Stable Frank-Kasper phases of self-assembled, soft matter spheres

Abhiram Reddy, Michael B. Buckley, Akash Arora, Frank S. Bates, Kevin D. Dorfman, Gregory M. Grason

Research output: Contribution to journalArticlepeer-review

89 Scopus citations


Single molecular species can self-assemble into Frank-Kasper (FK) phases, finite approximants of dodecagonal quasicrystals, defying intuitive notions that thermodynamic ground states are maximally symmetric. FK phases are speculated to emerge as the minimal-distortional packings of space-filling spherical domains, but a precise measure of this distortion and how it affects assembly thermodynamics remains ambiguous. We use two complementary approaches to demonstrate that the principles driving FK lattice formation in diblock copolymers emerge directly from the strong-stretching theory of spherical domains, in which a minimal interblock area competes with a minimal stretching of space-filling chains. The relative stability of FK lattices is studied first using a diblock foam model with unconstrained particle volumes and shapes, which correctly predicts not only the equilibrium lattice but also the unequal volumes of the equilibrium domains. We then provide a molecular interpretation for these results via self-consistent field theory, illuminating how molecular stiffness increases the sensitivity of the intradomain chain configurations and the asymmetry of local domain packing. These findings shed light on the role of volume exchange on the formation of distinct FK phases in copolymers and suggest a paradigm for formation of FK phases in soft matter systems in which unequal domain volumes are selected by the thermodynamic competition between distinct measures of shape asymmetry.

Original languageEnglish (US)
Pages (from-to)10233-10238
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number41
StatePublished - Oct 9 2018

Bibliographical note

Funding Information:
ACKNOWLEDGMENTS. We are grateful to R. Gabbrielli and J.-F. Sadoc, R. Mosseri for valuable input on geometric models of cellular packings, as well as to A.-C. Shi, M. Mahanthappa, and A. Travesset for additional discussion. This research was supported by Air Force Office of Scientific Research under Asian Office of Aerospace Research and Development Award 151OA107 and National Science Foundation Grants DMR-1719692 and DMR-1359191 (Research Experiences for Undergraduates Site: B-SMaRT). G.M.G. also acknowledges the hospitality of the Aspen Center for Physics, supported by NSF Grant PHY-1607611, where some of this work was completed. SCFT calculations were performed using computational facilities at the Massachusetts Green High Performance Computing Center and the Minnesota Supercomputing Institute.

Publisher Copyright:
© 2018 National Academy of Sciences. All rights reserved.


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