Exploiting floppy modes in geometrically-frustrated assemblies*

Geometrically-frustrated assembly, when the local preferred ordering of building blocks is not globally embeddable without distortion, has recently emerged as a paradigm for programming and controlling the morphologies and, in particular, the sizes of self-assembled structures.  The accumulation of stresses due to the local misfit of building blocks in principle results in equilibrium assemblies whose growth terminates at a finite size.  However, the ability to design misfitting particle-based assemblies that realize self-limiting sizes much larger than that of a single building block remains a challenge.  

We here explore a strategy for enhancing the self-limiting size of frustrated assemblies by introducing a floppy dilational mode into the misfitting particle design.  In particular, we focus on the case of negatively-curved ``trumpet" assemblies [1,2] and introduce a simple model of a cuboid building block that allows us to couple a square-twist dilational model with the prescribed negative Gaussian curvature of the trumpet.  Such building blocks are potentially realizable through the DNA origami framework.  We show through theory [3] and simulation that this dilational mode alters how the stresses within the frustrated trumpet accumulate as it grows in length.  We surprisingly find that this mode can potentially lead to an order-of-magnitude enhancement in size of the trumpets, with self-limiting size growing in proportion to the target curvature radius of the assembly.  We propose how alternative designs such as triangular building blocks on a Kagome lattice may lead to further improvements to size.  Finally, we explore how the changes to vibrational entropy due to the presence of the floppy dilational mode may also affect the size selection of such assemblies.

[1] B. Tyukodi, F. Mohajerani, D. M. Hall, G. M. Grason, M. F. Hagan, ACS Nano 16, 9077 (2022)

[2] D. M. Hall, M. J. Stevens, G. M. Grason, Soft Matter 19, 858 (2023)

[3] S. Roy, C. Santangelo, Soft Matter (2023)