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Do hierarchical structures assemble best via hierarchical pathways?

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.