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

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Contributed by Frank S. Bates, August 6, 2018 (sent for review June 5, 2018; reviewed by Scott Milner and Gerd Schroeder-Turk)

## Significance

Formation 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.

## 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 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 σ 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.

Spherical assemblies occur in nearly every class of supramolecular soft matter, from lyotropic liquid crystals and surfactants to amphiphillic copolymers (1). In concentrated or neat systems, self-assembled spherical domains behave as giant “mesoatoms,” adopting periodically ordered crystalline arrangements. While superficially similar to lattices formed in atomic or colloidal systems—which are stabilized largely by bonding or translational entropy—the periodic order in soft materials is governed by distinctly different principles because lattice formation occurs in thermodynamic equilibrium with the formation of the mesoatoms from the constituent molecules themselves. Thus, the equilibrium sizes and shapes of mesoatoms are inextricably coupled to the lattice symmetry and vice versa.

In this article, we address the emergence of noncanonical, Frank–Kasper (FK) lattices in soft materials, characterized by complex and large unit cells yet formed by assembly of a single molecular component. Initially constructed as models of metallic alloys (2, 3), FK lattices are a family of periodic packings (4, 5) whose sites are tetrahedrally close packed (i.e., sitting on the vertices of nearly equilateral tetrahedra, the densest local arrangement of equal radius spheres) and can be decomposed into polyhedral (e.g., Voronoi or Wigner Seitz) cells surrounding each site containing 12, 14, 15, or 16 faces. Known as the FK polyhedra, these cells (Z12, Z14, Z15, and Z16) possess variable volume and envelope spheres of distinct radii. Hence, FK lattices are natural candidates to describe ordered, locally dense packings of spherical elements of different radii such as atomic alloys (3, 5) or binary nanoparticle superlattices (6). Once considered anomalous in soft matter systems, the past decade has seen an explosion in the observation of FK lattices in a diverse range of sphere-forming assemblies. These include (A15, σ) liquid–crystalline dendrimers (7, 8), (A15, σ) linear tetrablock (9, 10), (σ, C14, C15) diblock (11⇓–13) and (A15) linear–dendron (14) block copolymer melts, (A15) amphiphilic nanotetrahedra (15, 16), (A15, σ, C14, C15) concentrated ionic surfactants (17, 18), and (C14) monodisperse, functionalized nanoparticles (19). The central puzzle surrounding the formation of FK lattices in these diverse systems is understanding why single components assemble into phases composed of highly heterogeneous molecular environments.

A common element distinct to FK formation in soft systems is the thermodynamic cost of asphericity imposed by incompatibility between uniform density and packing of perfectly spherical objects (Fig. 1). In soft assemblies, the ideally spherically symmetric domains are warped into lower symmetry, polyhedral shapes that fill space without gaps. Intuitively, one expects that the minimal free-energy state is the one for which the quasi-spherical domains (qSDs) remain “most spherical.” The most commonly invoked notion of sphericity in this context is the dimensionless cell area A to volume V ratio,

While the role of volume asymmetry has been implicated previously in the formation of FK lattices by soft qSD assemblies (29), critical questions remain unanswered. First, what are the relevant measures of sphericity optimized by the assembly thermodynamics? Second, how do these in concert determine the optimal balance between shape asymmetry (nonspherical domains) and volume asymmetry (molecular partitioning among domains) for a given qSD lattice? Finally, how does this balance select the equilibrium lattice and determine the scale of thermodynamic separation between the many competing FK lattices? We address these questions in the context of what we call the diblock foam model (DFM), which describes the thermodynamic competition between interdomain surface energy and chain stretching. For optimal mean qSD size, the DFM quantifies the thermodynamic cost of asphericity in terms of a geometric mean of reduced cell area and dimensionless radius of gyration of the cells and thus integrates elements of both the Kelvin and lattice Quantizer problems (30). These geometric proxies for interblock repulsion and intramolecular stretching in qSD exhibit qualitatively different dependencies on cell shape, a factor that we show, based on this model and SCFT analysis, to be critical to the volume partitioning among distinct qSD and optimal lattice selection.

Among the various classes of FK-forming soft matter (7⇓–9, 11⇓⇓⇓–15, 17, 19), we posit that diblock copolymers represent the optimal starting point for investigating the selection of low-symmetry FK phases by soft matter spheres. Diblock copolymers are a relatively simple chemical system, consisting of two flexible chains bonded together at their ends, and there exist robust theoretical methods for studying their phase behavior in the context of universal physical models (31, 32). The fundamental mechanisms underlying assembly of diblock copolymers that we elucidate here furnish the foundation for subsequent investigations of other soft matter systems, where these basic principles are conflated with additional phenomena emerging from electrostatics, hydrophobic interactions, and detailed packing of the complicated (non-Gaussian) configurations of their constituents.

## DFM of FK Lattice Selection

We adopt what we call the DFM, first developed by Milner and Olmsted (33, 34), in which the free energy of competing arrangements is reduced to purely geometric measures of the cellular volumes enclosing the qSD. To a first approximation, these cells are the polyhedral Voronoi cells for a given point packing, whose faces represent coronal brushes flattened by contact with neighboring qSD coronae. The model is based on strong-stretching theory (SST) of diblock copolymer melts, in which interblock repulsions drive separation into sharply divided core and coronal domains and the chains are well-extended. We also consider the case of large elastic asymmetry between core and coronal blocks, which itself derives from asymmetry of the block architecture or the segment sizes. This corresponds to the polyhedral-interface limit (35), in which the core/coronal interface in each qSD adopts a perfect, affinely shrunk copy of the cell shape (see Fig. 1, *Bottom Right*). Polyhedral warping of the interface is favored when the stiffness of the coronal blocks, which favors a more uniform extension from the interface to the outer cell wall, dominates over entropic stiffness of core blocks and interblock surface energy, both of which favor round interfaces.

In this limit, the free energy per chain (26, 34), *SI Appendix*, *1A*. Polyhedral Interface Limit of Strongly-Segregated Diblock Sphere Lattices for details). The first term represents the enthalpy of core–corona contact and hence is proportional to the (per volume) interfacial area, which itself is proportional to the cell area ^{†} While minimal area partitions (at constant volume) are associated with Kelvin’s foam problem, lattice partitions that optimize

The Milner and Olmsted model has been studied for flat-faced Voronoi cells of face-centered cubic (FCC), body-centered cubic (BCC), and A15 (27, 34), showing that the latter FK lattice of sphere-forming diblocks is favored over former two canonical packings in the polyhedral interface limit. Here, we analyze a vastly expanded class of 11 FK lattices, possessing up to 56 qSD per periodic repeat. Most critically, we use a Surface Evolver (37)-based approach that minimizes *SI Appendix*, *2* for detailed methods and tabulated results).

To assess the importance of relaxing volume and shape, consider the three distinct ensembles of qSD cells, shown for C15 in Fig. 2 *A* and *B*. We have computed results for equal-volume, relaxed-shape cells, which cannot exchange mass, and centroidal Voronoi cells, which have fixed flat-face shapes but unequal volumes (also fixed). The former ensemble neglects the possibility of mass exchange between micelles, while the second optimizes stretching (36) but is suboptimal in terms of cell area.^{‡}Neither model is realistic, but they provide useful points of comparison with the unconstrained, relaxed-volume and shape cells, which strictly minimize *C* shows that allowing both volume and shape to relax leads to a complete inversion of the trend of *E*, which all lie within 0.08% in

The interplay between area and stretching underlies the emergent asymmetry in equilibrium qSD volumes. Comparing the equal-volume to unconstrained DFM results in Fig. 2 *C* and *D* shows that volume relaxation has a far more significant effect on relaxation of *SI Appendix*, Fig. S1). Volume exchange for lattices with large proportions of lower area Z12 cells (e.g., C14 and C15) achieve relatively large (

Cell volume asymmetry in equilibrated DFM structures pushes well beyond that of the “natural” geometry of Voronoi cells, which is strictly optimal for stretching but not for its product with the square of dimensionless area. Fig. 2*F* shows that both unconstrained and Voronoi models of qSD cell geometry exhibit an increase volume dispersion with decreasing mean coordination (or with increasing fraction of Z12s). However, optimal unconstrained DFM cells are nearly twice as polydisperse in volume as the Voronoi distribution. This massive volume asymmetry among qSD (up to

Previous SCFT studies (12, 28) have shown that the canonical BCC sphere phase is overtaken by a stable σ lattice when the elastic asymmetry, embodied by ratios of statistical segment lengths, *SI Appendix*, Fig. S16 *A–C*, where we compare the free energies, scaled enthalpies, and entropies for σ, Z, C14, C15, and A15 predicted by the unconstrained cell DFM to AB diblock SCFT calculations using methods described in ref. 32 at somewhat strong segregation conditions *F*.

## Molecular Mechanism of Aspherical Domain Formation

To probe the molecular mechanism that underlies the selection of FK lattices in block copolymers, we analyze two order parameters that quantify the respective asymmetric shapes and volumes of qSD, computed from the volumes enclosing A-rich cores in SCFT composition profiles of diblocks at *SI Appendix*, *3B*. Geometrical *A*nalysis of *S*pherical *D*omains). The first parameter,

It has been argued previously (27) that the polyhedral warping, or faceting, of core–corona interfaces should increase with ϵ, which controls the ratio of corona- to core-block stiffness, due to the relatively lower entropic cost of more uniformly stretched coronae achieved by polyhedral interfaces. This expectation is consistent with the observed monotonic increase of α from 0 at *A*.^{§} As shown in *SI Appendix*, Fig. S17, the polyhedral warp of the interface grows also with increasing f, due to the increased proximity of the qSD cell boundary to the interface and relatively shorter coronal blocks at larger core fractions. While clearly far from a sharply faceted shape, the increase in core shape anisotropy is obvious from 2D cuts through the qSD shown in Fig. 3*B*, showing a visible warp of A/B interface toward the truncated-octahedral shape of the BCC cell at

For the FK phases, which are composed of distinct-symmetry qSD, areal distortion exhibits a markedly different dependence on increased coronal/core stiffness, as illustrated by the plots of *C*. Z12 domains exhibit a monotonic, albeit modest, increase in distortion with ϵ. Surprisingly, for the Z14 domains, the excess area drops from its maximal value of

The origin of this counterintuitive drop in dimensionless area of the Z14 cells with increased outer block stiffness is illustrated in Fig. 3*D*, which compares 2D sections of the Z14 qSD of A15 at *D* and *E*. For larger ϵ, the preference for more uniform coronal block stretching drives the quasi-polyhedral domain shape, with radial chain trajectories extending from the center of the domain, a point at which core block ends are concentrated. In contrast, for the case of *F*).

*SI Appendix*, Fig. S18 shows evidence of this same discoidal → polyhedral transition qSD within the most oblate cells of other FK phases, C15 and Z, leading to a corresponding drop in excess area *G*, we find this intradomain shape transition with increasing corona-/core-block stiffness is coupled to a transition in volume asymmetry among qSD. Discoidal domains of the conformationally symmetric diblocks (*F*.

Notwithstanding the broad agreement between SCFT and DFM predictions, the degree of polyhedral warping of qSD shape is both arguably modest (i.e.,

## Concluding Remarks

We anticipate that the emergence of optimal FK lattice structure and thermodynamics via a balance of competing measures of domain asymmetry highlighted here for high-molecular weight diblock copolymers will extend to other copolymer systems where these phases have been observed, including architecturally asymmetric copolymers, linear multiblocks, low-molecular weight/high-χ systems, and blends. In particular, lower molecular weight polymers drive the system closer toward the strong segregation limit and away from the mean-field limit. Each of these materials exhibits different molecular mechanisms through which the relative stiffness of the coronal domain transmits the asymmetry of the local qSD packing into the core shape. For example, the observation of polygonal/polyhedral warping of outer zones of core–shell domains of linear mulitblock polymers (39) provides a plausible mechanism to stabilize the σ phase observed in linear tetrablocks (9). On the other hand, accurately modeling the formation of σ by low-molecular weight conformationally asymmetric diblocks (11, 13) likely requires a non-Gaussian (finite extensibility) model of chain stretching but one that nevertheless, like the dimensionless radius of gyration

The present results for the DFM also shed light on the nonequilibrium pathways for stabilizing metastable FK competitors, as has been demonstrated for conformational asymmetric linear diblocks quenched from high-temperature disorder sphere phases to low-temperature metastable, C14, and C15 phases (12, 41). The low-temperature quench is suspected to freeze out the interdomain chain exchange needed to achieve the equilibrium σ state; thus, the kinetically trapped quenched state inherits the volume distribution of the high-temperature micelle liquid state. The DFM suggests a way to analyze the stability of FK states when domain volumes are out of equilibrium, suggesting the observation of C14 and C15 may be selected among the low-temperature kinetically trapped arrangements because it inherits a volume distribution that is both smaller in average cell size and possibly more polydisperse than the equilibrium state at the low temperature and hence a better fit to the “aggregation fingerprint” of low-

## 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.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: grason{at}mail.pse.umass.edu, dorfman{at}umn.edu, or bates001{at}umn.edu.

Author contributions: A.R., M.B.B., A.A., F.S.B., K.D.D., and G.M.G. designed research; A.R., M.B.B., A.A., and G.M.G. performed research; A.R., M.B.B., A.A., and G.M.G. analyzed data; and A.R., F.S.B., K.D.D., and G.M.G. wrote the paper.

Reviewers: S.M., The Pennsylvania State University; and G.S.-T., Murdoch University, Australia.

The authors declare no conflicts of interest.

↵*The tetrahedral coordination of FK lattices implies that their partitions closely approximate the geometric constraints of Plateau borders and are therefore near to minimal-area partitions. In addition to A15, at least two more partitions of FK lattices, σ and H, have also been shown previously (21, 23) to beat the area of optimum originally conjectured by Kelvin, the BCC partition. In

*SI Appendix*,*2B*. Minimal*A*rea*C*ells: Kelvin Problem, we report that the FK lattice P also belongs to this category.↵

^{†}Assembly thermodynamics depends on the dimensionless ratios of structure-averaged area and stretching of cells, as opposed to averages of dimensionless cell area and stretching.↵

^{‡}Centroidal Voronoi cells have generating points at the centers of volume of the cell and, hence, for a given X minimize the mean-square distance of all points to their corresponding central point (see*SI Appendix*,*2A*. Minimal*S*tretching*C*ells: Quantizer*P*roblem for additional details).↵

^{§}While this extends beyond what is realized with most flexible linear diblocks, bulky side chains including bottlebrush configurations and miktoarm polymers would make the upper limit accessible.This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1809655115/-/DCSupplemental.

- Copyright © 2018 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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