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Title:Statistics, geometry, and mechanical states in cellular matter
Author(s):Kim, Sangwoo
Director of Research:Hilgenfeldt, Sascha
Doctoral Committee Chair(s):Hilgenfeldt, Sascha
Doctoral Committee Member(s):Saif, Taher; Ostoja-Starzewski, Martin; Tang, Vivian
Department / Program:Mechanical Sci & Engineering
Discipline:Theoretical & Applied Mechans
Degree Granting Institution:University of Illinois at Urbana-Champaign
Biological Tissues
Abstract:Cellular matter such as foam, emulsions, granular materials, and biological tissues is ubiquitously observed in nature. Structural similarities in cellular matter are rather surprising because the governing physics of these systems vastly differ from each other. Identifying common structural features as well as understanding mechanical properties in cellular matter have been the subject of extensive research, but linking the two aspects has not received enough attention. In this dissertation, we elucidate the relations between domain statistics, geometry, and mechanical states in two-dimensional cellular matter using mathematical and mechanical modeling. In the 1920s, F.T. Lewis proposed an empirical linear correlation between the average area of n-sided cells and their topologies based on observation of epidermal layers of Cucumis and many supporting and dissenting findings have been reported in ensuing decades. Extensive data on the cells in epidermal layer of Cucumis confirms systematic deviations in size-topology correlations compare to many other two-dimensional cellular matter and anisotropic shapes in Cucumis cells explain the observed deviation. We develop a local geometric model that takes into account shape anisotropy and it successfully predicts both linear Lewis law and distinctive size-topology correlations. To connect statistics and geometry to mechanical properties, we develop a simple mechanical model with a leading-order interfacial energy functional and simulate typical ground states. Increasing relative adhesion strength induces shape change from isotropic domains to anisotropic domains that coincides with the rigidity transition from tense states to relaxed states. The quenched disorder on cell properties can significantly alter the position of the transition and statistics of Cucumis experiments indicate that the systems are right at the transition. The wildtype Drosophila compound eye exhibits regular hexagonal structure of facets called ommatidia but certain genetic mutations disrupt regulation and lead to generation of topological defects. With conventional order measures in statistical mechanics, we demonstrate that translational order disruption in Drosophila eye is sufficient for the onset of topological disorder. The statistical model based on perturbation of a regular lattice is developed to predict size-topology correlations. We identify a previously unknown large-scale systematic area variation across the eye, which has no effect on defect generation. Internal structure of ommatidia also significantly influences the topological order in Drosophila eye. In the last two chapters, we investigate the relationship between domain structure and mechanical properties in metastable states. A universal correlation between local edge length and energy difference between two adjacent metastable states is obtained and the correlation is shown to be independent of the exact form of energy functional. An efficient algorithm to approach ground states is proposed based on the universal correlation. The initial edge length distribution is sufficient to predict global energy difference between metastable states and the ground states. We further show that limited information on domain statistics yields an accurate prediction of energy level for metastable states. The defect density determines energetic states in monodisperse systems. The cross-correlation between size and topology in conjunction with the defect density describes the inherent structure energy of cellular matter in polydisperse systems. These relations are invariant across a large class of energy functionals and can therefore be applied to a wide variety of important systems.
Issue Date:2018-07-10
Rights Information:Copyright 2018 Sangwoo Kim
Date Available in IDEALS:2018-09-27
Date Deposited:2018-08

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