Understand the Relative Stability of Single-Gyroid\nto Double-Gyroid in AB-Type Block Copolymer: Chain Packing and Geometric\nAnalysis

It\nhas been commonly accepted that the bicontinuous double-gyroid\n(DG) network structure is the usual stable phase formed in pure AB-type\nblock copolymer melts. Thus, the emergence of the single-network counterpart,\ni.e., single-gyroid (SG), is mysterious. In previous work, it was\nproposed that the SG phase was stabilized by the synergistic effect\nof the local segregation between two unequal end B-blocks and the\nstretched bridging middle B<sub>2</sub>-block of the linear B<sub>1</sub>AB<sub>2</sub>AB<sub>3</sub> pentablock; however, this was\nnot sufficiently verified. Here, we reexamine the stabilization mechanism\nof the single-gyroid phase and reveal that the bridging middle B-blocks\ndo not play a key role in stabilizing the SG phase over DG due to\ntheir negligibly low bridging fraction. Instead, the middle B<sub>2</sub>-blocks mainly concentrate near the interface in the form\nof loops, helping the short end B<sub>1</sub>-block to push the long\nB<sub>3</sub>-block to fill the further space and thus relieving the\npacking frustration of B-blocks in the matrix of SG on the one hand.\nOn the other hand, the middle B-block is distributed more uniformly\non the interface in the SG structure than in the DG structure due\nto the more uniform interfacial curvature of the former. These two\nfactors lower the relative interaction energy of SG compared to DG,\nconsequently stabilizing SG. Clearly elucidating the stabilization\nmechanism of SG may provide guidance for further improving its stability\nor obtaining other single-network phases (e.g., single-diamond).