Scaling Laws for Block Copolymer Surface Micelles

Polymers under confinement exhibit different structures and properties from the bulk. While block copolymers (BCPs) create well-defined micelles in solution, two-dimensional (2D) spatial confinement at the air-water interface constrains the chain conformations and deforms the micellar structure, thus forming a surface micelle. The BCP surface micelles open up an opportunity in nanoscience and engineering by serving as an interfacial modifier and structural platform. Nevertheless, a scaling law, a principle governing the micellar structure, is absent. Herein, we report a unified scaling relation to describe the combinational structure of BCP surface micelles in two and three dimensions and further reveal their formation mechanism in line with the suggested scaling relation. We investigated the intrinsic scaling relations in a surface pressure-free environment by introducing a concept of excluded volume-dependent scaling exponent based on the scaling theory of 2D polymers. In addition, an extrinsic scaling relation is derived for the surface pressure-dependent corona scaling.