Supplementary information files for: Self-assembly in soft matter with multiple length scales

Spontaneous self-assembly in molecular systems is a fundamental route to both biological and engineered soft matter. Simple micellization, emulsion formation, and polymer mixing are well understood. However, the principles behind emergence of structures with competing length scales in soft matter systems remain unknown. Examples include droplet-inside-droplet assembly in many biomacromolecular systems undergoing liquid-liquid phase separation, analogous multiple emulsion formation in oil-surfactant-water formulations, and polymer core-shell particles with internal structure. We develop here a microscopic theoretical model based on effective interactions between the constituents of a soft matter system to explain self-organization both at single and multiple length scales. The model identifies how spatial ordering at multiple length scales emerges due to competing interactions between the system components, e.g., molecules of different sizes and different chemical properties. As an example of single and multiple length scale assembly, we map out a generic phase diagram for a solution with two solute species differing in their mutual and solvent interactions. We further connect the phase diagram to a molecular system via molecular simulations of a block-copolymer system that has a transition from regular single-core polymer particles to multicore aggregates that exhibit multiple structural length scales. The findings provide guidelines to understanding the length scales rising spontaneously in biological self-assembly but also open venues to the development and engineering of biomolecular and polymeric functional materials and pharmaceutical formulations.