Accelerated Method of Self-Consistent Field Theory for the Study of Gaussian Ring-Type Block Copolymers

Abstract

Although self-consistent field theory (SCFT) has been become one of the most successful methods for the study of block copolymers, its application to ring copolymers has been hindered for years, especially for three-dimensional structures. Here, we have developed an accelerated pseudospectral algorithm to solve SCFT equations of ring copolymers by taking advantage of the symmetry of ordered structures. By combining various advanced techniques, speed-up of orders of magnitude has been achieved, making the construction of a complete phase diagram affordable. This accelerated algorithm has been applied to constructing the phase diagrams of AB ring and AB tadpole copolymers, consisting of the double gyroids, O 70, and even Frank–Kasper spherical phases. This algorithm is applicable for the studies of other ring copolymers that will add a piece of puzzle to the understanding of architectures on the self-assembly behavior of block copolymers.

Architectural Design of Block Copolymers

“…Very recently, Qiang et al have developed an accelerated pseudospectral algorithm to solve SCFT equations of cyclic copolymers by taking advantage of the symmetry of ordered structures, in which speed-up of orders of magnitude has been achieved, making the construction of a complete phase diagram affordable. 130 Using this accelerated algorithm, they have constructed the phase diagram of AB tadpole copolymer, demonstrating the expanded region of sphere and cylinder phases as well as the formation of FK phases, which indicates an enhanced effect of spontaneous curvature. With the advances in the synthesis strategies and simulation methods for the cyclic block copolymers, a deeper understanding on the effect of molecular architecture on their selfassembly behavior will be obtained in the near future.…”

“…As another interesting architecture, the cyclic block copolymer is able to form structures with smaller domain size than the linear counterpart, endowing it with great potential for application in nanoscale patterning. , Focusing on this outstanding feature, most of the previous reports studied the scaling behavior with domain size, − while much less on the phase behavior except for limited researches of the AB cyclic diblock copolymers. − Moreover, related theoretical researches have been hindered for years while there was a lack of efficient algorithm for solving SCFT equations of block copolymers with ring topology. Very recently, Qiang et al have developed an accelerated pseudospectral algorithm to solve SCFT equations of cyclic copolymers by taking advantage of the symmetry of ordered structures, in which speed-up of orders of magnitude has been achieved, making the construction of a complete phase diagram affordable . Using this accelerated algorithm, they have constructed the phase diagram of AB tadpole copolymer, demonstrating the expanded region of sphere and cylinder phases as well as the formation of FK phases, which indicates an enhanced effect of spontaneous curvature.…”

Molecular Simulations Revealing Effects of Non-concatenated Ring Topology on Phase Behavior of Symmetric Diblock Copolymers

“…However, both γ sim are larger than the predicted γ RPA = 1.7 of the mean-field RPA theory. Future development of a phase-separation theory for diblock ring polymers needs to incorporate the topological invariants of nonconcatenation as well as the effects of fluctuations in order to match the sophistication of the field theories for linear , and Gaussian ring polymers. , The more significant deviation of the structure factor S 0 ( q ) in the disorder phase from the predictions of Marko’s RPA theory (Figure b), when compared with the counterpart for diblock linear polymers (Figure a), also call for a new theory for diblock rings. In the strong segregation regime, the increases of the lamellar spacing d with the strength of enthalpic repulsion and the molecular weight of polymer follow the respective power laws d ∼ ϵ̃ α and d ∼ N ν . The theoretical argument F enthalpic ≈ F entropic predicts α = 1/6, regardless of the copolymer topology.…”

“…However, both γ sim are larger than the predicted γ RPA = 1.7 of the mean-field RPA theory. Future development of a phase-separation theory for diblock ring polymers needs to incorporate the topological invariants of nonconcatenation as well as the effects of fluctuations in order to match the sophistication of the field theories for linear , and Gaussian ring polymers. , The more significant deviation of the structure factor S 0 ( q ) in the disorder phase from the predictions of Marko’s RPA theory (Figure b), when compared with the counterpart for diblock linear polymers (Figure a), also call for a new theory for diblock rings.…”

Miscibility and exchange chemical potential of ring polymers in symmetric ring–ring blends

“…31,37,38 In addition to the rheological and mechanical properties, the phase diagrams of polymer blends and block copolymer melts are expected to be affected by ring topology. 13,[39][40][41][42][43][44][45][46][47][48] When ring polymers are used as additives to alter material properties, miscibility is an important factor for their applicability. In studies focusing on phase behavior within the framework of the random phase approximation, 49 scattering functions derived from the assumption of Gaussian statistics for ring polymers have often been used 40,41 while ignoring topological constraints.…”