Oscillatory interaction between two like-charged nanoparticles induced by polyelectrolyte brush–solvent interface

We use two-dimensional (2D) self-consistent field theory to study the effective interactions between two like-charged cylindrical nanoparticles mediated by an oppositely weakly charged polyelectrolyte brush in a solvent solution. In a poor solvent, where a sharp brush–solvent interface forms, an oscillatory interaction is observed when two nanoparticles are both located at the brush–solvent interface. This oscillatory interaction depends on the penetration depths of the particles and their geometric orientations with respect to the substrate. When the particles are both immersed in the brush and/or the particles are oriented vertically or diagonally with large angles to the substrate, the oscillatory behavior disappears. We interpret our findings by analyzing in detail the contributions to the free energy from electrostatic interaction, nonelectrostatic interaction, and entropies, separately. Briefly, the deformations of the interface and the ion layers formed in the vicinity of the interface are responsible for this oscillatory behavior. In a good solvent, where the narrow brush–solvent interface vanishes, the effective particle–particle interactions behave like that for both particles immersed into the brush with poor solvent. They are found to be repulsive. The influences of the particle size, grafting density, and amount of charges and ions are also briefly discussed.