Shells, Sheets, and Shapes

Date of Award

August 2016

Degree Type


Degree Name

Doctor of Philosophy (PhD)




Teng Zhang

Subject Categories

Physical Sciences and Mathematics


Topology and geometry provide a unique perspective to study many systems. In this thesis we discuss three distinct soft condensed matter systems: (i) elastic shells, including topological defects on an elastic shell and how the symmetry of defect structures affects a shell’s ability to sustain external pressure; (ii) droplet sheets, including how defects and curvature drive a square-to-triangular lattice transition in a droplet network; (iii) atomically-thin membranes, including the effective bending rigidity of a fluctuating ribbon and a spontaneous symmetry breaking phenomenon of a vibrating flap. In (i), as perfect crystalline order is incompatible with spherical topology, defects need to be introduced for any triangulation of a 2-sphere. We study how the stability of spherical crystalline shells under external pressure is influenced by the defect structure. In particular, we compare stability for shells with a minimal set of twelve +1 disclinations to shells with extended defect arrays (grain boundary “scars” with non-vanishing net disclination charge). We find that the critical pressure at which shells collapse is lowered for scarred configurations that break icosahedral symmetry and raised for scars that preserve icosahedral symmetry. The particular shapes which arise from breaking of an initial icosahedrally-symmetric shell depend on the F ̈oppl-von K ́arm ́an number. In (ii), motivated by a recent experiment that prints tens of thousands of micron-sized aqueous droplets that bind by forming single lipid bilayers to form a cohesive network in bulk oil, we propose that this system is a model system to study lattice transitions that take place, for example, in colloidal crystals. In particular, we study the square-to-triangular lattice transition in 2D networks, where the transition is driven either by the explicit insertion of defects or by curving an initially flat lattice. In (iii), we use molecular dynamics to study the vibration of a thermally fluctuating 2D elastic ribbon clamped at both ends. We identify the eigenmodes from peaks in the frequency domain and track the dependence of the eigen-frequency of a given mode on the bending rigidity of the ribbon. With thermal contraction being taken care of, we define an effective bending rigidity using eigen-frequency and discuss the applicability of previous renormalization bending rigidity theory to the effective bending rigidity. Experimental realizations include two-dimensional atomically thin membranes such as graphene (see a recent experiment) and molybdenum disulfide or polymerized membranes. In the simulation of a flap, we find a spontaneous symmetry breaking phenomenon where the flap vibrates about a tilt plane in the upper or lower space.

This thesis contains some unpublished results. Some problems need further exploration.


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