Theory of Programmable Hierarchic Self-Assembly

We present a theoretical analysis of the inverse problem in self-assembly. A particular scheme is proposed for building an arbitrary desired nanostructure out of self-assembled building blocks ("octopus" nanoparticles). The conditions for robust self-assembly of the target structure are identified. This includes the minimal number of "colors" needed to encode interparticle bonds, which are to be implemented as pairs of complementary DNA sequences. As a part of this analysis, it is demonstrated that a floppy network with thermal fluctuations, in a certain range of coordination numbers [Z], possesses entropic rigidity and can be described as a traditional elastic solid. The onset of the entropic rigidity, [Z] = d+1, determines the minimal number of bond types per particle needed to encode the desired structure. Thermodynamic considerations give additional conditions for the implementation of this scheme.