Effieient Numerical Methods For Self-consistent Mean Field Equations And Its Applications

In this thesis, we study the numerical method of self-consistent mean ?eld(SCMF) equations for block copolymers and its applications. Block copolymersare macromolecules composed of two or more chemically distinct polymer sub-chains linked by covalent bond. The self-assembly of block copolymers can formmany various and fascinating ordered microstructures. Potential applications ofthese microstructures include optics, electronics, construction material and biol-ogy. How to search the ordered structures is an important research direction ofblock copolymers as well as a challenging problem of the theoretical, experimentalresearch and industry application.Theoretically, the self-consistent mean ?eld (SCMF) theory is one of themost wildly used frameworks for investigating the equilibrium phase behavior ofblock copolymers. The essential idea of the SCMF theory is to transfer a many-body system with interactions to a one-body problem in an e?ective potential?eld. The polymeric SCMF theory is based upon coarse-grained polymer chainmodel. In this thesis, we foucs our attention on ?exible chain and use continuousGaussian chain to describe polymer con?guration. The conventional SCMF the-ory is introduced based upon continuous Gaussian chain model, then followingthe latest development of mean ?eld theory, we derive the optimization SCMFtheory informally.SCMF equations is a set of highly non-linear and strongly non-local equa-tions with multi-solutions and multi-parameters. Generally, there are three as-pects that a?ects the e?ciency of the the numerical scheme: the appropriateinitial values, the discretize schemes for the model and the non-linear iterativemethod. The initial values in solving SCMF equations not only in?uence the algo-rithm e?ciency, but also decide the ?nal morphology of solutions. We introducethe space group for the ?rst time to estimate initial values because the structuresin microphase separated block copolymer exhibit periodic symmetry. In this the-sis, we discrete the SCMF equations in reciprocal space, and use pseudo-spectral method to solve partial di?erential equations in SCMF theory to reduce com-putational complexity. The non-linear iterative methods are dependent on theinformation provided by SCMF equations. Through asymptotic expansion, weextend the semi-implicit modi?ed Newton method based on optimization SCMFequations, which was only available in diblock copolymer system.Using the numerical method, we study the diblock copolymer system andlinear triblock copolymer system. For two systems, the e?ciency of two kindsof SCMF theory has been studied respectively, i.e., the e?ciency of di?erentiterative methods. We give the initial values to solve the SCMF equations ofdiblock copolymers in detail, and ?nd lots of new meta-stable structures as wellas those known structures already known including stable and meta-stable onesin numerical result. For the diblock copolymer system, by comparing with con-ventional spectral method in block copolymer domain, we show the advantagesof pseudo-spectral method.The parameter space of SCMF theory for triblock copolymer system is a5-dimension space. In addition to the symmetry of microstructures, a good un-derstanding of block copolymers'phase behavior and results of experiments andtheories is also an important part to estimating the initial values. Based uponthe basic structures in diblock copolymers, the thesis proposes a series of strate-gies to search patterns for di?erent kinds of microstructures. By those strategies,a large number of ordered microstructures emerge from the solutions. Thesenumerical results not only cover existing experimental and theoretical results,but also predict many new structures. It should be emphasied that, althoughthese strategies to search structures are applied in block copolymer systems inthis thesis, they are a general approach for other systems which have periodicsymmetric structures.