This thesis investigates the design of sharp interface level set methods in the context of self-consistent field theory (SCFT) in polymer physics. SCFT computes the structure and energy of inhomogeneous self-assembling polymers at thermodynamic equilibrium. Level set methods are based on an implicit representation of free boundaries, which enable motions with arbitrary change in topology. In addition, recent advances on how to impose Robin boundary conditions enables the study of free boundary problems of interest in the community interested in self-assembly.

We first present a computational framework, encoded on a forest of quad/oct-trees in a parallel environment. We then present results of imposing sharp Neumann boundary conditions as was first proposed by de Gennes, which enables SCFT computations of meaningful quantities at the boundary of irregular geometries. We then introduce the concept of functional level-set derivative in the context of SCFT and rigorously derive expressions for the change of energy of a diblock copolymer with respect to an enclosing shape. The level-set derivative is then used to embed SCFT into a variable shape simulator, where the internal structure and the enclosing shape are coupled together and evolve in tandem in order to reduce the energy of the diblock copolymer. Finally an algorithm for solving the inverse problem for directed self-assembly is presented.

Continuum formulation of interfacial transport phenomena views interfaces as sharp 2D surfaces separating contiguous 3D bulk phases. In this view, interfacial processes are described by phenomenological models for the underlying microscopic processes or in analogy to conservation/constitutive laws that regulate transport in surrounding bulk phases. Mathematically, these surfaces act as boundary conditions for the governing partial differential transport equations in the bulk and determine discontinuities in solution of the bulk phase as well as the solution gradient across the interfaces. From a computational perspective, accurate numerical solution of these equations poses several algorithmic and computational challenges that require application of advanced numerical algorithms as well as high performance computing techniques. In the first part of my dissertation I will present our design and implementation of two parallel simulation engines based on the level-set method and using finite difference and finite volume discretization techniques on adaptive Voronoi and Cartesian (Quad-/Oc-tree) grids. These simulations are capable to realize large enough computational domains at high resolutions that allow for realization of the mound formation phenomenon in epitaxial growth of materials systems, as well as detailed membrane permeabilization statistics emerging in electroporation of multicellular systems. In the second part of my dissertation, I will derive a general reduced-order state-space mathematical model for dynamics of probability density of interfacial polarizations in heterogeneous systems and compare its predictions with that of my direct numerical simulations.