Pre-submission Thesis Presentation

Two-dimensional soft materials such as flexible membranes offer an ideal testing ground for fundamental concepts involving order(symmetry), low-energy excitations, topological defects, and fluctuations. This thesis studies the interplay between geometry, topology, and elasticity in two-dimensional soft materials.

The theoretical studies presented in this thesis are separated into two parts. In the first part, we address the stability of singular, topological wall defects on spheres, catenoids, and helicoids. Unlike soliton-type wall configurations, these wall defects are singular lines. However, wall defects are topologically unstable on two-dimensional surfaces. Within the mean-field approximation, we show that singular, topological wall defects can be stabilized on curved surfaces because of their Gaussian (intrinsic) curvature. We attribute their stability to free-energetic considerations, which override those of topological stability.

In the second part of the thesis, we discuss the role of topological defects in determining the observed morphologies of polymer crystallites. Polymer crystals (solution-grown, as well as melt-grown) are significantly different from atomic and molecular crystals because of their connectivity. Interestingly, observed morphologies of polymer crystallites are lamellar. These lamellar structures display spherulitic, sectored, tent-like, or scroll structures. In our work, we use concepts borrowed from liquid-crystal physics, and the physics of crystalline membranes to study the sector-, and especially the tent morphologies. We construct ``phase diagrams" in parameter space, indicating the ranges of stability of the sector- and tent configurations over a range of dimensionless, phenomenological parameters involving bending rigidity, and (isotropic-, and anisotropic) line tensions of polymer lamellae.  To our knowledge, this is the first attempt at a detailed theoretical modelling that addresses the stability of the tent morphology.