**4. Conclusions**

This work presents the results of coarse-grained simulations for different polymerfunctionalized spherical nanoparticles immersed in an explicit fluid consisting of small particles P. We considered the particles with ligands permanently anchored at randomly chosen points at the core surface. We studied nanoparticles with attached flexible chains and rigid ligands.

First, hairy particles modified with perfectly flexible chains were considered. The simplest possible model in which only particle–segment interactions are attractive while the remaining pair interactions are repulsive was discussed. For this model, we demonstrated that, in a wide region of particle densities, the excess adsorption on a hairy nanoparticle increased for longer chains and higher grafting densities. The adsorption properties of hairy nanoparticles of different sizes and flat surfaces modified with grafted chains were compared. When the grafting density was kept constant, the normalized excess adsorption on the flat brush was considerably smaller.

However, the effect of the core size was negligible in a wide region of densities. This was significant only in the case of very low or relatively high bulk densities of adsorbed particles. For the systems with assumed attractive particle–particle and segment–segment interactions, the excess adsorption was greater than in our "basic" model system, where these interactions were repulsive. Our conclusions regarding the adsorption on hairy particles are in line with previous experimental observations [56–58].

Adsorption on hairy particles changes the morphology of the polymer corona. Our simulations elucidated the mechanism of adsorption-induced nanoparticle shell reconfiguration of the nanoparticle shell. Adsorbed particles form the bridges between segments belonging both to the same chain and different chains [50,51]. The chains form coils with the particles trapped inside and joined together. We computed the average thickness of polymer coatings for different fluid densities and stated that this is strictly correlated with the excess adsorption isotherms.

We analyzed the dependencies of the brush thickness on the density of adsorbed particles and compared them with the theoretical predictions [19,62,63] and experimental results [56]. Depending on the assumed parameters core-shell or cone-like structures of hairy particles were found in the model systems. In the case of the cone-like particles, the segmen<sup>t</sup> cloud was highly asymmetrical. The chains with adsorbed particles accumulated on a part of the core surface, while the remainder was uncovered. The polymer layer could be more or less compact. We found continuous adsorption-induced structural transitions in polymer coatings. The cone-like structure transformed into the core-shell structure at higher densities.

The adsorption-induced local bundling of chains on the nanoparticle surface was observed in the simulations carried out for other systems [48,50] as well as in experiments [56]. In the case of short chains, "the superadsorption" was observed at very low particle densities. The excess adsorption isotherm had a low local maximum at a very low density. For higher densities, however, the excess adsorption isotherm had a typical course. Likely, this follows from the superposition of the effect associated with the confinement of particles near the curved surface between ligands (which act as obstacles) and the effect of the interplay between all interactions in the system. This anomaly occurs only for special combinations of parameters and disappears for stronger attractive interactions.

Finally, we discuss adsorption on the hedgehog-like particles with attached rigid ligands. The adsorption on such particles is higher than on the corresponding hairy particles with flexible chains. If particle–particle interactions are attractive, we observe a "condensation" of fluid in "pores" between ligands.

In summary, we demonstrated how the model parameters affected adsorption on hairy nanoparticles. We analyzed the mechanism of adsorption and the structure of the polymer coating with details. The adsorption on hairy particles follows from competition between interactions near the surface and in the bulk phase and the entropy effects associated with the limiting of possible chain configurations near the core. Hairy particles with flexible chains are "living" adsorbents, which makes the research difficult.

Future studies of adsorption on ligand-tethered nanoparticles can proceed on several fronts. The density functional theory of adsorption on flat brushes [26–29,33,34] can be adapted to describe the adsorption on hairy nanoparticles. The theoretical predictions can be compared with the results of our simulations. Moreover, the molecular dynamics simulation of adsorption from different explicit solvents will be performed. We hope that our results will help to rationally design hairy particles that could be carriers of bioactive compounds.

**Author Contributions:** The manuscript was written through equal contributions from both of the authors. Both authors have given approval to the final version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
