  Hierarchically structured natural materials possess functionalities
unattainable to the same components organized or mixed in simpler ways. For
instance, the bones and teeth of mammals are far stronger and more durable than
the mineral phases from which they are derived because their constituents are
organized hierarchically from the molecular scale to the macroscale. Making
similarly functional synthetic hierarchical materials will require an
understanding of how to promote the self-assembly of structure on multiple
lengthscales, without falling foul of numerous possible kinetic traps. Here we
use computer simulation to study the self-assembly of a simple hierarchical
structure, a square crystal lattice whose repeat unit is a tetramer. Although
the target material is organized hierarchically, it self-assembles most
reliably when its dynamic assembly pathway consists of the sequential addition
of monomers to a single structure. Hierarchical dynamic pathways via dimer and
tetramer intermediates are also viable modes of assembly, but result in general
in lower yield: these intermediates have a stronger tendency than monomers to
associate in ways not compatible with the target structure. In addition,
assembly via tetramers results in a kinetic trap whereby material is
sequestered in trimers that cannot combine to form the target crystal. We use
analytic theory to relate dynamical pathways to the presence of equilibrium
phases close in free energy to the target structure, and to identify the
thermodynamic principles underpinning optimum self-assembly in this model: 1)
make the free energy gap between the target phase and the most stable fluid
phase of order kT, and 2) ensure that no other dense phases (liquids or
close-packed solids of monomers or oligomers) or fluids of incomplete building
blocks fall within this gap.
