## 3 Results and Discussion
The \(\tau_{\rm c}\) of free PMA chains was not significantly affected by \(N\) (Figure 2c) in the range of molecular weight studied. The grafting of the PMA chains to the PS core resulted in a decrease of in the local subsegmental mobility of the PMA chains resulting in larger \(\tau_{\rm c}\) values for the grafted PMA chains in comparison to the free PMA chains. As the \(\sigma\) increased, the constrain on each chain created by the neighboring chains increased and led to the increase in \(\tau_{\rm c}\) observed especially at low \(N\). At every \(\sigma\) as \(N\) increased, the difference in \(\tau_{\rm c}\) of the free and grafted PMA chains decreased. The \(\tau_{\rm c}\) measured by NMR relaxation is the average \(\tau_{\rm c}\) for the chains, and the decrease in the average \(\tau_{\rm c}\) of the PMA chain with increasing \(N\) suggested that the reduced dynamics experienced by the PMA segments was more important close to the interface with the PS core, in keeping with the decrease of monomer concentration as the distance from the surface increases, which reduced the crowding generated by neighboring chains [13]. Furthermore, as the length of the chains increased the difference observed for the nanoparticles with different \(\sigma\) decreased, because the segments of the grafted chains farther away from the core are less constrained at every \(\sigma\), in keeping with observation of polymer chains grafted to hard nanoparticle cores [16]. Furthermore, the grafting of the PMA chains to the PS core influenced the local dynamics of the PS cores. After the grafting of the PMA chain at the interface of the PS NPs, a limited decrease in the \(\tau_{\rm c}\) of the PS core was observed, and this effect was stronger when the surface of the PS NPs was functionalized with long PMA chains (Figure 2d). Furthermore, as \(\sigma\) increased, the impact of the PMA chains on the dynamics of the PS core increased. This result can be attributed to an interplay between  
Figure 2: (a) \({}^{1}\)H spin–spin relaxation time of methoxy group of the PMA canopy and (b) \({}^{1}\)H spin-lattice relaxation time of aromatic ring of the PS core in the PS–PMA NPs with different degrees of polymerization of the PMA chains (\(N\)) grafted on PS cores with different grafting density, measured at 298 K by using an NMR spectrometer at a Larmor frequency of 700.02 MHz; (c) Correlation time of \({}^{1}\)H of methoxy group of PMA canopy obtained from \(T_{2}\) relaxation and (d) correlation time of \({}^{1}\)H of the aromatic ring of PS core obtained from \(T_{1}\) relaxation, calculated from BPP theory. For PS-PMA NPs with low (0.17 chains nm\({}^{-2}\), \(\square\)), medium (0.8 chains nm\({}^{-2}\), \(\triangle\)), high (2.5 chains nm\({}^{-2}\), \(\Circle\)) grafting density and free PMA chain (\(\Diamond\)). the slow dynamics of PS core and faster dynamics of PMA canopy. A similar behavior can be observed for block copolymer when a component with a high relaxation rate, increases the local dynamics of a slower block, because of the increased local fluctuations of the junction between the two segments initiated by the faster component [37; 38]. Additionally, the effect of \(\sigma\) suggests that the number of grafted chains, rather than the dynamics of the grafted chains themselves, was the driving force for the faster dynamics observed with the PS core functionalized with a high number of long PMA chains. The variation of the \(\tau_{\rm c}\) of a solvated polymer at different temperatures (Figure 3) can be described as an Arrhenius relation, where the activation energy associated with the subsegmental relaxation of the chains (Figure 4a,b) is a measure of the cooperativity of the segmental movement [39; 40; 41]. As the temperature increased, the \(\tau_{\rm c}\) of the PMA chains decreased as expected (Figure 3a,b), since higher temperatures increase the local subsegmental mobility of the polymer. However, in the case of the PS core, the \(\tau_{\rm c}\) increased with an increase in the temperature (Figure 3c,d).