The Geometry of Soft Materials: A Primer

Abstract

We present an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.

45 pages, RevTeX, 12 eps figures

References in corpus (15)

• Molecular Chirality and Chiral Parameters
• Torsional Directed Walks, Entropic Elasticity, and DNA Twist Stiffness
• Entropic Elasticity of Twist-Storing Polymers
• Twirling and Whirling: Viscous Dynamics of Rotating Elastica
• Conformations of Linear DNA
• The Viscous Nonlinear Dynamics of Twist and Writhe
• Direct Determination of DNA Twist-Stretch Coupling
• Minimal Surfaces, Screw Dislocations and Twist Grain Boundaries
• Stretching an heteropolymer
• Sequence Effects on DNA Entropic Elasticity
• Conformations of closed DNA
• n-atic Order and Continuous Shape Changes of Deformable Surfaces of Genus Zero
• Singularities, Structures and Scaling in Deformed Elastic m-Sheets
• Local Writhing Dynamics
• Disclination loop behavior near the nematic-isotropic transition

Cited by in corpus (150)

• Reconfigurable knots and links in chiral nematic colloids
• Topological active matter
• Topological colloids
• Disclination Loops, Hedgehogs, and All That
• Dynamics of Self-Propelled Particles Under Strong Confinement
• Dynamics of filaments and membranes in a viscous fluid
• Nematic textures in spherical shells
• Self-organized dynamics and the transition to turbulence of confined active nematics
• Vortices in Superfluid Films on Curved Surfaces
• Perspective: Geometrically-Frustrated Assemblies
• Geometric theory on the elasticity of bio-membranes
• Topologically stable magnetization states on a spherical shell: curvature stabilized skyrmions
• Stable nematic droplets with handles
• Multiplet of skyrmion states on a curvilinear defect: skyrmion lattices as a ground state
• Orientational properties of nematic disclinations
• Flory theory for Polymers
• Diffusion on Ruffled Membrane Surfaces
• Lensing effects in a nematic liquid crystal with topological defects
• 3-dimensional structure of a sheet crumpled into a ball
• Cross-talk between topological defects in different fields revealed by nematic microfluidics
• Strain and the optoelectronic properties of non-planar phosphorene monolayers
• Nematic braids: topological invariants and rewiring of disclinations
• Minimal surfaces bounded by elastic lines
• A geometrical approach to the dynamics of spinor condensates II: Collective modes
• A change in stripes for cholesteric shells via anchoring in moderation
• A geometrical approach to the dynamics of spinor condensates I: Hydrodynamics
• Magnetization in narrow ribbons: curvature effects
• Braided Bundles and Compact Coils: The Structure and Thermodynamics of Hexagonally-Packed, Chiral Filament Assemblies
• Defects in nematic membranes can buckle into pseudospheres
• Model for Polymorphic Transitions in Bacterial Flagella
• Topological defects in cholesteric liquid crystal shells
• Geometry and Optimal Packing of Twisted Columns and Filaments
• Waltzing route towards double-helix formation in cholesteric shells
• First-order patterning transitions on a sphere as a route to cell morphology
• Pore formation in fluctuating membranes
• Models for twistable elastic polymers in Brownian dynamics, and their implementation for LAMMPS
• Defect generation and deconfinement on corrugated topographies
• Environmental bias and elastic curves on surfaces
• Theory and Monte Carlo simulations for the stretching of flexible and semi-flexible single polymer chains under external fields
• The geometry of thresholdless active flow in nematic microfluidics
• Smectic Phases with Cubic Symmetry: The Splay Analog of the Blue Phase
• Umbilic Lines in Orientational Order
• Extrinsic Curvature, Geometric Optics, and Lamellar Order on Curved Substrates
• Self-assembly of hard helices: a rich and unconventional polymorphism
• Lassoing saddle splay and the geometrical control of topological defects
• Shape instabilities in vesicles: a phase-field model
• Defects in Crystalline Packings of Twisted Filament Bundles: I. Continuum Theory of Disclinations
• Spin eigen-excitations of an antiferromagnetic skyrmion
• Patterning of morphogenetic anisotropy fields
• Twist/Writhe Partitioning in a Coarse-Grained DNA Minicircle Model
• Data-driven discovery of active nematic hydrodynamics
• BPS Configurations in Smectics
• Bijels Containing Magnetic Particles: A Simulation Study
• Achiral symmetry breaking and positive Gaussian modulus lead to scalloped colloidal membranes
• Vesicle shape, molecular tilt, and the suppression of necks
• Structural motifs of biomolecules
• Helfrich-Hurault elastic instabilities driven by geometrical frustration
• Effect of supercoiling on formation of protein mediated DNA loops
• Nonequilibrium fluctuations of an interface under shear
• Helical Tubes in Crowded Environments
• Functionals linear in curvature and statistics of helical proteins
• Robust fadeout profile of an evaporation stain
• Dynamic supercoiling bifurcations of growing elastic filaments
• Curvature-induced frustration in the $XY$ model on hyperbolic surfaces
• The Topology of Dislocations in Smectic Liquid Crystals
• Effective stiffness and formation of secondary structures in a protein-like model
• Smectic blue phases: layered systems with high intrinsic curvature
• Ring-o-rings: a new category of supramolecular structures with topologically tunable properties
• Statistical Mechanical Analysis of the Electromechanical Coupling in an Electrically-Responsive Polymer Chain
• Defect Spirograph: Dynamical Behavior of Defects in Spatially Patterned Active Nematics
• Phase transitions in superconductor liquid crystals
• Topological elasticity of non-orientable ribbons
• Mechanochemical action of the dynamin protein
• Paramagnetic filaments in a fast precessing field: Planar versus helical conformations
• Effect of Bending Anisotropy on the 3D Conformation of Short DNA Loops
• Cholesteric shells: two-dimensional blue fog and finite quasicrystals
• Intradomain Textures in Block Copolymers: Multizone Alignment and Biaxiality
• Revisiting the foundations of the quantum theory of atoms in molecules: Some open problems
• Learning developmental mode dynamics from single-cell trajectories
• The Geometry of the Cholesteric Phase
• Metric approach for sound propagation in nematic liquid crystals
• Superfluid vortex dynamics on a torus and other toroidal surfaces of revolution
• Topological and geometric decomposition of nematic textures
• Softly Constrained Films
• Geometry and scaling of tangled vortex lines in three-dimensional random wave fields
• Elastic interactions between topological defects in chiral nematic shells
• Stiff knots
• Curvature-controlled defect dynamics in active systems
• Colloidal Surfaces with Boundaries, Apex Boojums and Nested Elastic Self-Assembly of Nematic Colloids
• Chiral edge fluctuations of colloidal membranes
• A blend of stretching and bending in nematic polymer networks
• Continuum model for polymers with finite thickness
• Phase transition of $q$-state clock models on heptagonal lattices
• DNA Elasticity : Topology of Self-Avoidance
• Topology and geometry of spin origami
• The elixir phase of chain molecule
• Emergence of Secondary Motifs in Tube-Like Polymers in a Solvent
• Geometry of Bend: Singular Lines and Defects in Twist-Bend Nematics
• Local symmetry determines the phases of linear chains: a simple model for the self-assembly of peptide
• Electrostatic Contribution to the Persistence Length of a Semiflexible Dipolar Chain
• Smectic Liquid Crystals: Materials with One-Dimensional, Periodic Order
• The distribution of extremal points of Gaussian scalar fields
• Hyperbolic Interfaces
• Role of Gaussian curvature on local equilibrium and dynamics of smectic-isotropic interfaces
• Superfluid Vortex Dynamics on Planar Sectors and Cones
• Stresses in curved nematic membranes
• Tensorial conservation law for nematic polymers
• Cationic vacancies as defects in honeycomb lattices with modular symmetries
• Vortex and disclination structures in a nematic-superconductor state
• Stochastic curvature of enclosed semiflexible polymers
• Singular Values, Nematic Disclinations, and Emergent Biaxiality
• Fracton-elasticity duality on curved manifolds
• End exclusion zones in strongly stretched, molten polymer brushes of arbitrary shape
• Membrane stress and torque induced by Frank's nematic textures: A geometric perspective using surface-based constraints
• Topological Defects in Gravitational Lensing Shear Fields
• Stochastic geometry and topology of non-Gaussian fields
• Topological Phases and Curvature-Driven Pattern Formation in Cholesteric Shells
• Phase Separation on Bicontinuous Cubic Membranes: Symmetry Breaking, Re-entrant and Domain Facetting
• Helfrich-Canham bending energy as a constrained non-linear sigma model
• Triply-Periodic Smectics
• Geometry of flexible filament cohesion: Better contact through twist?
• Nonlinear Dynamics of Moving Curves and Surfaces: Applications to Physical Systems
• Force induced formation of twisted chiral ribbons
• Topological defects of dipole patchy particles on a spherical surface
• Confining spheres within hyperspheres
• Marginally compact phase and ordered ground states in a model polymer with side spheres
• Mechanics of nematic membranes: Euler-Lagrange equations, Noether charges, stress, torque and boundary conditions of the surface Frank nematic field
• Defect-driven shape instabilities of bundles
• Vortex interaction on curved surfaces
• Effects of orientational order on modulated cylindrical interfaces
• Rogue breather modes: Topological sectors, and the `belt-trick', in a one-dimensional ferromagnetic spin chain
• Good Fibrations: Packing Rules for Diabolic Domains
• Constraint percolation on hyperbolic lattices
• Poisson-Bracket Approach to the Dynamics of Bent-Core Molecules
• Knots in Physics
• Topological transport of vorticity on curved magnetic membranes
• Umbilical points of a non-Gaussian random field
• Topological dipoles of quantum skyrmions
• Topological Defect Propagation to Classify Knitted Fabrics
• Current-driven motion of magnetic topological defects in ferromagnetic superconductors
• Motility and self propulsion of active droplets
• Doubly-periodic instability pattern in a smectic A liquid crystal
• Synchronization of the Frenet-Serret linear system with a chaotic nonlinear system by feedback of states
• Dislocations and Fibrations: The Topological Structure of Knotted Smectic Defects
• Machine Learning Forecasting of Active Nematics
• Berry Phase Enforced Spinor Pairing Order
• Geometrical dynamics of edge-driven surface growth
• Intrinsic defects, fluctuations of the local shape, and the photo-oxidation of black phosphorus
• Quantum Buckling
• Topological defects and shape of aromatic self-assembled vesicles