Entropic Effects in Solvent-Free Bidisperse Polymer Brushes Investigated Using Density Functional Theories

Solvent-free polymer-functionalized nanoparticles form a special type of colloid composed of inorganic cores self-suspended by their grafted coronas. In the absence of intervening solvent molecules, the fluidity of the system is provided by these tethered polymers as they fill the space. Here, we study the structure and interaction of neighboring polymer-grafted surfaces in the solvent-free condition using mean-field density functional theories. For opposing flat surfaces, the brush configuration and the associated energy landscape are semianalytically investigated given the incompressibility of the tethered entropic chains. The effect of brush polydispersity (including variations in both chain length and surface grafting density) is considered by two bidisperse models corresponding to different physical scenarios: one for opposing brushes uniformly mixed with two species at a fixed grafting density, and the other for opposing brushes with distinct chain lengths and grafting densities. The space-filling capabilities of the neighboring coronas differ not only by their ratio of radii of gyration for the composing polymers but also by their ratio of grafting densities. We show that the system energy depicts a steric repulsion as the brushes are compressed, which is typical for hairy particles in a solvent. However, as the interwall separation increases, the cooperative stretching of the chains leads to an entropic attraction between them, a unique characteristic of solventless systems. The corresponding brush profiles change from a bell-like shape to a more step-function-like feature as the interwall spacing increases significantly. The interwall separation associated with the overall free energy minimum therefore characterizes the favorable interparticle spacing for solvent-free polymer-functionalized particles. The limiting accessible parameter space of polymer sizes and grafting densities subjected to the space-filling constraint is comprehensively explored for representative interparticle spacing characterizing the compressed, relaxed, and stretched regimes for a given polymer species, respectively. Such information would be useful for guiding the design of experimental solvent-free polymer-functionalized nanoparticles.