Evolution of Curvatures Between Lamellar and Bicontinuous Phases: Formation of Saddle-Shaped Hierarchical Lamellar Structures in Binary Self-Assembly System.

Curvatures are fundamental parameters governing the topological changes and phase separations in soft matter systems. Explorations of Gaussian curvature, mean curvature, and their combination (as interfacial curvature linked to packing parameter) are essential for understanding the dynamic self-assembly processes, especially for the critical and classic lamellar-to-bicontinuous transition. However, whether intermediate states arising from the changes in these curvatures exist remains a long-standing controversy. Herein, this issue is addressed by identifying a unique saddle-shaped hierarchical lamellar phase in a binary self-assembly system containing polystyrene-b-poly(acrylic acid) (PS-b-PAA) and stearyltrimethylammonium bromide (STAB). This intermediate phase exhibits characteristics with negative Gaussian curvature similar to bicontinuous phase while maintaining lamellar topologies. Its existence attributes to the competing effects of the inserted STAB micelles imposed on PAA segments: electrostatic screening reduces PAA effective volume, while volume expansion increases it. This competition balance promotes the lamellae toward saddle-shapes with enhanced molecular capacity and modulates their interfacial curvature to a critical intermediate value that is rarely stable in block copolymer systems. The thermostability of this intermediate phase serves as compelling evidence for the smooth interfacial curvature evolution pathway. This discovery provides insights into phase transformations in biological membranes, marking a significant advance in understanding complex soft matter systems.