5,616 research outputs found
Encoding Color Sequences in Active Tile Self-Assembly
Constructing patterns is a well-studied problem in both theoretical and experimental self-assembly with much of the work focused on multi-staged assembly. In this paper, we study building 1D patterns in a model of active self assembly: Tile Automata. This is a generalization of the 2-handed assembly model that borrows the concept of state changes from Cellular Automata. In this work we further develop the model by partitioning states as colors and show lower and upper bounds for building patterned assemblies based on an input pattern. Our first two sections utilize recent results to build binary strings along with Turing machine constructions to get Kolmogorov optimal state complexity for building patterns in Tile Automata, and show nearly optimal bounds for one case. For affinity strengthening Tile Automata, where transitions can only increase affinity so there is no detachment, we focus on scaled patterns based on Space Bounded Kolmogorov Complexity. Finally, we examine the affinity strengthening freezing case providing an upper bound based on the minimum context-free grammar. This system utilizes only one dimensional assemblies and has tiles that do not change color
Binary pattern tile set synthesis is NP-hard
In the field of algorithmic self-assembly, a long-standing unproven
conjecture has been that of the NP-hardness of binary pattern tile set
synthesis (2-PATS). The -PATS problem is that of designing a tile assembly
system with the smallest number of tile types which will self-assemble an input
pattern of colors. Of both theoretical and practical significance, -PATS
has been studied in a series of papers which have shown -PATS to be NP-hard
for , , and then . In this paper, we close the
fundamental conjecture that 2-PATS is NP-hard, concluding this line of study.
While most of our proof relies on standard mathematical proof techniques, one
crucial lemma makes use of a computer-assisted proof, which is a relatively
novel but increasingly utilized paradigm for deriving proofs for complex
mathematical problems. This tool is especially powerful for attacking
combinatorial problems, as exemplified by the proof of the four color theorem
by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and
Thomas) or the recent important advance on the Erd\H{o}s discrepancy problem by
Konev and Lisitsa using computer programs. We utilize a massively parallel
algorithm and thus turn an otherwise intractable portion of our proof into a
program which requires approximately a year of computation time, bringing the
use of computer-assisted proofs to a new scale. We fully detail the algorithm
employed by our code, and make the code freely available online
Block Copolymer at Nano-Patterned Surfaces
We present numerical calculations of lamellar phases of block copolymers at
patterned surfaces. We model symmetric di-block copolymer films forming
lamellar phases and the effect of geometrical and chemical surface patterning
on the alignment and orientation of lamellar phases. The calculations are done
within self-consistent field theory (SCFT), where the semi-implicit relaxation
scheme is used to solve the diffusion equation. Two specific set-ups, motivated
by recent experiments, are investigated. In the first, the film is placed on
top of a surface imprinted with long chemical stripes. The stripes interact
more favorably with one of the two blocks and induce a perpendicular
orientation in a large range of system parameters. However, the system is found
to be sensitive to its initial conditions, and sometimes gets trapped into a
metastable mixed state composed of domains in parallel and perpendicular
orientations. In a second set-up, we study the film structure and orientation
when it is pressed against a hard grooved mold. The mold surface prefers one of
the two components and this set-up is found to be superior for inducing a
perfect perpendicular lamellar orientation for a wide range of system
parameters
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