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Yi Jiang, Cellular Pattern Formation, PhD thesis, University of Notre Dame, 1998


This thesis studies the formation and evolution of cellular patterns in foams and living organisms using the extended large-$Q$ Potts model. Specific problems include grain growth, foam drainage, foam rheology, and patterning and cell sorting in the mound phase of the slime mode Dictyostelium discoideum.

In a wide range of cellular materials, surface-energy-driven diffusion leads to boundary motion which causes some grains to expend and others to shrink. Two-dimensional large-$Q$ Potts model simulation of the evolution of a disordered cluster developed from a hexagonal grain array with a defect shows that abnormal grain growth can occur without strong anisotropy of surface energy. The grains at the boundary of the disordered cluster reach a special scaling state with no scale change.

In three-dimensional liquid foams, drainage occurs due to gravity. Large-$Q$ Potts model simulations, extended to include gravity in three dimensions, agree with both experimental and analytical results for various kinds of foam drainage, and also predict new phenomena.

Foams exhibit a unique rheological transition from solid-like to fluid-like. Simulations using the large-$Q$ Potts model, extended to apply shear to a two-dimensional foam, show three different types of hysteresis in foam's stress-strain relationship, which correspond to the elastic, viscoelastic and viscous fluid properties. This wide-ranging mechanical response depends on the structure and dynamics of local topological rearrangement of foam cells.

Biological tissues resemble foams and the large-$Q$ Potts model can also simulate sorting in biological cell aggregates. In {\it Dictyostelium} mound, two types of cells are initially randomly distributed. In time, one cell type sorts to form a tip. Simulations show that both differential adhesion and chemotaxis are required for sorted tip formation. With only differential adhesion, no tip forms. With only chemotaxis, a tip forms containing both cell types. Thus simulations can provide a method to determine the processes necessary for biological patterning.

BibTeX Entry

author = {Yi Jiang},
title = {Cellular Pattern Formation},
year = {1998},
urlpdf = {http://math.lanl.gov/~yi/Papers/Jiangthesis.pdf},
school = {University of Notre Dame}