Columnar structures, many of which are helical, refer to dense cylindrical packings of particles. They are ubiquitous, for example, they exist in the contexts of botany, foams, and nanoscience. There have been in-depth investigations of columnar structures of both hard spheres (e.g., ball bearings) and soft spheres (e.g., wet foams), through computer simulations, analytic derivations, or simple experiments. This monograph serves as a comprehensive guide for scientists, engineers, or artists who would like to have a good grasp of the fundamentals and applications of such aesthetically appealing structures for their own professional interests.
The book begins with an introduction to the field of packing problems, where such problems are closely related not only to the columnar structures presented in the book but also to the structures of condensed matter systems in general. It then discusses about columnar structures of spheres and overviews their classifications and applications. It reviews the models and concepts employed in the authors’ studies on columnar structures of spheres. It also details the method of sequential deposition for generating columnar structures of hard spheres computationally or experimentally. Lastly, it presents some latest findings on the columnar structures of soft spheres and on the structures obtained from the longitudinal compression of a hard-sphere chain in a cylindrical harmonic potential.
Author(s): Jens Winkelmann, Ho-Kei Chan
Publisher: Jenny Stanford Publishing
Year: 2023
Language: English
Pages: 213
City: Singapore
Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Acknowledgements
Chapter 1: An Introduction to Packing Problems
1.1: Packing Problems in Daily Life
1.2: Packing Problems in Physics
1.3: Computational Aproaches to Packing Problems
1.4: Random Packings of Particles
1.5: Applications in the Physical Sciences
1.6: Packing Problems as a Growing Research Field
Chapter 2: An Introduction to Columnar Structures
2.1: A Friday‐Afternoon Experiment: Packing Golf Balls into a Tube
2.2: What Are Columnar Structures?
2.3: The Phyllotactic Notation: Categorising Columnar Structures
2.4: Applications of Columnar Structures: From Botany and Foams to Nanoscience
2.4.1: Examples from Botany
2.4.2: Dry and Wet Foam Structures
2.4.3: Nanoscience: Microrods and Optical Metamaterials
2.5: Advantages of Generic‐Model Simulations
Chapter 3: Models and Concepts for Columnar Structures
3.1: The Packing Fraction ϕ
3.2: Hard Spheres vs. Soft Spheres
3.2.1: The Hard‐Sphere Model
3.2.2: The Soft‐Sphere Model
3.3: Different Types of Columnar Structures
3.3.1: What Is a Uniform Structure?
3.3.2: What Is a Line‐Slip Structure?
3.4: Densest Hard‐Sphere Packings inside Cylinders
3.5: Simulation Techniques: Minimisation Algorithms
3.5.1: Local Minimisation Routines
3.5.2: Global Minimisation Routines
Chapter 4: Packing of Hard Spheres by Sequential Deposition
4.1: Mathematical Description of Sequential Deposition
4.1.1: Equations for the Geometric Relation between Two Spheres in Contact
4.1.2: Surface Representation of a Single Sphere
4.1.3: Simultaneous Deposition of Spheres
4.1.4: Empirical Packing Fractions for an Infinitely Long Cylinder
4.2: Empirical Trials of Sequential Deposition
4.2.1: Successful Trial for a Densest Zigzag Structure
4.2.2: Successful Trial for a Densest Achiral Structure
4.2.3: Unsuccessful Trial for a Densest Single‐Helix Structure
4.2.4: Successful Trial for a Densest Single‐Helix Structure
4.3: Problem with a Flat Base at D > 2
4.4: Sequential Deposition: The Packing Algorithm
4.4.1: Packing Algorithm for D ∈ [1,2)
4.4.2: Packing Algorithm for D ≥ 2
4.5: Columnar Structures from Sequential Deposition
4.5.1: Examples of Structures at D = 2.35
4.5.2: Examples of Structures at D = 2.25
4.6: Conclusions
Chapter 5: Soft‐Sphere Packings in Cylinders
5.1: Introduction to Soft‐Sphere Packings in Cylindrical Confinement
5.2: Simulations: Minimisation of Enthalpy H
5.3: Simulation and Observation of Line‐Slip Structures in Soft‐Sphere Packings
5.3.1: Phase Diagram of All Uniform and Line‐Slip Structures without Internal Spheres
5.3.2: Structural Transitions in the Phase Diagram
5.3.3: Experimental Observation of Line‐Slip Structures
5.4: Hysteresis and Metastability in Structural Transitions
5.4.1: Enthalpy Curves at Constant Pressures for a Reversible Transition
5.4.2: Stability Diagram for a Reversible Transition
5.4.3: Directed Network of Structural Transitions
5.5: Conclusions
Chapter 6: Rotational Columnar Structures of Soft Spheres
6.1: Introduction to Rotational Columnar Structures
6.2: Lee et al.’s Lathe Experiments
6.3: Columnar Structures from Rapid Rotations: A Theoretical Analysis
6.3.1: Energy of Hard‐Sphere Packings
6.3.2: Analytic Energy Calculation of Soft‐Sphere Packings
6.4: Columnar Structures from Rapid Rotations: Simulations of Finite‐Sized Systems
6.4.1: Method of Simulation: Energy Minimisation
6.4.2: Line‐Slip Structures of Finite‐Sized Systems
6.5: Conclusions
Chapter 7: Hard‐Sphere Chains in a Cylindrical Harmonic Potential
7.1: Sphere Chains in a Cylindrical Harmonic Potential
7.1.1: Localised Buckling in Compressed Sphere Chains
7.1.2: Compression Δ
7.2: Simulation Methods
7.2.1: Iterative Stepwise Method
7.2.2: Simulations Based on Energy Minimisation
7.3: Numerical Results
7.3.1: Typical Profiles
7.3.2: Bifurcation Diagrams
7.3.3: Maximum Angles
7.4: Linear Approximation
7.5: Comparison with Experiments
7.6: Conclusions
Chapter 8: Summary and Outlook
8.1: Chapter Outline
8.2: Hard‐Sphere Packings from Sequential Deposition
8.2.1: Summary
8.2.2: Outlook
8.3: Soft‐Sphere Packings in Cylinders
8.3.1: Summary
8.3.2: Outlook: An Exhaustive Investigation of Hysteresis
8.4: Rotational Columnar Structures of Soft Spheres
8.4.1: Summary
8.4.2: Outlook: Further Investigations of Finite‐Size Effects
8.5: Hard‐Sphere Chains in a Cylindrical Harmonic Potential
8.5.1: Summary
8.5.2: Outlook: Extensions of Current Simulations and Experiments
8.6: Soft‐Disk Packings Inside a Two‐Dimensional Rectangular Channel
8.7: Limitations of the Soft‐Sphere Model
8.7.1: Soft Disks vs. Two‐Dimensional Foams
8.7.2: Average Contact Number Z(ϕ)
8.8: The Morse–Witten Model for Deformable Spheres
Appendix A: Tabulated Hard Sphere Results
Appendix B: Minimisation Routines
Appendix C: Energy of (l, l, 0) Structures
Bibliography
Index
