  Controlling the topology of structures self-assembled from a set of
heterogeneous building blocks is highly desirable for many applications, but is
poorly understood theoretically. Here we show that the thermodynamic theory of
self-assembly involves an inevitable divergence in chemical potential. The
divergence and its detailed structure are controlled by the spectrum of the
transfer matrix, which summarizes all of self-assembly design degrees of
freedom. By analyzing the transfer matrix, we map out the phase boundary
between the designable structures and the unstructured aggregates, driven by
the level of cross-talk.
