Columnar Crystals:Mechanism Of Competition For Space In Multi-layer Structures

In any busy modern society,people would take any efficient utilization of space and time as top of their priorities.Problems of particle packing belong to a class of problems that concern the optimal utilization of space,and have interested researchers from a variety of disciplines.Such problems,which exhibit great complexities and diversities,can generally be classified into problems of ordered packings and of disordered packings,where the former include problems of finding densest possible packings of particles(spheres,ellipsoids,tetrahedra,etc.)in space.Applications of packing problems span a variety of length scales,from celestial bodies to microscopic particles,and they can be found in almost all walks of life,from high-end scientific research to the mundane in your daily life.Any packing problem can be dealt with by analytical,experimental or computational means.With the limitations of conventional theoretical and experimental methods and with the rapid developments of computing power and numerical methods in recent years,many complex packing problems have been studied in a computational manner.Based on results from computer simulations,this thesis presents an in-depth theoretical analysis on the densest possible structures of identical spheres in cylindrical confinement.This research focuses on a structural analysis of the densest possible ordered packings of identical spheres inside a cylinder,where such periodic ordered structures are referred to as “columnar crystals”.Consider the cylinder-to-sphere diameter ratio D.Within the regime D?(1,2.7148),the system exhibits a variety of densest possible structures,such as the planar zigzag,single helix,double helix,triple helix,and quadruple helix,where the spheres of each structure are all in contact with the inner wall of the cylinder.At D>2.7148,however,the densest possible structures consist of spheres that are not in contact with the cylindrical wall.Such structures are here referred to as “multi-layer structures”,where any sphere in contact with and away from the cylindrical wall is respectively thought of as part of an outermost layer and of an inner layer.The complex densest possible structures at this regime of D are results of competition for space between the outermost layer and the inner layers.This research focuses on the columnar crystals at D?[2.7013,2.8612],where there exists only a single inner layer in each densest possible structure.It addresses the problems of how such multi-layer structures vary for changing D and of how the outermost and inner layers compete with each other for space.For each densest possible structure,the positional coordinates of a unit cell are obtained from numerical simulations,such that a three-dimensional visualization of the structure can be constructed and a two-dimensional surface analysis of the cylinder-touching spheres can be carried out.The contact condition between any pair of adjacent spheres is determined using numerical data,and a corresponding “skeleton” that displays the whole network of inter-sphere contacts is constructed.Based on continuity analyses of first-order derivatives of packing fractions,surface analyses of cylinder-touching spheres in the outermost layer,and constructed “skeleton” networks of inter-sphere contacts,four types of densest possible structures,named respectively as “quintuple helix”,“tilted pentagon”,“doublelayer pentagon”,and “sextuple helix” in ascending order of D,have been identified,where inner spheres exist for the last three structures.For D?[2.7013,2.8612],this research includes a detailed analysis of how spheres in a densest possible structure are “displaced” as D increases,and provides insights into the mechanism of interlayer competition for space.For both the “tilted pentagon” and “sextuple helix”,the inner spheres are densely packed while the outermost cylinder-touching spheres are loosely packed.The opposite holds for the “double-layer pentagon”,with loosely packed inner spheres and densely packed outermost spheres.The results would help us understand how those complex densest possible structures come into existence,and would serve as a theoretical basis for the designing and fabrication of lowdimensional materials.