## 3 Results and Discussion
Thus, \(T_{1}\) relaxation time, although providing less precise measurement of the local subsegmental chain relaxation, was also use to probe the PS-PMA NPs. The \(T_{2}\) of PMA was used to probe the relaxation dynamics of the tethered chains (Figure 2a). The \(T_{2}\) of free PMA chains depicted ca. 0.77 s at \(N=46\) and slightly decreased with an increase of \(N\). The short tethered PMA chains (\(N=35\)) at \(\sigma_{\mathrm{low}}\) showed of \(T_{2}\) of 0.42 s and as chain length increased, the \(T_{2}\) value also moderately increased. The increase in the \(T_{2}\) value indicates an enhanced subsegmental mobility of the chains. Thus, it can be concluded that the local mobility of the tethered PMA chains with \(\sigma_{\mathrm{low}}\) increased when the \(N\) of the chain increased. Even though the PMA chains on the PS-PMA NPs at \(\sigma_{\mathrm{low}}\) displayed a collapsed brush conformation, given their chain stretching factor (Table 2), their local subsegmental mobility was reduced in comparison to untethered PMA chains of the same length, likely because the presence of the PS substrate restricted the subsegmental motion of the chains. As \(\sigma\) increased from low to high, the \(T_{2}\) of the short tethered PMA decreased due to the transition from a collapsed conformation to a stretched brush regime (Table 2) generated by the crowding produced by the adjacent chains. The relaxation of the free and untethered PMA chains was not influenced by the \(N\) in the range of molecular weights studied. However, for the grafted chains, as the chain length increased, the \(T_{2}\) of the grafted PMA chains systematically increased for all \(\sigma\). The increase in the local mobility of the grafted PMA chains as \(N\) increased was attributed to the reduction in the spatial crowding observed as the distance from the core increased with increasing \(N\)[13]. These observations are in keeping with what has been observed for the local subsegmental dynamics of polymer chains grafted to the rigid nanoparticles [19; 36]. In the PS-PMA NPs system, both the PMA and the PS were undergoing relaxation. Addressing the effect of the grafting of polymer chains on the dynamics of the soft core can be the key to explain the divergences observed when comparing the macroscopic properties of soft and hard polymer-functionalized particles [18]. However, the local dynamics of the crosslinked PS core was too slow to be probed efficiently by the measurement of the \(T_{2}\) (Figure S4, Supplementary Material). In the slow regime, the \(T_{1}\) of a proton will increase as the local chain \(\tau_{\mathrm{c}}\) decreases. Furthermore, the \(T_{1}\) values are significantly influenced by the local magnetic field. Consequently, to get a clear picture of the chain relaxation in the PS cores \(T_{1}\) relaxation times were measured at different magnetic fields (Figure S5, Supplementary Material). The relaxation of all the unfunctionalized PS cores exhibited similar \(T_{1}\) relaxation times. The grafting of PMA chains of different \(N\) at \(\sigma_{\mathrm{low}}\) resulted in no significant changes in the \(T_{1}\) value observed. However, when PS cores were functionalized with PMA chains at \(\sigma_{\mathrm{medium}}\) and \(\sigma_{\mathrm{high}}\), the \(T_{1}\) of the PS core decreased with the increase of \(N_{\mathrm{PMA}}\). (Figure 2b) Consequently, at \(\sigma_{\mathrm{medium}}\) and \(\sigma_{\mathrm{high}}\) the grafting of the PMA chains increased the mobility of the PS core, and this effect was more pronounced with longer chains. The relaxation time \(T_{1}\) and \(T_{2}\) obtained by NMR spectroscopy are measures of the relaxation of the spins system, an indirect measurement of the local dynamics, those values can be used to calculate the \(\tau_{\rm c}\) of subsegmental motions of the chains [35]. The average \(\tau_{\rm c}\) of the PMA chains and of the PS core were independently measured from the \(T_{2}\) for the PMA and \(T_{1}\) for the PS.