Interpenetration of Solvent-Free Mutually Attracting Brushes

The equilibrium structure and mutual interpenetration of two opposing solvent-free polymer brushes with a negative Flory–Huggins interaction parameter χ are investigated using numerical self-consistent field calculations and an analytical approach. It is numerically demonstrated that strong mutual attraction fundamentally alters the brush structure, transforming the monomer density profiles from a sigmoidal to a linear shape and the overlap function into a broad inverted parabola. An analytical theory is developed that accurately describes these linear density profiles and the overlap integral, seamlessly bridging weak entropy-driven fluctuation interpenetration and strong energy-driven attraction interpenetration. Based on these scaling laws, a comprehensive diagram is constructed, clearly identifying the crossover boundaries between fluctuation-dominated and attraction-dominated regimes, as well as between partial and full brush interpenetration states. It is shown that for a fixed chain length, the overlap integral exhibits a pronounced non-monotonic dependence on the grafting density, passing through a distinct maximum. An exact analytical expression for the optimal grafting density that maximizes inter-brush adhesion is derived, revealing that this optimal value is a universal function of the interaction parameter χ and is practically independent of the chain length.