We investigate hydrodynamic interactions between two three-sphere swimmers analytically and numerically. Hydrodynamic forces exerted on the swimmers as well as the translational and angular velocities of them are obtained in the far field regime. We demonstrate that the active term of the translational velocity is along the intrinsic direction of swimming (n) and has no component along the direction of relative positions of swimmers (r[over ̂]) as reported in previous papers. Using numerical simulations we investigate the long-time swimming paths of swimmers in two general situations of swimming in the same and opposite directions. The former reveals four swimming states for symmetric swimmers-attractive, repulsive, parallel, and oscillatory-and only three swimming states for asymmetric swimmers-attractive, repulsive, and contracting-oscillatory, confirming that the expanding-oscillatory state reported in previous papers is not stable. The latter shows that there are rotative bound states in hydrodynamic scattering of the swimmers.

Starting from statistical mechanics of polymers and using density functional theory (DFT) method, I review theory of polymer brush bilayers at thermal equilibrium thorough a novel point of view. Density profiles, brush heights and interpenetration length in terms of molecular parameters and distance between substrates are obtained at thermal equilibrium. With a transparent analytic approach, it is being shown, how polymer brushes balance compression and elasticity when bilayers of brush come into contact. Pressure is calculated and it is shown that substrates might repel, attract or even do not see each other upon varying system parameters. Uniqueness of the results, based on taking deformations of both brushes into account, suggests that existing theories about polymer brush bilayers (blob picture) need highly likely to be reviewed.

Statistical mechanics is employed to tackle the problem of polymer brush bilayers under stationary shear motion. The article addresses, solely, the linear response regime in which the polymer brush bilayers behave very much similar to the Newtonian fluids. My approach to this long-standing problem split drastically from the work already published Kreer, T, Soft Matter, 12, 3479 (2016). It has been thought for many decades that the interpenetration between the brushes is source of the friction between the brush covered surfaces sliding over each other. Whiles, the present article strongly rejects the idea of interpenetration length in that issue. Instead, here, I show that structure of the whole system is significant in friction between brush covered surfaces and the interpenetration is absolutely insignificant. The results of this research would blow one's mind about how the polymer brush bilayers respond at small shear rates.

By means of the density functional theory framework I tackle the long-standing problem of a polymer star interpenetrating with a polymer brush at thermal equilibrium. Remarkably, the star is repelled to the outside of the brush once it sucks into the brush. It turns out that there could be a highly fluctuating region at the brush edge. The highly fluctuating region would be responsible for discontinuous absoption transitions by brushes. However, up to an small interpenetration length, below which asphericity of the star is maintained, the star gets collapsed by sucking more and more into the brush.

The present article addresses the long-standing problem of the polymer brush bilayers under stationary shear ow at non-linear response regime where the system gets a non-Newtonian uid. The main idea behind this research would be the fact that the immense lubricity of the polymer brush bilayers originates from a global restructuring that takes place at large shear rates. It is shown here that physical quantities like, stress tensor, viscosity tensor, the friction coefficient and the chain extensions could become dependent on the shear rate at non-Newtonian regime. Apparently, the sub-linear scaling of the physical quantities at large shear rates is solely due to the fact that the hains stretch in the shear direction.