The research supported by this award is concerned with the analysis and numerical modeling of problems that arise in the basic and applied physics of liquid crystals. Specifically, this award will support research in two interrelated areas: (1) development of more efficient numerical methods for characterizing the equilibrium orientational properties of liquid crystals in the large-scale regime (three space dimensions) and (2) refined analysis of the nature of the coupling between liquid crystal orientational order and electric fields. The most commonly used models for liquid crystals at experimental and device scales are macroscopic continuum director models, which model local average orientation in terms of a unit-length vector field (the "director field"). The principal investigator will use techniques from numerical linear algebra and numerical optimization (saddle point problems, nullspace methods, reduced Hessian methods, preconditioning) to develop efficient solvers that deal effectively with the pointwise unit-length constraint associated with discretizations of such models, for which current approaches are inadequate in large-scale settings. Most liquid crystal devices and experiments utilize electric fields to control liquid crystal order, and the interaction between liquid crystal orientation and local electric field poses both numerical and analytical challenges (due to coupled equilibrium equations, free energies failing to be positive definite, and anomalous behavior of instability thresholds). The principal investigator will develop a variationally based, unified stability framework capable of providing computable local stability criteria for numerical solutions (in the discretized setting) and also capable of explaining anomalous instability thresholds (in the continuum setting).
Liquid crystal materials exist in a complex fluid phase that exhibits orientational order of molecular axes at the microscopic level, just like a crystal. Virtually all of their material properties are anisotropic, that is, they are direction dependent (unlike ordinary liquids which look and behave the same in all directions). The orientational properties can therefore be coupled to a variety of external fields (electromagnetic, optical, temperature gradients, acoustic waves, etc.). The well-known use of liquid crystals in optical displays makes use of this behavior. However, the special properties of these materials make them useful in a much wider range of technological applications, including also drug delivery systems, soft actuators, biosensors, and others. Scientific computing is widely used in this area to optimize the performance of devices, to analyze experiments, and to validate the theory that is used to describe such devices. This award will support the development of advanced computational techniques for the numerical exploration of equilibrium orientational properties of large, complicated, modern liquid crystal systems associated with realistic devices and experiments, especially those involving coupled electric fields. Special attention is given to the nature of the coupling between the liquid crystal order and an applied electric field and to electric-field-induced instabilities of orientational order, which underlie the functioning of most devices and for which a deeper understanding is needed. The research will be carried out in collaboration with interdisciplinary and international partners, and it will impact applications and technologies using liquid crystals through improved performance analysis tools and techniques for interpreting experiments. The impact is expected to extend to the numerical modeling of ferromagnetic materials such as magnetic films used in hard drives, where models with similar features arise.
