A Theoretical Study Of Columnar Crystals

Packing problems are a class of optimization problems concerning a search for the optimal?e.g.densest possible?arrangements of objects in space.In particular,problems concerning optimal packings in spatial confinement have attracted much interest and attention,thanks to their potential applications in almost all walks of life.This research focuses on the densest possible arrangements of equal-sized hard spheres in cylindrical space,where the periodic structures of such confined systems,known as columnar crystals,have also been observed across a variety of scenarios,from condensed matter physics all the way to structural biology.The densest possible structures of such systems are independent of absolute length scales and depend solely on the diameter ratio D = cylinder diameter / sphere diameter.In particular,the densest possible helical structures of such quasi-one-dimensional systems have been observed experimentally for a variety of physical systems,such as colloidal crystal wires,nanotube-confined fullerenes and self-assembled chiral photonic crystals.Most results reported in the literature for the densest possible structures of equal-sized hard spheres in cylindrical confinement are either numerical or experimental in nature,and there has been a lack of rigorous analytical solutions to this problem.This research focuses on the development of a general analytical theory that explains the origin of densest possible columnar structures,in particular those that exhibit helicity,in narrow cylindrical confinement.A two-dimensional version of this problem,involving the packing of equal-sized circular disks into a narrow rectangular channel,was first studied experimentally.The densest possible structures were studied as a function of the ratio S between the channel width and the spheres' diameter.Based on experimental results of packing fractions,it was conjectured that the densest possible structures consist of staggered triplets?but not any other multiples?of mutually touching spheres.Back to the original problem,the single-and double-helix structures were both studied in terms of their repeating triplet configurations of mutually touching spheres.It was found that a local optimization of such triplet configurations would result in the densest possible single-and double-helix structures at 1+3^1/2/2?D?1+4(3^1/2)/7 and 1+4(3^1/2)/7?D?2 respectively,where D is the cylinder-to-sphere diameter ratio.In addition,the emergence of these densest possible helical structures has been interpreted as results of perturbations from neighboring achiral structures.This ideaproves to be crucial in our understanding of how the densest possible structures of columnar crystals emerge and vary at given values of D.For 2?D?1+3(3^1/2)/5,it was found that the densest possible columnar structures are outcomes of a structural optimization of repeating quadruplet configurations of mutually touching spheres.It is expected that this ‘microscopic' approach might be extended to larger D values that involve larger multiples of mutually touching spheres.