This dissertation focuses on the development of numerical methods for the solidification of multicomponent alloys and a number of moving boundary problems in physics of block copolymer (BCP) materials: the self-assembly of free surface BCP melts, the co-assembly of BCP nanocomposites, and the inverse design problem for the Directed Self-Assembly (DSA). While these processes have different physical nature, they share several computational challenges: the nonlinearity of the governing equations, the diffusion-dominated character, and the presence of moving boundaries/interfaces that are themselves part of the solution. Accurate and efficient simulations of these phenomena require advanced fundamental numerical capabilities as well as solving conceptual challenges specific to each application.
The first part of this work describes the general numerical methods for solving partial differential equations (PDEs) that are necessary for the simulation of the considered processes and which, at the same time, have significance beyond these applications. Specifically, we present novel second-order accurate finite-volume discretizations on Cartesian grids for Poisson-type equations in irregular domains subject to Robin boundary conditions and/or with discontinuities across immersed interfaces. In addition, we provide a PDE-based approach for smooth extensions of scalar fields across piecewise smooth interfaces.
In the second part, we present a computational approach for the simulation of multialloy solidification in the sharp-interface limit. The main challenge -- solving a non-linearly coupled system of PDEs at each time step -- is solved by a novel Newton-type approach. Further, a combination of adaptive Cartesian quadtree grids and the Level-Set Method is used to address the highly complex evolution of the solidification front and the multiscale nature of the process. The proposed method is validated on cases with known analytical solutions and applied to study the segregation behavior of a Co-Al-W ternary alloy.
Finally, the third part considers the moving boundary problems related to the self-assembly of BCPs. First, we augment the Self-Consistent Field Theory with a consistent approach for imposing surface energies, which is crucial for accurate modeling of polymer-air and polymer-wall interactions. Second, the machinery of PDE-constrained shape sensitivity analysis is applied to derive the equations governing the free surface shape of BCP melts and the placement of nanoparticles as well as the shape derivative of the cost functional associated with the inverse design problem for DSA. The obtained numerical methods are then used to study the self-assembling behavior of substrate supported BCP droplets, to investigate the co-assembly of BCP and nanoparticles of complex shapes, and to design confining masks for nanolithography applications.
