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

TL;DR

This paper demonstrates how Frank-Kasper phases in self-assembled soft matter spheres emerge from principles of strong-stretching theory, highlighting the role of shape asymmetry and volume exchange in their thermodynamic stability.

Contribution

It provides a quantitative analysis linking FK phase formation to strong-stretching theory and molecular stiffness, offering a molecular interpretation via self-consistent field theory.

Findings

FK phases are stabilized by shape asymmetry and volume exchange.

The diblock foam model accurately predicts equilibrium FK lattices.

Molecular stiffness influences local packing asymmetry.

Abstract

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 quantitation 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 minimal inter-block area competes with 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 {\sigma} 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 regulates the coupling between intra-domain chain configurations and the asymmetry of local packing. These findings shed new 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.