Computational Modeling of Complex Interfacial Structures with Nonlinear and Nonlocal Interactions

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Modeling interfacial structures and dynamics is of great importance in many applications such as biology, physics, and materials science. Many existing computational models can sometimes predict unphysical structures and also suffer from inefficiency in large scale computations, especially when the system of interest involves nonlinear and nonlocal interactions. On the other hand, some powerful numerical methods have been designed to approximate the interfacial structures in a stable and efficient manner. This can only happen if numerical methods are designed to preserve the underlying physical structures of the system of interest. The purpose of this project is to bring together researchers with complementary backgrounds to come up with a unified computational model to investigate the interfacial structure and dynamics with nonlocal and nonlinear interactions. The model will largely improve the efficiency of computations for interfacial structures, as well as correctly describe key quantities of interest in equilibria and dynamic structures of an underlying interfacial system. In addition, the mathematical modeling techniques and computational methods of this project will address key scientific challenges in applied mathematics, and meet basic research needs and provide necessary modeling tools for the applications to other systems involving interface problems. Besides, the computational model can provide theoretical guidance on producing nanostructured materials, which will ultimately promote a wide range of contemporary engineering applications such as materials synthesis, nanomedicine, and nanotechnology. Further important impacts will be research oriented curriculum development, mentoring undergraduate/graduate students to take part in the project, and the engagement of various outreach activities.This project focuses on developing a unified computational phase field model to investigate the complex interfacial structures with nonlinear and nonlocal interactions. Though some existing phase field approaches can be applied to model simple periodic structures such as lamellar, spherical, bicontinuous syroidsin block copolymers, some other interesting patterns are overlooked and have not been well studied theoretically. Therefore one needs to examine the variational problem in its full generality from a mathematically more sophisticated point of view, one which in particular allows for a fuller analysis of the competition between different terms in the system of interest. The inclusion of the general nonlocal and nonlinear interactions in this research project can characterize a broader class of features of microphase separation and pattern formation for interfacial structures, and provide more insights on theoretical studies of these subjects. The PIs will develop efficient, stable and accurate numerical methods for the system of interest. More specifically, asymptotically compatible, maximum principle preserving and energy stable schemes will be explored to preserve specific physical structures of the system of interest at the level of numerical approximations. Additionally, the designed numerical solvers will be used for the systematic study of materials science applications such as block copolymer melts and the bubble assemblies of the block copolymer system.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.