Scaling Theory of Diblock Copolymer Surface Micelles

This work deals with the self-assembly of AB diblock copolymers into micelles with 2D corona and 3D core at the air-liquid interface. Flat and laterally swollen conformations of A blocks in the corona are provided by their strong interfacial adsorption and repulsions in 2D, while attractions between the insoluble B blocks result in the formation of non-spherical 3D core. A scaling theory of circular spherical micelles is developed to predict the dependencies of their dimensions and the aggregation number, p, on the block lengths of the diblock copolymer, N_A and N_B. The power laws are derived for the starlike (st) and crew-cut (cc) limits, which correspond to the micelles with a low and high core-to-corona size ratio, respectively. For 2D theta solvent for corona chains, scaling predictions are p_st ~ N_A⁰ N_B^0.5 and p_cc ~ N_A^-0.67 N_B^0.83 for starlike and crew-cut micelles, respectively. Theoretical exponents are in reasonable agreement with the experimental power law, p ~ N_A^-0.48 N_B^0.7. [1] The empirical exponents, which have been obtained by numerical fitting the data with the single power law, fall within the range theoretically found herein. A similar agreement takes place for the core and corona dimensions and holds in the case of athermal solvent for the corona A blocks. Theory generalization to 2D+2D micelles shows that scaling laws are weakly affected by the geometry of the core. The developed theory reveals principles of diblock copolymer self-assembly under partial confinement.

[1] Kim, D. H.; Kim, S. Y. Scaling Laws for Block Copolymer Surface Micelles. J. Phys. Chem. Lett. 2022, 13 (24), 5380–5386. https://doi.org/10.1021/acs.jpclett.2c00979.