Soft Self-assembly and Densest Packings in Colloidal Models

Inspired by the beauty of various materials with distinct structures and functions in nature, researchers have been dedicating themselves to discover new ways of creating functional artificial materials to fulfill the increasingly various needs in life. Self-assembly, categorized into the ‘bottom-up’ method, is an important approach in building man-made crystals. Elegant and useful structures have been obtain by colloidal self-assembly, including triangular, kagome and square lattices in two- dimensional systems and hexagonal layered structures and color-tunable structures in three-dimensional systems. However, the self-assembly process itself has not yet been fully understood, both thermodynamically and dynamically. Three major challenges in this field are: a) given a set of conditions and parameters of the system, what is the equilibrium assembly structure? b) given a predetermined structure, how should we design particles to self-assemble it? c) how to avoid possible kinetic barriers to assembly complex structures? This dissertation will focus on answering some of the above questions using statistical mechanics approaches and then provide some guidance to colloidal experiments. More specifically, we first study a quasi- two-dimensional, binary colloidal alloy that exhibits liquid—solid and solid—solid phase transitions, focusing on the kinetics of a diffusionless transformation between two crystal phases. Experiments are conducted on a monolayer of magnetic and nonmagnetic spheres suspended in a thin layer of ferrofluid and exposed to a tun- able magnetic field. A theoretical model of hard spheres with point dipoles at their centers is used to guide the choice of experimental parameters and characterize the underlying materials physics. When the applied field is normal to the fluid layer, a checkerboard crystal forms; when the angle between the field and the normal is sufficiently large, a striped crystal assembles. As the field is slowly tilted away from the normal, we find that the transformation pathway between the two phases de- pends strongly on crystal orientation, field strength, and degree of confinement of the monolayer. In some cases, the pathway occurs by smooth magnetostrictive shear, while in others it involves the sudden formation of martensitic plates. Secondly, we examine the densest packing structures and their assembly dynamics for hard spheres of diameter σ within cylinders of diameter D. We extend the identification of close packings up to D = 4.00σ by adapting Torquato—Jiao’s adaptive-shrinking-cell formulation and sequential-linear-programming (SLP) technique. We identify 17 new structures, almost all of them chiral. Beyond D ≈ 2.85σ, most of the structures consist of an outer shell and an inner core that compete for being close packed. In some cases, the shell adopts its own maximum density configuration, and the stack- ing of core spheres within it is quasiperiodic. In other cases, an interplay between the two components is observed, which may result in simple periodic structures. In yet other cases, the very distinction between the core and shell vanishes, resulting in more exotic packing geometries, including some that are three-dimensional extensions of structures obtained from packing hard disks in a circle. Although in such a system phase transitions formally do not exist, marked structural crossovers can nonetheless be observed. Over the range σ ≤ D ≤ 2.82σ, we find in simulations that structural crossovers echo the structural changes to the densest packing sequence. We also observe that the out-of-equilibrium self-assembly depends on the compression rate. Slow compression approximates equilibrium results, while fast compression can skip intermediate structures. Crossovers for which no continuous line-slip exists are found to be dynamically unfavorable, which is the source of this difference. Results from colloidal sedimentation experiments at low diffusion rate are found to be consistent with the results of fast compressions, as long as appropriate boundary conditions are used. The similitude between compression and sedimentation results suggests that the assembly pathway does not here sensitively depend on the nature of the out-of-equilibrium dynamics. We also examine the behavior of different correlation lengths in such quasi-one-dimensional systems via transfer matrix method. Non-monotonicity of the correlation lengths is observed, as has been identified in the assembly simulations. For the quantities that have a Delta distribution function at in- finite pressure, the corresponding correlation lengths vanish as the pressure increases, due to the suppression of fluctuations. For quantities that grow non-monotonically, the decrease of correlation lengths always corresponds to the structural crossovers in the system. As another approach to obtain quasi-one-dimensional assemblies, we examine the focusing of nanoparticles in acoustic standing waves. To perform this study, we build an acoustic focusing chamber containing opposing piezoelectric transducers to rapidly focus particles of different size into highly parallel patterns and visualized this process in real time using dark field microscopy. We select gold as a model material because its high density and low compressibility, making it an ideal candidate for investigating the limits of particle acoustophoresis. To extend our results, we use our theoretical model to estimate, for the first time, the minimum pressure amplitude necessary to concentrate nanoparticles of any composition. Finally, to overcome the limitations of focusing of acoustic standing waves, we develop a simple UV light-based method to controllably induce the aggregation of particles for their rapid concentration below the theoretical size limit at a given pressure amplitude. Lastly, we study the Gardner transition in polydisperse crystals and identify the aging effect by measuring the mean squared displacement as a function of time.