Two-Dimensional Matter: Order, Curvature and Defects

Many systems in nature and the synthetic world involve ordered arrangements of units on two-dimensional surfaces. We review here the fundamental role payed by both the topology of the underlying surface and its detailed curvature. Topology dictates certain broad features of the defect structure of the ground state but curvature-driven energetics controls the detailed structured of ordered phases. Among the surprises are the appearance in the ground state of structures that would normally be thermal excitations and thus prohibited at zero temperature. Examples include excess dislocations in the form of grain boundary scars for spherical crystals above a minimal system size, dislocation unbinding for toroidal hexatics, interstitial fractionalization in spherical crystals and the appearance of well-separated disclinations for toroidal crystals. Much of the analysis leads to universal predictions that do not depend on the details of the microscopic interactions that lead to order in the first place. These predictions are subject to test by the many experimental soft and hard matter systems that lead to curved ordered structures such as colloidal particles self-assembling on droplets of one liquid in a second liquid. The defects themselves may be functionalized to create ligands with directional bonding. Thus nano to meso scale superatoms may be designed with specific valency for use in building supermolecules and novel bulk materials. Parameters such as particle number, geometrical aspect ratios and anisotropy of elastic moduli permit the tuning of the precise architecture of the superatoms and associated supermolecules. Thus the field has tremendous potential from both a fundamental and materials science/supramolecular chemistry viewpoint.Comment: Review article, 102 pages, 59 figures, submitted to Advances in Physic

DNA Conformational Changes and Phase Transitions Induced by Tension and Twist

DNA is a double stranded helical molecule with an intrinsic right handed twist. Its structure can be changed by applying forces and torques in single molecule experiments. In these experiments DNA has been seen to form super-helical structures (supercoils), collapse into tightly condensed states (toroids) and undergo structural changes (phase transitions). Our work focuses on studying all these phenomena by accounting for DNA elasticity, entropic effects due to thermal fluctuations and electrostatics. First, we study the DNA compaction problem in super-helices and toroidal structures. To do so we combine a fluctuating elastic rod model of DNA with electrostatic models for DNA-DNA interactions. Our models are able to predict the onset of the transition to supercoils and toroids under a wide range of experimental conditions. Next, we address DNA phase changes in the presence of mechanical loads.A phenomenon well known from experiments is the overstretching transition associated with the sudden change of DNA extension at high tensions. Depending on the ionic concentration, temperature and pulling rate, DNA can either transform into a melted state (inner strand separation) or S-DNA. Motivated by this, we study the equilibrium and kinetics of the DNA overstretching transitions making use of a quartic potential and non-gaussian integrals to evaluate the free energy of the system. We find that the cooperativity of the transition is a key variable that characterizes the overstretched state. In a separate study we make use of a heterogeneous fluctuating rod model to examine the hypothesis that a newly discovered left-handed form called L-DNA is a mixture of two relatively well-characterized DNA phases - S-DNA and Z-DNA. L-DNA is stable at high tensions and negative twist. We show that if the idea of a mixed state is correct, then the content of S-DNA and Z-DNA varies as a function of the ionic concentration. Finally, we also use our fluctuating rod model to study the mechanical properties of drug-DNA complexes. We show that our methods can predict the results of experiments from various labs if we use only one set of experiments to fit the data to our model

Microstructure and Phase Behavior in Colloids and Liquid Crystals

This thesis describes our investigation of microstructure and phase behavior in colloids and liquid crystals. The first set of experiments explores the phase behavior of helical packings of thermoresponsive microspheres inside glass capillaries as a function of volume fraction. Stable helical packings are observed with long-range orientational order. Some of these packings evolve abruptly to disordered states as the volume fraction is reduced. We quantify these transitions using correlation functions and susceptibilities of an orientational order parameter. The emergence of coexisting metastable packings, as well as coexisting ordered and disordered states, is also observed. These findings support the notion of phase-transition-like behavior in quasi-one-dimensional systems. The second set of experiments investigates cross-over behavior from glasses with attractive interactions to sparse gel-like states. In particular, the vibrational modes of quasi-two-dimensional disordered colloidal packings of hard colloidal spheres with short-range attractions are measured as a function of packing fraction. A crossover from glassy to sparse gel-like states is indicated by an excess of low-frequency phonon modes. This change in vibrational mode distribution appears to arise from highly localized vibrations that tend to involve individual and/or small clusters of particles with few local bonds. These mode behaviors and corresponding structural insights may serve as a useful signature for glass-gel transitions in wider classes of attractive packings. A third set of experiments explores the director structures of aqueous lyotropic chromonic liquid crystal (LCLC) films created on square lattice cylindrical-micropost substrates. The structures are manipulated by modulating of the concentration-dependent elastic properties of LCLC s via drying. Nematic LCLC films exhibit preferred bistable alignment along the diagonals of the micropost lattice. Columnar LCLC films form two distinct director and defect configurations: a diagonally aligned director pattern with local squares of defects, and an off-diagonal configuration with zig-zag defects. The formation of these patterns appears to be tied to the relative free energy costs of splay and bend deformations in the precursor nematic films. The observed nematic and columnar configurations are understood numerically using a Landau-de Gennes free energy model. This work provides first examples of quasi-2D micropatterning of LC films in the columnar phase and the first micropatterning of lyotropic LC films in general, as well as demonstrating alignment and configuration switching of typically difficult-to-align LCLC films via bulk elastic properties

Applied Mathematics to Mechanisms and Machines

This book brings together all 16 articles published in the Special Issue "Applied Mathematics to Mechanisms and Machines" of the MDPI Mathematics journal, in the section “Engineering Mathematics”. The subject matter covered by these works is varied, but they all have mechanisms as the object of study and mathematics as the basis of the methodology used. In fact, the synthesis, design and optimization of mechanisms, robotics, automotives, maintenance 4.0, machine vibrations, control, biomechanics and medical devices are among the topics covered in this book. This volume may be of interest to all who work in the field of mechanism and machine science and we hope that it will contribute to the development of both mechanical engineering and applied mathematics