## 1 Introduction  
Nanoparticles (NP) possess a large surface area to volume ratio, and the control of the interface has a critical influence on the properties of the NP systems by tuning their interaction with their surrounding environment. Controlling the interfacial properties is critical since the incompatibility of the NPs with their surrounding media can result in the agglomeration of the NPs. The interfacial properties of the NPs can be controlled by tethering polymer chains at the interface, to improve the NPs compatibility with a solvent or a polymer matrix [1; 2; 3; 4; 5]. The layer of tethered polymers, whether the chains are in a brush conformation or not, forms a canopy around the core particle, and this layer of grafted chains defines the interaction between the NPs and between the NPs and their environment [6]. In such cases, the final behavior of the NPs is influenced by the chemical composition of the grafted chains, but also by their degree of polymerization (N) and their grafting density, the number of grafted chains per unit of area (\(\sigma\)). Variations in \(N\) and \(\sigma\) in systems where the polymer chains are tethered to hard and rigid nanoparticle core, such as silica or gold, influence the dynamics of the grafted polymer chains.And the architecture of the canopy of grafted polymer chains defines the interfacial properties of the NPs, such as particle/particles interaction or NPs/solvent interaction [6; 7; 8; 9; 10; 11]. However, the behavior of systems where the core is itself soft and deformable like a nanogel has not been addressed, and the dynamics of polymer chains grafted on the surface of soft NPs need to be thoroughly understood to improve the design of such systems [12]. The architecture of the grafted polymer system, defined by the \(\sigma\) and \(N\) of the canopy, affects the conformation [13; 14; 15] and the dynamics of the grafted chains [16]. Both of which, in turns, largely affect the macroscopic behavior of systems based on polymer-functionalized NPs, such as the mechanical properties of nanocomposites [16; 17] or the rheology of colloidal suspensions [11; 16; 18]. In a polymer-grafted NPs system, usually, the polymer chains experience a deceleration of the relaxation in comparison to free chains as a consequence of the stretched chain conformation near the surface of nanoparticles and increased confinement generated by the proximity of adjacent chains [19] at least at the scale of the local segmental motion. On a larger scale, the increased confinement experienced by densely grafting chains usually resulted in an increase of glass transition temperature of tethered chains in comparison to the free chains ascribed to the presence of a reduced number of chain ends and smaller free volume in the grafted system [20; 21; 22]. The investigations of the behavior of polymer grafted nanoparticle systems have mostly dealt with rigid core NPs like silica or gold. However, when soft and swollen polymer NPs are used as the substrate, the intrinsic dynamics of the polymer chains in the core also needs to be taken into account. In such swollen polymer nanoparticles, the dynamics of the chains is typically constrained and sparsely crosslinked nanogel displayed a soft interface with polymer chains having a rapid relaxation, while densely crosslinked nanogels exhibited glassy surface with constrained local mobility [23; 24; 25]. In nanocolloids resulting from the self-assembly of block copolymers, this arrested core is surrounded by a canopy of swollen and mobile polymer chains. In such a case, the polymer chains within the core experienced a soft confinement, where the dynamics is influenced by the fluctuation of the interface where the two polymer components are connected [26]. Such phenomena affect the design of soft particles for different applications, for example, theranostic systems [23; 25; 27].