Mathematics of solid and liquid crystals
Start date: Apr 1, 2012,
End date: Mar 31, 2018
The project combines two closely related themes in which nonlinear analysis addresses central issues of material behaviour. The first is the prediction and analysis of microstructure arising from solid phase transformations in alloys. Such microstructure largely determines the macroscopic properties of the material, but the prediction of its morphology remains poorly understood, and is related to deep unsolved problems in the calculus of variations. The aim is to make advances in this area using appropriate static and dynamic continuum models of nonlinear elasticity type, thus helping to create a predictive theory. The second theme is to develop the mathematical theory of the Landau - de Gennes theory of liquid crystals, in which the distribution of molecular orientations is described by a matrix order parameter. Regarded by physicists as a theory of choice for liquid crystals, the Landau - de Gennes model has been little studied by mathematicians. The aim is to understand more about its validity and properties of solutions, with potential gains for the prediction of the behaviour of new generations of liquid crystal displays.Linking and underpinning the two themes are common mathematical and conceptual challenges, such as understanding the existence and singularities of minimizers in the multi-dimensional calculus of variations, the approach to equilibrium of thermomechanical systems, and the passage from atomic and molecular to continuum descriptions of materials. An expectation of the project is that the simultaneous study of problems from the two themes will lead both to new understanding of these fundamental scientific questions and to beneficial cross-fertilization between the themes.
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