SignificanceFormation of complex Frank–Kasper phases in soft matter systems confounds intuitive notions that equilibrium states achieve maximal symmetry, owing to an unavoidable conflict between shape and volume asymmetry in space-filling packings of spherical domains. Here we show the structure and thermodynamics of these complex phases can be understood from the generalization of two classic problems in discrete geometry: the Kelvin and Quantizer problems. We find that self-organized asymmetry of Frank–Kasper phases in diblock copolymers emerges from the optimal relaxation of cellular domains to unequal volumes to simultaneously minimize area and maximize compactness of cells, highlighting an important connection between crystal structures in condensed matter and optimal lattices in discrete geometry.

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Block copolymer (BCP) melts are a paradigm for pluripotent molecular assembly, yielding a complex and expanding array of variable domain shapes and symmetries from a fairly simple and highly expandable class of molecular designs. This Perspective addresses recent advances in the ability to model and measure features of domain morphology that go beyond the now canonical metrics of D spacing, space group, and domain topology. Such subdomain features have long been the focus of theories seeking to explain and understand mechanisms of equilibrium structure formation in block copolymer melts, from inhomogeneous curvatures of an intermaterial dividing surface to variable domain thickness. Quantitative metrics of variable subdomain geometry, or packing frustration, are central to theoretical models of complex BCP phase formation, from bicontinuous networks to complex (e.g., Frank–Kasper) crystals, and new experimental methods bring the possibility of their quantitative tests into reach. Here we not only review generic approaches to quantify local domain morphologies that both connect directly to thermodynamic models of BCP assembly but also generalize to domains of arbitrary shape and topology. We then overview experimental methods for characterizing BCP morphology, focusing on recent advances that make accessible detailed and quantitative metrics of fine features of subdomain geometry. Beyond even the critical comparison between detailed predictive models and experimental measurements of complex BCP assembly, validation of these advances lays the foundation to “mold” morphology in BCP assemblies at ever finer subdomain scale, through controlled architectures and processing pathways.

Triply-periodic networks are among the most complex and functionally valuable self-assembled morphologies, yet they form in nearly every class of biological and synthetic soft matter building blocks. In contrast to simpler assembly motifs – spheres, cylinders, layers – networks require molecules to occupy variable local environments, confounding attempts to understand their formation. Here, we examine the double-gyroid network phase by using a geometric formulation of the strong stretching theory of block copolymer melts, a prototypical soft self-assembly system. The theory establishes the direct link between molecular packing, assembly thermodynamics and the medial map, a generic measure of the geometric center of complex shapes. We show that “medial packing” is essential for stability of double-gyroid in strongly-segregated melts, reconciling a long-standing contradiction between infinite- and finite-segregation theories. Additionally, we find a previously unrecognized non-monotonic dependence of network stability on the relative entropic elastic stiffness of matrix-forming to tubular-network forming blocks. The composition window of stable double-gyroid widens for both large and small elastic asymmetry, contradicting intuitive notions that packing frustration is localized to the tubular domains. This study demonstrates the utility of optimized medial tessellations for understanding soft-molecular assembly and packing frustration via an approach that is readily generalizable far beyond gyroids in neat block copolymers.

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