Title: Design Rules for Self-assembled Block-copolymer Patterns Using Tiled-templates

Directed self-assembly of block copolymers has been used for fabricating various nanoscale patterns, ranging from periodic lines to simple bends. However, assemblies of dense bends, junctions, and line segments in a single pattern have not been achieved by using sparse templates, because no systematic template-design methods for achieving such complex patterns existed. To direct a complex pattern by using a sparse template, the template needs to encode the key information contained in the final pattern, without being a simple copy of the pattern. Here we develop a set of topographic template tiles consisting of square lattices of posts with a restricted range of geometric features. The block-copolymer patterns resulting from all tile arrangements