## 2 Experimental
### SANS Measurements  
SANS experiments were conducted on both diffractometers: PACE at LLB, CEA-Saclay, Gif-sur-Yvette, France; and D22, at ILL, Grenoble, France. Measurements on PACE were done using three distinct configurations (sample-to-detector distances \(D\) = 1, 3, and 4.6 m with wavelengths \(\lambda\) = 6, 10, and 13 A, respectively) covering the \(q\)-range 0.003 \(<q<\) 0.322 A\({}^{-1}\). Those on D22 used two configurations (\(D\) = 1.5 and 15 m with \(\lambda\) = 6 A) leading to the \(q\)-range 0.004 \(<q<\) 0.544 A\({}^{-1}\). \(q\) is the magnitude of the scattering vector, defined by the wavelength of the incident beam \(\lambda\) and the angle between incident and scattered beams \(\theta\) such that \(q=(4\pi/\lambda)\cdot\sin(\theta/2)\). The scattered intensity was recorded by: a multidetector with 30 concentric rings of 1 cm width, on PACE; a 2-dimensional \({}^{3}\)He gas detector, on D22. PE solutions were held in circular and calibrated quartz cells of distinct thickness 1, 2.5, or 5 mm according to their concentration. Multiple scattering was then negligible. Also, the cells were mounted in a temperature-controlled sample holder. Most of the measurements were done at 20 \({}^{\circ}\)C, but a few others at 25 and 65 \({}^{\circ}\)C.  
The scattering data were treated according to the usual procedures for isotropic SANS [38; 39; 40]. They were first corrected for electronic noise of the detector, empty cell, sample transmission, and sample thickness. Geometrical factors and detector cells efficiency were then accounted for, using the (flat) incoherent scattering of H\({}_{2}\)O. Finally, the normalization to the unit incident flux was carried out through either the direct measurement of the number of incident neutrons (PACE) or the scattering of H\({}_{2}\)O 1 mm thick (D22). In this way, the differential cross section per unit volume \(\sum(q)\) of each solution is obtained in absolute units (cm\({}^{-1}\)). It is the sum of a coherent cross section term containing all the information needed to describe the structure of the solution, \(I(q)\), and a background term, \(\sum^{B}\), which has to be removed from \(\sum(q)\) to obtain \(I(q)\). \(\sum^{B}\) involves the scattering of the solvent and the incoherent scattering of the macromolecules (macro-ion plus counterions for a PE). For PE solutions, it can be readily replaced by the scattered intensity of a specific mixture of D\({}_{2}\)O and H\({}_{2}\)O solvent molecules. The D\({}_{2}\)O volume fraction of the equivalent D\({}_{2}\)O-H\({}_{2}\)O mixture is then assessed via a method described in references [7; 27; 28; 38]. At this stage, to make meaningful the comparisons between distinct polymers and distinct solutions of the same polymer, we consider the dimensionless scattering function \(g(q,c)\) defined as:  
\[g(q,c)\ =\ S(q,c)/cN_{Av}\ =\ I(q,c)/<K>^{2}cN_{Av} \tag{1}\]  
\(I(q,c)\) (cm\({}^{-1}\)) corresponds to the quantity \(I(q)\) defined above when measured at a given concentration \(c\), \(<\)K\(>^{2}\) (cm\({}^{2}\)) to the squared contrast factor between the polymer unit and the solvent (see Equation (2) below); \(S(q,c)\) (cm\({}^{-3}\)), to the scattering function of the polymer per unit volume; \(c\) (mol\(\cdot\)cm\({}^{-3}\)), to the polymer concentration of the solution (mode of repeating unit par unit volume); \(N_{Av}\) (mol\({}^{-1}\)), to Avogadro's number. For solutions of parent PS stars in THF, the average contrast length \(<\)K\(>\) is obtained from the relation:  
\[<K>\ =\ \tau_{\rm DVB}\cdot\ K_{\rm DVB}\,+\,(1-\tau_{\rm DVB})\cdot\ K_{\rm PS} \tag{2}\]  
\(K_{\rm DVB}\) and \(K_{\rm PS}\) (cm) are the \"contrast lengths\" of the elementary scatterers DVB (C\({}_{10}\)H\({}_{10}\)) and PS unit (C\({}_{8}\)H\({}_{8}\)) in deuterated THF, respectively.