_{\rm B}c_{\rm trapped} \tag{12}\]  
As \(q_{2}\)* does not depend on concentration in the dilute regime (Figure 3a), we can assume, \(c_{\rm ion}\) or \(c_{\rm trapped}\) to be constant for \(c<c^{\star}\). Equation (12) is also valid in the semidilute regime (\(c>c^{\star}\)) where \(q_{2}\)* varies as \(c^{1/2}\) (trapped ions can now be viewed as classical \"free ions\" in a semidilute solution). Since \(q_{2}\)* gradually evolves from a constant value to the \(c^{1/2}\) regime, we can argue that the concentration of trapped ions in the dilute state coincides with that of the free ions just above \(c^{\star}\). This indicates that most counterions are confined inside the stars and show a condensation process similar to that of linear chains, as expected from the scaling approach of [PERSON] and [PERSON] for osmotic star PEs [14].  
In the dilute state, the fixed concentration of trapped counterions corresponds to a fraction of the internal monomer concentration \(c_{\rm i}\) and is thus related to the overlap concentration \(c^{\star}\). Therefore, Equation (12) becomes:  
\[K_{\rm i}{}^{2}\propto 4\pi l_{\rm B}c*\ \propto 4\pi l_{\rm B}c_{\rm i}\propto\ <f>\ N_{\rm a}/R^{3} \tag{13}\]  
around the arms. The observed existence of Manning-Oosawa condensation agrees with findings of simulations [10; 11; 12; 13], but at the same time no effect of center-peaked counterion distribution is shown by \(q_{2}\)*. Observing an average can be linked to the fact that the inner ionic strength is not low enough to yield a sufficiently detectable \(l_{\rm p}\) increase. Then its value would be irrelevant: the main trend towards chain stretching remains repulsion between all charged repeating units inside the star, even in the strong coupling limit. This agrees also with our finding that \(R/N_{\rm a}\) varies as 1.4 A: the chain is rather strongly elongated. Above \(c^{\star}\), \(q_{2}\)* varies as in linear chain semidilute solutions; we cannot detect any onset of the progressive compaction suggested by simulations in the strong coupling regime [10].  
A more extensive discussion would require more information about counterion distribution. This will be studied more precisely, via the variation of the counterion valence in a forthcoming paper.