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Heuristic algorithms for agnostically identifying the globally stable and competitive metastable morphologies of block copolymer melts


Block copolymers are composed of chemically distinct polymer chains that can be covalently linked in a variety of sequences and architectures. They are ubiquitous as ingredients of consumer products and also have applications in advanced plastics, drug delivery, advanced membranes, and next generation nano-lithographic patterning. The wide spectrum of possible block copolymer applications is a consequence of block copolymer self-assembly into periodic, meso-scale morphologies as a function of varying block composition and architecture in both melt and solution states, and the broad spectrum of physical properties that such mesophases afford.

Materials exploration and discovery has traditionally been pursued through an iterative process between experimental and theoretical/computational collaborations. This process is often implemented in a trial-and-error fashion, and from the computational perspective of generating phase diagrams, usually requires some existing knowledge about the competitive phases for a given system. Self-Consistent Field Theory (SCFT) simulations have proven to be both qualitatively and quantitatively accurate in the determination, or forward mapping, of block copolymer phases of a given system. However, it is possible to miss candidates. This is because SCFT simulations are highly dependent on their initial configurations, and the ability to map phase diagrams requires a priori knowledge of what the competing candidate morphologies are. The unguided search for the stable phase of a block copolymer of a given composition and architecture is a problem of global optimization. SCFT by itself is a local optimization method, so we can combine it with population-based heuristic algorithms geared at global optimization to facilitate forward mapping. In this dissertation, we discuss the development of two such methods: Genetic Algorithm + SCFT (GA-SCFT) and Particle Swarm Optimization + SCFT (PSO-SCFT). Both methods allow a population of configurations to explore the space associated with the numerous states accessible to a block copolymer of a given composition and architecture.

GA-SCFT is a real-space method in which a population of SCFT field configurations ``evolves" over time. This is achieved by initializing the population randomly, allowing the configurations to relax to local basins of attraction using SCFT simulations, then selecting fit members (lower free energy structures) to recombine their fields and undergo mutations to generate a new ``generation" of structures that iterate through this process. We present results from benchmark testing of this GA-SCFT technique on the canonical AB diblock copolymer melt, for which the theoretical phase diagram has long been established. The GA-SCFT algorithm successfully predicts many of the conventional mesophases from random initial conditions in large, 3-dimensional simulation cells, including hexagonally-packed cylinders, BCC-packed spheres, and lamellae, over a broad composition range and weak to moderate segregation strength. However, the GA-SCFT method is currently not effective at discovery of network phases, such as the Double-Gyroid (GYR) structure.

PSO-SCFT is a reciprocal space approach in which Fourier components of SCFT fields near the principal shell are manipulated. Effectively, PSO-SCFT facilitates the search through a space of reciprocal-space SCFT seeds which yield a variety of morphologies. Using intensive free energy as a fitness metric by which to compare these morphologies, the PSO-SCFT methodology allows us to agnostically identify low-lying competitive and stable morphologies. We present results for applying PSO-SCFT to conformationally symmetric diblock copolymers and and a miktoarm star polymer, AB$_{4}$, which offers a rich variety of competing sphere structures. Unlike the GA-SCFT method we previously presented, PSO-SCFT successfully predicts the double gyroid morphology in the AB-diblock. Furthermore, PSO-SCFT successfully recovers the the A$_{15}$ morphology at a composition where it is expected to be stable in the miktoarm system, as well as several competitive metastable candidates, and a new sphere morphology belonging to the hexagonal space group 191, which has not been seen before in polymer systems. Thus, we believe the PSO-SCFT method provides a promising platform for screening for competitive structures in a given block copolymer system.

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