A Theoretical Study Of Sphere Packing In Confined Space

The densest possible spatial arrangements of rigid bodies have always been a subject of interest for both mathematicians and condensed matter physicists.Packing problems,such as those concerning the closest packed structures of hard spheres,are among the oldest and most challenging problems in science.A notable example is the Kepler conjecture,which concerns the densest possible packings of equal-sized spheres in open space.A well-accepted mathematical proof of this conjecture arrived almost four centuries later after it was proposed by Johannes Kepler in 1611.In comparison with the densest possible packings of rigid bodies in open space,problems that involve a search of densest possible packings in confined space present even greater challenges to scientists,where the extra complications come from the introduction of a boundary.A notable example is the problem of packing equal-sized spheres into an infinitely long cylinder,where most of the results reported so far are numerical in nature and a rigorous mathematical solution is still lacking.This research aims at providing theoretical solutions to this problem for cases of narrow confinement.In this work,the density possible packings of equal-sized hard spheres in an infinitely long cylinder were studied,where the cylinder is assumed to be hard as well.Let D be the ratio of the cylinder's diameter to the spheres5 diameter.The maximum possible packing density depends only on this diameter ratio,and has nothing to do with the absolute size of neither the cylinder nor the spheres.This research focuses on the densest possible structures and the corresponding packing densities at specific values of D,and how these structures and their chirality vary with D,and mathematical solutions for the densest possible single-and double-helix structures at D<2 have been obtained.If the diameter ratio D is less than 2.7013,all spheres of densest possible packings are in contact with the cylindrical wall,according to simulation results,and the focus of this work is on cases of D<2.In such cases,each sphere is in contact with two or four other spheres.At 1+(?)/2<D<1+4(?)/7,we have a single helix structure where each sphere is in contact with two other spheres above it and two below it.This helical structure has a repeating local structure with a triplet of mutually touching spheres.At 1+4(?)/7<D<2,we have a double helix structure where each sphere is in contact with two other spheres above it and two below it.This helical structure consists of two repeating local structures,each with a triplet of mutually touching spheres.On the other hand,the method of Taylor expansion has been employed for a study of how the densest possible structures vary with D,providing a theoretical understanding for the origin of structural transitions,such as transitions between different helical structures and those between achiral structures and chiral structures.It is worth pointing out that all these derivations are based on repeating units of the densest possible structures under study,for example repeating units with only two or three spheres.This approach simplifies our problem of global quasi-one-dimensional structures into a problem of local structures,and may be applicable for cases of larger D values.