ASSEMBLE TIME FOR SELF-ASSEMBLING SQUARE TILES - JAMES SOLBERG 1 Assemble time for self-assembling square tiles

As technology drives components smaller, traditional methods of assembling these components become ever more impractical. Self-assembly is a viable solution in which the individual components autonomously organize themselves without the external guidance of a supervising agent. In this paper we look at the assembly times for various one and two dimensional objects that are assembled with square tiles in the plane. Theoretical planar objects were explored to determine the fewest number of steps needed to construct them. In general, the assembly times for linear objects were found to be O(tiles), and 2-D objects were found to be O( √ tiles), where tiles is the number of tiles in the final assembly. But, when these same types of structures were assembled in a simulated environment, the 2-D objects only exhibited O(tiles). One factor that could cause these slower assembly times is the diffusion of the ‘free’ tiles to the stationary assembly. The manner in which the tiles move in the simulation world affects the order of assembly time as a function of assembly size. Three tile movement protocols are investigated. When we use a movement protocol that is not affected by diffusion, we are able to match the actual (simulated) assembly time for a 2× 2 with its calculated expected time.