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Static and Dynamic Properties of Block-Copolymer Based Grafted Nanoparticles Across the Non-Ergodicity Transition

We present a systematic investigation of static and dynamic properties of block copolymer micelles with crosslinked cores, representing model polymer-grafted nanoparticles, over a wide concentration range from dilute regime to an arrested (crystalline) state, by means of light and neutron scattering, complemented by linear viscoelasticity. We have followed the evolution of their scattering intensity and diffusion dynamics throughout the non-ergodicity transition and the observed results have been contrasted against appropriately coarse-grained Langevin Dynamics simulations. These stable model soft particles of the core-shell type are situated between ultrasoft stars and hard spheres, and the well-known star pair interaction potential is not appropriate to describe them. Instead, we have found that an effective brush interaction potential provides very satisfactory agreement between experiments and simulations, offering insights into the interplay of softness and dynamics in spherical colloidal suspensions.


Introduction
Soft colloidal particles can be thought of as hybrids interpolating between polymers and hard spheres, 1 and offering a plethora of possibilities for designing systems with tunable dynamic response. Examples of spherical soft colloidal systems are vesicles, dendrimers, 2 microgels, [3][4][5] block copolymer micelles, [6][7][8][9] polymer-grafted nanoparticles, [10][11][12][13][14] and star polymers. [15][16][17][18][19] Unlike hard-spheres, for which the phase diagram and associated dynamic properties have been exhaustively investigated, 20 the respective consequences of softness have not been fully explored. For instance, bridging the gap between ultrasoft stars and polymer-grafted nanoparticles by identifying similarities and distinct features in the behavior still presents formidable challenges. The pioneering work of McConnnel et al. 21 established the phase diagram of polystyrene-polyisoprene diblock copolymers by means of small angle X-ray diffraction studies. Transition between face-centered cubic (fcc) and body-centered cubic (bcc) lattices was observed as the ratio of corona layer thickness to core radius increased. The resulting stable crystalline structure was found to be strongly dependent on the length scale of interactions: short-range repulsion favors fcc lattice over bcc and vice versa, as it was also demonstrated through previous experiments 22 and Monte Carlo simulations. 23 Particular attention has been paid on star polymers, whose density profile and conformation are described by the Daoud-Cotton model. [24][25][26] Such ultrasoft colloids represent one of the widest investigated model systems, where the softness can be tuned by varying the number of polymer chains anchored to a common center (functionality) and described accurately by a coarse-grained repulsive pair potential that varies logarithmically with the core-core distance. 27 Likos and co-workers 28 established the phase diagram of star polymer solutions in athermal solvent over a wide range of number densities and functionalities. The observed phases included bcc, fcc, body centered orthogonal (bco) and diamond crystals. Over a limited range of number of arms, a reentrant melting of the bcc phase was observed as the packing fraction increased. Additionally, at relatively high functionality a fcc phase was detected instead, and the effect of increasing packing fraction translated into different ordered lattices until the diamond crystal was reached. Interestingly, this interaction potential was found to be suitable also for several star-like systems, such as diblock copolymer micelles. 8,[29][30][31] The common denominator of such systems seems to be the relatively large ratio of hydrodynamic radius of the particle to the core radius (between 7 and 10). Gupta et al. 32 have presented a thorough experimental and theoretical phase diagram of soft colloids made of tunable block copolymers which allowed adjusting the softness by altering the solvophobic-to-solvophilic block ratio, hence bridging the starto-micelle regimes. An important finding relates to the threshold in functionality above which crystalline phase is not formed and the resulting non-ergodic state is characterized by a random distribution of the particles forming a glass. On the other hand, below the threshold value, fcc lattice or fcc-to-bcc transition is observed as the previous works mentioned above. Note that these micelles were stable due to the combination of solvophobicity and large interfacial tension. In general, block copolymer micelles are characterized by arm exchange kinetics which impacts their structure and dynamics properties. 8,30,[33][34][35][36][37][38][39] Stable micelles can be obtained in a relatively straightforward way by chemical crosslinking of their cores, and this type of soft colloids will be discussed below. 5,10,40,41 An important challenge is the fact that the internal particle microstructure may give rise to completely different features with respect to the star-polymers or star-like micelles, in a way that is not fully understood yet. For instance, Ohno et al. 42 and Morinaga et al. 43,44 showed by means of confocal laser scanning microscopy in fluorescent mode, that for core-shell systems (silica core grafted with polymethylmethacrylate) the crystalline phase was strongly dependent on the brush length but the bcc crystalline structure was not observed at all. Instead, a hexagonal close-packed (hcp) structure was detected. When short chains were grafted, the probability of having a fcc lattice was found to be the same as for the hcp organization. A nearly random stacking of fcc and hcp was formed. The authors called this region concentrated polymer brush (CPB) where the excluded volume region effect is unimportant. In a region of large chain lengths, the CPB was followed by the excluded volume region, or semidilute polymer brush regime (SDPB), whose effect on the crystalline structure becomes important. Note that, the brush conformation regime proposed by Ohno et al. 43 derives from the well-known Daoud-Cotton model for star polymers. 24 The probability of finding a fcc increases sharply as the interaction potential softens due to the increase in length of the brushes. However, during the crystallization process, some hcp arrangements might be frozen in an irreversible way.
This non-equilibrium process is more likely to happen for systems with a shorter range of interparticle potential such as hard-spheres and CPB systems. The theoretically deduced entropy difference between fcc and hcp phases may be too small to realize an equilibrium system experimentally. 43 On the other hand, as the interparticle potential range becomes longer-ranged, not only the nearestneighbor but also the second-nearest (and higher-order) interactions can play a role in crystallization, yielding much larger energetic and entropic differences between fcc and hcp phases. Therefore, crystalline processes are less likely to occur in systems with a longer range of interparticle potential.
Given their static features, soft colloids also exhibit very rich dynamics, and numerous experimental works have revealed intriguing properties. 40,[45][46][47][48][49][50][51][52] Semenov et al. 48 investigated the dynamics of polybutadiene multiarm star polymers in good solvency conditions by means of dynamic light scattering. Three relaxation modes in the semidilute regime were probed: a fast cooperative diffusion due to the polymeric nature of the systems (star-arms interpenetration taking place at the overlap concentration c* and beyond), a slow self-diffusion of the star-polymers (due to finite functionality, 4 hence size polydispersity) and an intermediate structural mode reflecting the collective structural rearrangements of ordered stars. With increasing concentration, the fast-cooperative diffusion speeds up, [53][54][55] whereas the slowest mode slows-down, while the collective structural mode is concentrationindependent. Along the same lines, Loppinet et al. 40 investigated concentrated suspensions of diblock copolymer micelles with crosslinked cores and also detected three relaxation modes analogous to the star polymer case. The study of the dynamics of core-shell (silica-polystyrene) particles by Voudouris et al., 50 revealed similar behavior. However, by varying the molecular architecture (grafting density, aggregation number, and molar mass of the grafted chains) it was found that high-grafting density particles exert a large osmotic pressure which makes the grafted chain interpenetration more difficult, shifting the emergence of a polymeric cooperative diffusion to concentrations well above c*. In addition, it was shown that the decoupling of the modes becomes more visible for systems with low grafting density and long brushes. 50 From the above brief summary, it is evident that the slowest mode, i.e., the self-diffusion of the particle, can be observed in hard-spheres, multiarm star-polymers and core-shell particles. On the other hand, the cooperative diffusion and arm retraction modes represent unique signature of interpenetration of hairy particles. In this respect, we note that microgels without dangling ends do not exhibit cooperative diffusion relaxation mode. 56 In spite of the numerous similarities in phase diagram and dynamics among soft spherical colloids, their realm remains quite broad and there is a need to link soft interactions to structure and dynamics over a wide range from stars to grafted spheres. 57 Clearly, this cannot be done with one type of interaction potential, as we show here. We thus need further experiments with model systems and appropriate description of their interactions.
In this work we use well-characterized polymer-grafted nanoparticles (PGNPs), which are stable (since their cores are crosslinked) and exhibit characteristics embracing core-shell particles and starpolymers. Suspensions of PGNPs are investigated by means of, neutron and light scattering, as well as rheology in the linear viscoelastic regime. Langevin Dynamics (LD) simulations, using the star polymer interaction potential, are performed and the results compared with the observed static and dynamic properties of the PGNPs. While the static structure factor can be captured by the star polymer potential, major issues arise when the same potential is used to test the dynamic behavior. In fact, as the concentration increases, the star-polymer potential is not able to capture the experimental intermediate scattering functions. To this end, the pair potential developed in the context of colloidal interfaces with adsorbed gelatin 58 (hereafter named "brush potential") is modified and applied to our PGNPs. It is found to be very effective in capturing both static and dynamic properties observed in experiments conducted at various concentrations and length scales. In addition to the above-5 mentioned finding, it is also reported that: i) an arrested state is detected by means of static light scattering and rheology, ii) the solid-like behavior has features of a crystalline phase and given the polymer chain being in the SDPB regime, a fcc structure is expected according to Morinaga et al. 44 Indeed, LD simulations confirm the presence of a fcc structure. iii) Dynamic light scattering at various concentrations revealed three different modes assigned to the cooperative diffusion of the polymer chains, center of mass motion (collective diffusion) and self-diffusion. Our results show the importance to carefully combine different approaches to study these systems, the richness of softness and its link to structure and dynamic properties. The present PGNPs serve as another model soft colloidal systems, easier to prepare compared to star polymers, stable and spanning the regime from stars toward hard spheres.

Materials and Methods
Spherical grafted nanoparticles were prepared by microphase separation of diblock copolymers of poly(3-(triethoxysilyl)propyl methacrylate) (PTEPM) which represents the core, and polystyrene (PS) which represents the shell, in the presence of oligomers of both o-TEPM and/or o-S, followed by cross-linking and dispersion in good solvent for polystyrene. This technique, known as Assembly Cross-linking and Dispersion (ACD), allows to obtain precisely tailored nanoparticles which include different shapes with identical PS shells, different core sizes but the same shell, and particles with fixed shape but varied PS shell. Thus, among the main advantages of this technique there is the possibility to obtain nanoparticles with different shape, same shell and very similar grafting density.
Further details about these nanoparticles are reported elsewhere. 10,59 Table 1 reports the molecular characteristics of the spherical micelles used in the present work; core radius (Rc), polydispersity index (PD) of the polystyrene chains, number-average molar mass of the PS arm (Mn,arm), PS weight fraction respect to the total particle (PS fraction), grafting density (), aggregation number (Nagg), and the weight-average molar mass (Mw) of the whole nanoparticle calculated as Mw = (Mn,arm·PDI)/PS fraction.
Two solvents, toluene and chloroform were used for the scattering experiments. The hydrodynamic radius (RH), the radius of gyration (Rg), their ratio (Rg/RH) and the overlap concentration (c * ) for each solvent were measured by means of light scattering (see Figure S1 in the SI) and are reported in Table   2. Concentrations are expressed in terms of an effective volume fraction  (simply called volume fraction hereafter) estimated as the ratio between the actual and the overlap concentration defined as being the Avogadro number. Given the soft hairy nature of the particles, the effective volume fraction may reach values higher than the unit values due to its deformability and compressibility. The ratio between the radius of gyration and the hydrodynamic radius is an indication of the softness of the colloidal system. 60 Such characteristic particle sizes strongly depend on solvency conditions. 61 In the present study, Rg/RH decreases as the solvent is varied from toluene to chloroform. Notably, the Rg/RH ratio ranges from 0.77, for a homogeneous sphere such as a hard sphere, to 1.5 for monodisperse linear chains in good solvent. 60 In the present systems, as well as in core shell particles, 50 star-like 40,45 and star polymers, 62 such a ration is > 0.77. However, recent investigations on thermoresponsive ionic microgel systems showed a Rg/RH ratio < 0.77. 63 In our study, we find that the lowest value belongs to the solution in chloroform, where nanoparticles nearly approach the hardsphere limit. However, in the latter case the RH resulted in the highest value, more than that in toluene in spite of the fact that toluene is a good solvent for PS. This can be tentatively ascribed to the fact that chloroform favors better solvency conditions for the crosslinked core promoting its slight swelling. The procedure adopted in order to estimate the hydrodynamic radius and the radius of gyration is reported in Figure S1 of the SI. Note that, for the present systems, the ratio of overall hydrodynamic-to-core radii is about 5, smaller than the soft systems previously mentioned in the Introduction. Table 3 collects the refractive indices of the core, shell and the different solvents used in this work. 64 7 The interaction between PGNPs are purely repulsive. Static light scattering measurements in the dilute regime in toluene allowed the measure of the second virial coefficient, A2, that characterizes the particle-particle interaction. The Zimm representation 65 is presented in Figure S3 in the SI, together with the linear fit which yielded a value A2 =3.4 10 -5 mol ml g -2 . The second virial coefficient A2 resulted positive, implying good solvency conditions, but its value is almost one order of magnitude lower than that of pure PS chains in toluene at the same Mw, 1.1 10 -4 mol ml g -2 . 66 As expected, particle interactions are different from those between simple linear chains. This relates to the colloidal nature of the PGNPs.

Photon Correlation Spectroscopy
The experimental normalized light scattering intensity ( , ) used to compute the autocorrelation which is typical smaller than one. In the dilute regime ( , ) is a single decay function, and the effective short time diffusion coefficient is determined from the initial decay rate according to 68 where (1) ( , ) is the normalized intermediate scattering function ( , ). In the non-dilute regime, the presence of multiple relaxation processes requires the analysis by inverse Laplace transformation: The distribution of relaxation times L(ln ) can be decomposed as a sum of distributions Here we adopted a three-mode analysis that has been used for other soft colloidal particles. 40,45,48,50 This can be understood as a simple visualization of the dense dispersion of cores in a sea of polymeric blobs. 48

Langevin Dynamics (LD) simulations
The properties of hairy particles are usually analyzed in the context of the star polymer potential developed by Likos. 27 The latter represents a coarse-grained potential for long-ranged repulsive interactions which depend on the functionality. For spherical particles with adsorbed polymers forming an effective brush, Likos et al. 58 developed a pair potential specifically accounting for the interactions between colloidal surfaces with adsorbed gelatin. This was accomplished through a naïve superposition of a simple electrostatic repulsion with an equally simple model for polymer steric 10 stabilization. By only adopting the term that considers the steric repulsion, in the specific case given by the polymeric shell of the soft particle, it is possible to reproduce with good agreement the static and dynamic properties of the systems considered in this work. A similar approach was also adopted by Loppinet et al. for stable diblock copolymer micelles with non-negligible core and ratio of hydrodynamic particle to core equal to 5.5. 45 The star polymer potential VSP (r) is where f is the functionality and is a characteristic length which, for large f, is found to be comparable to the hydrodynamic radius of the particle. 8,71,72 On the other hand, the brush potential VBrush (r) reads where y=(r-2Rc)/(2L) and f(y) with Rc the radius of the core, L the thickness of the brush layer and s the square-root of the inverse of the grafting density defined as = √ = 3.65, with A being the core surface and Nagg the aggregation number reported in Table 1. In both potentials,  represents the inverse thermal energy 1/ , with being the Boltzmann constant and T the temperature. The potential in Eq.4, for distances smaller than 2Rc, resembles that of hard-spheres, whereas the interpenetration of the brush layer for 2 < < 2( + ) is modelled by a smooth function f(y). For such a potential we define the total particle size 2( + ) = Brush Note the different definition of the characteristic length in the two potentials: for the brush model, Brush corresponds to the diameter of the particle, whereas for the star polymer potential SP corresponds to an effective radius, amounting to 1.3Rg, or similarly, to the hydrodynamic radius. 8,71,72 This is clearly visible in Figure 1, which depicts a comparison between star and brush potentials at the same number of arms (f = 212), highlighting their qualitative difference and the smoother divergence of the former potential as particle-particle distance is reduced. The brush potential appears to be closer to that of hard-spheres, so it is more relevant to grafted spheres with substantial core (as discussed below). Note that for the brush potential that we consider in this work we fix the parameters to experimental values, namely the radius of the core (Rc) and the square-root of the grafting density (1/s). In view of this, we only adjust the brush thickness (L), and consequently the particle size (Brush), and the packing fraction (Brush). Remarkably, we find that the brush potential provides more physical insights on the architectural characteristics of the experimental systems. potential correspond to those of the real system. Importantly, whereas Brush is the total particle diameter, SP is a characteristic length which corresponds to the hydrodynamic radius of the star polymer. 8,71,72 We performed Langevin Dynamics simulations of a system with N = 2000 particles with mass m = 1, where the total force on the ith particle is defined as: where represents the conservative force computed from the star polymer and brush interaction potentials. , defined as = − , where is the friction coefficient, fixed in our case at = 10 2 , and is the particle velocity. Random forces are characterized by < > = 0, and < ( ) ( ′) >= 6 ( − ′ ). The equation of motion were integrated using the self-adaptive OVRVO scheme, 73 which is suitable for both equilibrium and nonequilibrium dynamics. 73 qualitatively match the experimental observations (see SI for further details). Note that, since the particle size is defined in a different way in the two potentials, different numbers can be obtained.
However, the concentration regime probed is the same as the experimental one.

Structural properties
All experiments reported in this section were performed in toluene.
The effect of concentration was investigated through static light scattering measurements and the intensity, expressed in terms of R/KC, is plotted against the scattering wavevector q, as shown in Figure 2A. The smallest length scale that can be probed within the explored range of q is of the order  conditions. 85 At  = 0.034, still in the dilute non-interacting regime, the particle form factor can be obtained and a good overlap with light scattering data is attained, although there is a slight difference in concentration (concentrations are reported in the legend of Figure 3). As the volume fraction approaches 0.5, structural peaks emerge. However, the solution is still in an ergodic state and the length scale at the peak (green triangles) is 2π/0.07 = 89 nm, slightly larger than the hydrodynamic size of the particle in dilute regime (RH = 62nm). A further increase in volume fraction leads to an arrested state and the structural peaks shift towards higher q-values. In fact, the peak now corresponds to 2π/0.12 = 52 nm < RH, implying a strong packing of the particles (likely reflecting osmotic compression and interpenetration). A pseudo-structure factor was estimated by simply dividing the total scattering intensity I(q) at  = 1.26 by the form factor P(q) at  = 0.034 and plotted versus the scattering wavevector ( Figure 4). This approach entails the rough approximation to P(q) being concentration-independent, which is not entirely true. Indeed, as the particles deform and reduce their size with concentration it is not appropriate to define such a ratio as the static structure factor of the system. Nevertheless, it turns to be very useful to implement an effective interaction potential for the LD simulations. The value of structural peak detected in Figure 4 overcomes the empirical Hansen-Verlet crystallization threshold value of 2.85, 86,87 suggesting the possible presence of an ordered state or a mixture of ordered states. As a matter of fact, it is not possible to distinguish different crystalline phases as shown by the fitting curves in Figure 4, since the structure factors for an fcc, bcc as well as a hcp order would all fit reasonably well the experimental data with weak and noisy higher-order peaks. The presence of crystalline phases in experimental soft systems is of course possible 7,21,44,88,89 but less likely in comparison to monodisperse hard spheres 20,80 and spherical micelles with dynamic arm exchange. 8,33,34 According to Ohno et al., 43 the probability of finding an fcc structure increases as the degree of polymerization of the tethered chains increases. In fact, this occurs in the presence of an excluded volume region (semi-dilute polymer brush regime). By knowing the grafting density  at the core, the radius of the core Rc (see Table 1) and the monomer length lm (0.25 nm 90 ), a critical radius is estimated as = 1/2 * = 0.7 where * = 4 1/2 with being the excludedvolume parameter. 24 Assuming good solvency conditions, = 0.5. Given that Rc >> rcrit 43 , one may infer that the entire brush layer is in the SDPB regime, hence, the most likely structure is represented by an fcc order. Dashed lines represent light scattering data at similar volume fraction (see also Figure 2). The slope -2 at high q values represents the q-dependence of the intensity for Gaussian chains in good solvent. respectively. According to the Hansen-Verlet criterion the first peak overcomes the 2.85 threshold value for crystalline structure. 86 We simulated monodisperse particle suspensions and performed the local bond-order analysis introduced by Steinhardt et al. 91 . The order parameters are expressed in terms of angular correlations between a vector, characterized by a certain number of spherical harmonics identifying the local environment around a central particle, and its neighboring vectors. This leads to the rotationally invariant order parameter distributions ( ̅ ), with i being the number of spherical harmonics, which are used to discriminate between bcc, fcc and hcp structures. [92][93][94] The probability of finding either bcc or hcp/fcc structures is shown in Figure 5A in terms of ( ̅ 6 ) for two different volume fractions corresponding to regime III. Figure 5B shows that, when considering 4 spherical harmonics, ( ̅ 4 ), the most likely crystalline structure is a fcc, i.e., the same that was previously assessed, based on the brush conformation model proposed by Ohno et al. 43 Two important remarks follow in order.
Simulations were performed by using the effective brush potential, for reasons that will be clarified later, and star polymers with the same functionality as the PGNPs would also exhibit a fcc crystalline structure. 95 The system's tendency to crystallize is compatible with the formation of a fcc crystal. Additional studies would be needed to investigate the stable crystals in the full phase diagram of the brush potential. However, this goes beyond the scope of the present work.

Dynamic properties
Intermediate scattering functions at q = 0.025 nm -1 are shown in Figure 6 as a function of the volume fraction, along with the corresponding relaxation time distributions with amplitude rescaled by the volume fraction. In the dilute non-interacting regime ( = 0.076), or simply regime I, a single decay process is present. As the volume fraction increases to  = 0.46, within the previously defined regime II, three distinct and physically meaningful diffusive (see Figure S6 in the SI) relaxation modes emerge. The fast process becomes faster with increasing volume fraction, undoubtedly describing the cooperative diffusion of the interpenetrated arms, 48,50,96 hence, polymeric in nature. The intermediate mode is nearly concentration-independent and attributed to the center of mass collective diffusion, as was also observed by Voudouris and coworkers in core-shell particles. 50 The slow mode, which slows-down with  is ascribed to PGNP long-time self-diffusion, detectable by light scattering due to the finite polydispersity of the particles. 48,50 This mode is present in both soft 40,45,50,96  The concentration dependence of the light scattered intensity, I, normalized diffusion coefficient (by that in dilute regime), D/D0, and the inverse of the relative viscosity, 1 = , is shown in Figure   7. In regime I, dilute non-interacting regime, the total intensity relates to the particle diffusion D0 (see Figure 7A). The intensity of the polymeric cooperative mode constitutes a small contribution to the total intensity, and it decreases with I ~  − as reported in the literature.  Similarly, the intensity of the self-diffusion mode, decreases with increasing volume fraction, yet with a stronger dependence, I ~  − , as also reported in the literature. 48,53 Finally, the center of mass collective motion exhibits a weak dependence of the intensity on volume fraction, as also observed in block copolymer micelles. 45 In Figure 7B, until  = 0.3, in the dilute regime, the diffusion coefficient D0 is constant and equal to 6.2 10 -12 m 2 /s (see Figure S1 in the SI). At  = 0.46, in regime II, different modes appear. The fast mode speeds-up with concentration and follows the expected scaling for star and linear polymers in 19 good solvent, D ~  0.77 . 48,[53][54][55] When chloroform is used as a solvent, the refractive indices of the core and the solvent are nearly matched, hence the contrast between the polymeric shell and the solvent is increased. The result is that only two modes are detectable (see Figures S7 and S8 in the SI): 1), the polymeric cooperative diffusion, which was further confirmed, and 2) the center of mass collective structural mode (see black symbols in Figure 7B). The center of mass collective diffusion (blue symbols) is nearly concentration-independent, as was also observed in core-shell particles, 50 where is the maximum packing fraction and [] is the intrinsic viscosity equal to 2.5 for spheres. 102,104 The empirical fit yielded a value of equal to 0.82, still belonging to regime II. Such a value is smaller compared to that expected for arrested state (crystal) ( > 1.26). However, the fact that > 0.74, the maximum possible packing for hard spheres, reflects mild brush interpenetration, therefore, mild softness. This represents an additional signature that the system investigated in this work exhibits strong features of rigid spherical particles. Softer colloids typically exhibit much larger . 96 In this respect, a special note on the highest-self-diffusion data point of Figure 7B is in order.
It clearly deviates from the rest and the Krieger-Dougherty line. A tentative interpretation of this behavior calls for osmotic compression and/or arms interpenetration, as discussed in the context of other stable polymeric micelles and soft particles. 41,96,105,106 In this regime, the brush potential is likely not applicable. However, clarifying these interesting issues goes beyond the scope of the present investigation.
20 Figure 7. Volume fraction dependence of A) normalized intensity extrapolated at q→0 divided by the volume fraction, and B) normalized diffusion coefficients by that in the dilute non-interacting regime (regime I). Black solid symbols in panel B are measured in chloroform where the refractive index of the solvent nearly matches that of the core (see Table 3). The relative viscosity, green points in panel

Coarse-grained simulations
The static structure factor S(q) was calculated for both interaction potentials by solving the Ornstein-Zernike equation (OZ). 108 The latter involves an iteration procedure where different particle size values (hence, packing fractions) are tested in order to match the experimental pseudo-structure factor at  = 0.5, in the ergodic regime (see Figure 8). Figure 8 shows the comparison between the experimental and the calculated static structure factor by using both star polymer and brush potential. ). This agreement not only includes the slow relaxation process, but also the fast relaxation mode, a rather challenging task due to the presence of both self and collective modes, as shown in previous work with frozen block copolymer micelles. 101 On the other hand, the star polymer potential is not fully able to capture the experimental behavior at high volume fractions. This is evident in particular in Figs. 10 and 11. We note that several attempts to match the experimental data were performed for the star potential by varying either SP or SP, without being able to faithfully describe the full behavior of the experimental curve, as shown in Figure S9 of the SI for  = 1.26. Indeed, by varying SP and or SP, either the slow mode or the fast mode of the experimental intermediate scattering function can be separately captured, but not both at the same time. Conversely, the brush potential, characterized by the tunable parameter L, the brush layer thickness, which can also account for the osmotic compressibility of the shell, provides a better agreement with the experimental behavior. We believe that the key factor is the qualitatively different approach to repulsion as the interparticle distance decreases (i.e., different softness), which is depicted in Figure 1. Indeed, the star potential overestimates the softness of the PGNPs, resulting in a much smoother repulsion at particle contact.
The presence of a highly cross-linked core shortens the range of repulsion between particles, making them harder compared to stars (see Rg/RH ratio in Table 2). In other words, one may consider the ratio between the total particle radius and the core radius as a parameter to predict the length of the particle interactions range. Below a certain value of this ratio, the star polymer potential does not seem to capture the behavior of a soft spherical system, while the brush potential is found to be valid.
Although, this aspect deserves more investigation, literature works 8,[29][30][31] combined with the present case, suggest as a threshold value a size ratio in the range 7-10 (present case ~ 5).
Relaxation time distributions were also computed for the simulated correlation functions and are reported along with the experimental ones in Figures 9, 10 and 11. The same CONTIN analysis as the experimental correlation functions was used for a fair comparison. The self-diffusion is reasonably well-captured by the simulated relaxation time distributions for all volume fractions.
Surprisingly, the two fast processes seem to be present also in the simulated data at  = 1.26.
However, it would be unreasonable to consider them separately and associate the fastest mode to the cooperative diffusion of arms with this level of coarse-graining. To this end, it seems more appropriate, and actually more consistent with the simulation model, to consider only two relaxation 23 processes, the self-diffusion and the collective diffusion. The presence of a broad (at  = 0.46 and  = 0.76) and double-peak (at  = 1.26) in the faster mode is due to polydispersity. In fact, when the same simulations are performed without polydispersity, only one fast narrow relaxation process is observed (see Figure S10 of the SI).

Osmotic Compressibility
Star polymers are known to undergo osmotic de-swelling in a crowded environment, 83,109,110 an effect that can be pronounced depending on the functionality of the stars. In fact, for low to intermediate values of functionality (f < ~100) the osmotic de-swelling, quantified with the size reduction, resembles that of linear polymer chain solutions with R(C) ~ C -1/8 , with R being the star radius and C the concentration. 83 Star polymers with higher functionality (f~362) are more efficient osmotic compressors than linear chains since the osmotic pressure increases with functionality. 18,111 Osmotic pressure in high-functionality stars can also favor crystalline order. 1,28,112,113 In the present case, the shrinkage of the particles was estimated through the reduction of Brush needed to obtain good agreement between the simulated and experimental correlation functions when using the brush potential. Figure 12 depicts the shrinkage of the soft particles as a function of the volume fraction, using present data and star data from the literature. 18,29 Although only few data points are available, 25 it is clear that the dependence of the particle size on the volume fraction () is much stronger for PGNPs compared to low functionality stars, and very similar to high-functionality stars, with a slope around -0.2. Figure 12. Normalized particle size by that at  →0 (no particle de-swelling) against volume fraction for PGNPs (solid circles) and star polymers taken from literature. 18,29,78 Dashed black lines represent the -1/8 slope predicted for linear polymer chains and low-functionality star polymers in solution, and -0.2 the slope found for high-functionality star polymers and PGNPs.

Conclusions
A versatile, stable spherical model PGNP system with features encompassing core-shell particles and star polymers was investigated by means of experimental and simulation techniques with the goal of developing an experimental-simulations toolbox to describe the static and dynamic properties of such tunable colloidal systems over a large range of volume fractions from dilute to the non-ergodicity transition. In this particular case with negligible polydispersity and precisely controlled structure, the latter case is associated with crystallization of the PGNPs, marking a departure from the behavior of monodisperse multiarm star polymers which typically vitrify under similar conditions. This study allows testing the limits of the star-polymer interaction potential and obtain a better understanding on the dynamics of soft spherical colloids. The ergodic-to-non-ergodic transition upon increasing concentration was probed by light scattering and rheology. Neutron scattering experiments and 26 simulations revealed that the non-ergodic state has features of a fcc crystalline structure. This is also predicted by simple brush conformation arguments, according to which, when the tethered chains belong to the semi-dilute polymer brush regime, the probability of having an fcc structure becomes higher than a hcp because the excluded volume interactions enhance the energetic and entropic differences between fcc and hcp phases.
Despite the relatively long-ranged repulsive interactions of the PGNPS compared to hard spheres, the star potential was not particularly effective in capturing their dynamics. The reason is that particle interactions of the investigated systems are not as long-ranged as those of star polymers. Combining literature works with the present one, suggests that when the ratio between the particle size and the core size is below 7-10, the star polymer potential cannot capture the dynamics of spherical soft colloids. Consequently, a clear need for an alternative interaction potential arose.
To this end, we adapted an existing brush interaction potential that captures reasonably well both the static and dynamic properties of the PGNPs at various concentrations at a fixed value of the scattering wavevector (q = 0.025 nm -1 at 90°). Further investigations of this system and similar ones with different characteristic length scales, combined with targeted experimental and numerical studies on crystallization, will provide the possibility to have finesse the present results and come-up with a robust effective potential that describes a large class of core-shell particles, allowing the access to new insights on soft spherical nanoparticles.
We also showed that, particle shrinkage due to osmotic pressure was found to have a stronger dependence on volume fraction compared to simple linear chains or star polymers with relatively low functionality and long arms. For the PGNPs treated in this work, the dependence of the particle shrinkage on the volume fraction is about  -1/4 . This work sets the foundation for a better understanding of the link between soft particle interactions and dynamics, 114,115 paving the way for a more raffinate design of soft hairy nanoparticles with desired range of static and dynamic properties.

Supplementary Material
Static and dynamic light scattering in the dilute and semidilute regime in toluene and chloroform; comparison between simulations and experiments in terms of correlation functions by using the star polymer potential; effect of polydispersity in the simulated correlation functions.