Optimal Staged Self-Assembly of General Shapes

We analyze the number of tile types $t$, bins $b$, and stages necessary to assemble $n \times n$ squares and scaled shapes in the staged tile assembly model. For $n \times n$ squares, we prove $\mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{\log{n} - tb - t\log t}{b^2})$ are necessary for almost all $n$. For shapes $S$ with Kolmogorov complexity $K(S)$, we prove $\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{K(S) - tb - t\log t}{b^2})$ are necessary to assemble a scaled version of $S$, for almost all $S$. We obtain similarly tight bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201

Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large-scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example, we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in the length of the line. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time that is polylogarithmic in the size of the shape, plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus those monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the efficiency advantages of active self-assembly over passive self-assembly and motivates experimental effort to construct general-purpose active molecular self-assembly systems

Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability

We introduce a new model of algorithmic tile self-assembly called size-dependent assembly. In previous models, supertiles are stable when the total strength of the bonds between any two halves exceeds some constant temperature. In this model, this constant temperature requirement is replaced by an nondecreasing temperature function $\tau : \mathbb{N} \rightarrow \mathbb{N}$ that depends on the size of the smaller of the two halves. This generalization allows supertiles to become unstable and break apart, and captures the increased forces that large structures may place on the bonds holding them together. We demonstrate the power of this model in two ways. First, we give fixed tile sets that assemble constant-height rectangles and squares of arbitrary input size given an appropriate temperature function. Second, we prove that deciding whether a supertile is stable is coNP-complete. Both results contrast with known results for fixed temperature.Comment: In proceedings of ISAAC 201

Entropic Bonding in Nanoparticle and Colloidal Systems

Scientists and engineers will create the next generation of materials by precisely controlling their microstructure. One of the most promising and effective methods to control material microstructure is self-assembly, in which the properties of constituent “particles” guide their assembly into the desired structure. Self- assembly mechanisms rely on both inherent interactions between particles and emergent interactions resulting from the collective effects of all particles in the system. These emergent effects are of interest as they provide minimal mechanisms to control self-assembly, and thus can be used in conjunction with other assembly methods to create novel materials. Literature shows that complex phases can be obtained solely from hard, anisotropic particles, which are attracted via an emergent Directional Entropic Force. This thesis shows that this force gives rise to the entropic bond, a mesoscale analog to the chemical bond. In Chapter 3 I investigate the self- assembly of a system from a random tiling into an ordered crystal. Analysis of the emergent directional entropic forces reveal the importance of shape in the final self-assembled system as well as the ability for shape manipulation to control the final self-assembled structure. In Chapter 4, I investigate three-dimensional analogs of two-dimensional systems in Chapter 3, explaining the self-assembly behavior of these systems via understanding of the emergent directional entropic forces. In Chapter 5 I investigate the nature of the entropic bond, investigating two-dimensional systems of hexagonal nanoplatelets. The Entropic bond is quantified, and the ability to manipulate the bonds to produce similar self- assembly behavior to chemically-functionalized nanoparticles is demonstrated. Finally, Chapter 6 investigates the phase transitions of the general class of particle studied in Chapter 5, showing the ability for particle shape to change the type of phase transition present in a system of nanoparticles as well as stabilize phases otherwise not found. As a whole, this work details the nature of the entropic bond and its use in directing the self-assembly of systems of non- interacting anisotropic particles.PHDMaterials Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144096/1/harperic_1.pd