A mathematical study of periodic structures inspired by soft matter systems: The QTZQZD family of triplyperiodic minimal surfaces and doublyperiodic WeftKnitted textiles
Abstract
In this thesis, we study mathematical objects that are periodic and arise as ordered states in soft matter systems. Motifs resembling triplyperiodic minimal surfaces and their constant mean curvature variants are found to form spontaneously through selfassembly in soft matter systems, and doublyperiodic weftknitted textiles with repeating motifs of strings are constructed using needles and yarn according to a mechanical protocol. Despite the difference in their physical origins, a common underlying theme among these two systems is the role played by \emph{topological} constraints in determining the end states that minimize a prescribed energy functional.
Soap films form minimal surfaces due to the energy cost from the surface tension of the soap films. In case of compact soap films  soap films with a fixed boundary (a space curve)  topology comes into play through the number of pathconnected components and the knot or link type of the boundary curve. Another set of systems where interfaces resembling minimal surfaces can be found are cubic mesophases in lyotropic liquid crystals, lipidwater mixtures, lipid bilayers, diblock copolymer melts and other systems constituting amphiphiles  macro molecules with hydrophobic and hydrophilic components. In these systems, periodic minimal surfaces are formed as the system tries to minimize the energy cost locally due to local nature of the interactions subjected to some global constraints i.e., the set of accessible states of minimum energy are governed by the interactions between the constituent particles, their shape, volume fraction of different components in the mixture and the boundary condition (in case of systems of finite size). The first two factors listed above are local in nature versus the other two that are of global kind.
Despite having phenomenological models, it is not clear why and how triplyperiodic minimal surface like interfaces emerge as the states of minimum energy (most likely, a local minimum) without any direct connection to the nontrivial topology found in these ordered phases. For example, the fundamental translational unit of the gyroid and the diamond surface are compact surfaces of genus three embedded inside a three torus. Thus, the boundaries are identified by three periodicity and their lift to the euclidean three space gives rise to a pair of nonoverlapping, infinite, threeperiodic labyrinth of connected tunnels. Independent of the lack of understanding of all the complex interactions at play in selfassembly of triplyperiodic minimal surface motifs, from a mathematical standpoint there is no limitation to constructing and studying triplyperiodic minimal surfaces. Thus, guided by mathematical constructions we can explore and design triplyperiodic structures with different genus at different length scales and study their optical and mechanical responses. In this thesis, we take the initial steps in designing a general protocol for constructing triplyperiodic minimal surfaces, and we apply it to generate a family of chiral triplyperiodic minimal surfaces.
Even though there is no good theory to describe the mechanics of open ended knots, bends, hitches and textiles, we have been using them for centuries. The reason for their applicability in spite of limited understanding of their physics is that, it is easy to grasp the topological aspects of these quasi 1D structures. In other words, in most cases the function of a tied string or a rope is decided by its knotting or linking. Thus, topology is at the root of the matter and the geometric aspects play a role only at a higher order. Therefore, in order to develop a deeper understanding of any of these families of knotted objects, one must start by studying their topological properties. In this thesis, we develop a topological framework to study knotting and linking of yarn in weftknitted textiles. Textiles are interesting due to their hierarchical structure, and weftknitted textiles are the most appealing among textiles as they give rise to complex emergent properties.
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