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By making use of the developed potential theory, we investigate the polymer-mediated depletion interactions between nanoparticles and hard planar surfaces in the following settings: (i) interaction between two nanoparticles; (ii) interactions between nanoparticle and hard planar surface; (iii) interaction between two nanoparticles in the presence of hard planar surface; and (iv) interaction between nanoparticle and walls of the plane-parallel slab. For each of the listed systems, we have calculated the polymer end density, excess grand potential, and the potential of the depletion forces. On the basis of the obtained results, we have analyzed the effect of the geometric constraints imposed by hard walls on the polymer density structure and the depletion interaction between nanoparticles.

The effect of spherical and sphero-cylindrical fillers on the order–disorder transition (ODT) of a symmetric diblock copolymer melt is investigated. Self-consistent equations describing the copolymer density distribution in the presence of fillers are derived. Using these equations, we calculate the excess free energy due to the presence of the particles in the diblocks. The critical value of the segregation factor χN is recalculated with the effect of the fillers taken into account. We find that a relatively small volume fraction of fillers can cause a significant suppression of the ODT temperature. It is shown that smaller particles cause a greater suppression of the ODT temperature provided a constant particle volume fraction is maintained. The effect of the particle shape on the ODT is investigated. The ODT temperature shift is calculated for the sphero-cylindrical particles as a function of their aspect ratio at a given particle volume. It is found that sphero-cylinders with the smaller aspect ratio produce the bigger effect on the ODT. A scaling analysis of the presented results and a comparison with the experimental work are given.

We investigate the collective behavior of self-propelled particles (SPPs) undergoing competitive processes of pattern formation and rotational relaxation of their self-propulsion velocities. In full accordance with previous work, we observe transitions between different steady states of the SPPs caused by the intricate interplay among the involved effects of pattern formation, orientational order, and coupling between the SPP density and orientation fields. Based on rigorous analytical and numerical calculations, we prove that the rate of the orientational relaxation of the SPP velocity field is the main factor determining the steady states of the SPP system. Further, we determine the boundaries between domains in the parameter plane that delineate qualitatively different resting and moving states. In addition, we analytically calculate the collective velocity v of the SPPs and show that it perfectly agrees with our numerical results. We quantitatively demonstrate that v does not vanish upon approaching the transition boundary between the moving pattern and homogeneous steady states.

By developing and making use of the potential theory of the polymer-mediated interaction between spherical colloids, we investigate the many-body effects on the depletion interaction among these colloids in the colloid-polymer mixture. As our main results, we obtain analytic expressions for the polymer end density in the presence of the colloids and the many-body depletion potential acting between these colloids in the "protein" limit. We present a comparison of our theoretical findings with the results of recent computer simulations.

We analytically solve the problem of the reversible adsorption of Gaussian polymers onto the planar and spherical surfaces in the presence of the square well attractive potential. By making use of the obtained exact solution of the Edwards equation, we calculate the end density and surface excess of the polymers at the planar and spherical substrates. We derive the exact equation that determines the surface bound states that give rise to the dominant contributions to the polymer surface excess. In the case of the spherical substrate, the exact expression for the polymer surface excess is obtained in the remarkably simple form of a quadratic function of the radius of the substrate. Using the calculated polymer surface excesses, we obtain the adsorption-desorption diagrams of the polymers adsorbed onto the spherical and planar surface in terms of the introduced “effectiveness” of the adsorption potential. By performing the analogous calculation based of the standard boundary condition approach, we demonstrate that this method overlooks the effect of the spatial interplay between the depletion and adsorption forces acting on the adsorbed polymers. Based on the comparison with the obtained exact solutions, we propose a modification of the boundary condition for the spherical substrate that preserves, in particular, the correct “protein” limit.

We theoretically perform a comparative analysis of the adsorption of polymers onto the regularly and randomly nonuniform surfaces. By developing and making use of the self-consistent perturbation expansion we calculate the surface excesses of the polymers adsorbed on the random and periodically patterned surfaces. In both cases the enhancement of the polymer adsorption is indicated, as compared to the adsorption onto the homogeneous surface that has the same average affinity for polymers. Moreover, the results obtained for the randomly nonuniform and periodically patterned adsorbing surfaces show striking quantitative similarity, when compared at the same characteristic sizes of inhomogeneities of these surfaces. This finding leads to the conclusion that the adsorption ability of the nonuniform surface primarily depends on the characteristic size of the surface inhomogeneity, rather than on the spatial distribution of the inhomogeneities on this surface. In all cases, the calculated total surface excess is found to be a decaying function of the ratio of the radius of gyration of polymers to the characteristic size of the surface inhomogeneity. The effect of the excluded volume is found to reduce the polymer adsorption.