Date of Award
Doctor of Philosophy (PhD)
Mark J. Bowick
Defects, Shape, Topological Defects
Physical Sciences and Mathematics
How does shape emerge at macroscopic scales from the spontaneous self-organization of building blocks at smaller scales? In this thesis, I will address this question in the context of closed soft two-dimensional membranes with internal order. Soft matter represents a good arena to identify simple universal mechanisms for shape selection. In fact, a distinctive property of soft materials is that they can undergo dramatic changes in geometric conformation at relatively low energetic cost. Thus, shape itself becomes a statistically fluctuating degree of freedom, and in some cases it can be found as the ground state of an appropriate free energy functional.
The building blocks of soft materials typically have lower symmetry than elementary point particles and exhibit rich patterns of spontaneous ordering as the free energy of the system is lowered. Order is frustrated if the membrane’s topology is non-trivial, and topological defects, which typically arise as finite temperature excitations, are forced to exist even in the ground state. Topological defects can play a key role in the selection of the ground state shape. For example, the presence of defects in closed 2-dimensional membranes with liquid-crystalline order enable to predict the existence of surprisingly sharp, faceted yet extremely soft polyhedral ground-state shapes. Upon functionalization of the defect sites, these faceted ground states may provide soft self-assemblable building blocks analogue to atoms but micron-sized, whose valence could be selected according to the internal symmetry of the liquid crystal.
Topology plays an important role also in the statistical mechanics of fluid membranes. These closed fluid sacs can have the shape of deformed spheres, one-handled tori, or closed shapes with an arbitrary number of handles. The statistical mechanics of membranes with fluctuating number of handles, encoded in the ensemble's partition function, is dominated by shape fluctutations, and new techniques are required to extract physical information from the sum over arbitrary topologies.
Serafin, Francesco, "The Shape of Soft Matter: Geometry and Defects" (2019). Dissertations - ALL. 1070.