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

Abstract

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.

Stability of the A15 phase in diblock copolymer melts

“…Block polymer self-assembly is governed by a delicate balance of two competing energetic effects-interfacial energy and loss of conformational entropy due to chain stretching-that play a crucial role in the selection of various sphere phases (42). Why does A15 form in linear diblock copolymers at large volume fractions and low temperatures (high χN)?…”

Complex Crystals from Size-Disperse Spheres

“…It has been proposed that dynamic dispersity assists the formation of Frank-Kasper (FK) phases [19] and other topologically close-packed complex crystals [20][21][22]. Indeed, the FK phases A15, σ, and Laves C14 and C15 are found with micelles [23][24][25][26][27][28] and soft nanoparticles [29] where shape dispersity and size dispersity are dynamic because micelles and nanocrystal ligand shells can deform and exchange molecules.…”

An Exceptionally Stable and Scalable Sugar–Polyolefin Frank–Kasper A15 Phase

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One-component" soft material Frank-Kasper (FK) phases are an intriguing structural form of matter that possess periodically ordered structures arising from the self-reconfiguration and close packingo fa ni nitial assembly of identical "deformable" spheres into two or more size-or shape-distinct sets of particles.S ignificant challenges that must still be addressed to advance the field of soft matter FK phases further,h owever,i nclude their rare and unpredictable occurrence,u ncertain mechanisms of solid-state assembly,a nd low thermodynamic stability.H ere we show that ar eadily-accessible sugar-polyolefin conjugate quantitatively produces an exceptionally stable solid-state FK A15 phase through arapid and irreversible thermotropic order-order transition, which contrary to other prevailing proposed mechanisms,d oes not require mass transfer between particles or large structural reorganization in the bulk to establish unit cell non-equivalency.Our results provide the basis for arealistic strategy for obtaining practical and scalable quantities of adiverse range of sugar-polyolefin FK A15 phases with unique intrinsic physical properties and chemical reactivities not previously seen in such systems.Periodically-ordered, "one-component" soft matter Frank-Kasper (FK) phases,w hich arise from the topological close packing (TCP) of "deformable" sphere-shaped particles,have now been experimentally verified for amphiphilic liquid crystals and dendrons,b lock copolymers," giant" molecules and surfactants,a nd colloidal nanoparticles. [1][2][3][4][5][6][7][8][9] Frank and Kasper [10] originally developed the concept of TCP to rationalize how the complex crystal structures of certain intermetallic alloys arise through the packing of asymmetric polyhedra that are associated with different sets of atoms that define ap articular coordination number, CNx, where x = 12,14,15, or 16. Forexample,asshown in Figure 1, the unit cell of the cubic FK A15 (Pm3 n)p hase,w hich is found for some bimetallic alloys with A 3 Bstoichiometry (e.g.

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“…Periodically-ordered, "one-component" soft matter Frank-Kasper (FK) phases,w hich arise from the topological close packing (TCP) of "deformable" sphere-shaped particles,have now been experimentally verified for amphiphilic liquid crystals and dendrons,b lock copolymers," giant" molecules and surfactants,a nd colloidal nanoparticles. [1][2][3][4][5][6][7][8][9] Frank and Kasper [10] originally developed the concept of TCP to rationalize how the complex crystal structures of certain intermetallic alloys arise through the packing of asymmetric polyhedra that are associated with different sets of atoms that define ap articular coordination number, CNx, where x = 12,14,15, or 16. Forexample,asshown in Figure 1, the unit cell of the cubic FK A15 (Pm3 n)p hase,w hich is found for some bimetallic alloys with A 3 Bstoichiometry (e.g.…”

“…[11,12] Given the successful material science of "hard" sphere FK phases,w hich includes the discovery of superconducting properties for A15 bimetallics, [13] one of the primary interests in the occurrence of similar one-component soft matter FK phases is the desire to understand the underlying principles and mechanisms by which an initial set of identical spherical particles can be driven to reconfigure and self-sort into two or more sets with different size or shape, and importantly,w ith different free energies. [14,15] Unfortunately,b ased on the structures and properties of the extremely small number of soft matter FK phases reported to date,s ignificant barriers to answering this question, and others,c urrently exist, and these include,u ndefined or limiting mechanisms and processes for thermotropic selfassembly,l ow thermodynamic or kinetic stability,a nd lowyielding syntheses of structurally-complex precursor materials.F or the discovery of new physical properties and phenomena, and ultimately,n ew technological innovations that are beneficial to society,i ti sa lso attractive,o rp erhaps even critical, if the scalable production of practical quantities of new soft matter FK phases can be achieved within the range of afew grams to up to several kilotons using short and reproducible synthetic schemes that are based on inexpensive, readily-available,a nd sustainable feedstocks.H erein, we report the serendipitous discovery that the one-component sugar-polyolefin conjugate derivative represented by triazole-linked, cellobiose-atactic poly(4-methyl-1-pentene)(CB-aPMP) 1 of Figure 2i sc apable of forming as oft matter FK A15 phase through as elf-assembly process that proceeds through fast and irreversible thermotropic order-order tran- sitions involving an initial hexagonal cylindrical (C)phase and at ransient A15 intermediate.I mportantly,u pon cooling to room temperature,t his final A15 phase of 1 displays exceptional stability in the solid state by remaining unchanged after storage under ambient conditions for, at least, three months, and after repetitive thermal cycling between 25 8 8Cand 180 8 8C. Ap roposed new,a nd potentially general, mechanism that accounts for these observations is based on as ymmetryallowed epitaxial transition pathway that uniquely avoids the need to invoke either dynamic mass transfer of molecular components between spheres or large structural reorganizations in the bulk to occur in order to establish unit cell nonequivalency of particles.S ince aw ide variety of sugarpolyolefin conjugates,i ncluding 1,c an indeed be obtained through simple synthetic routes in practical quantities from abundant and sustainable precursors,o ur results serve to establish ap otential path to viable scalable production and technological innovations of stable sugar-polyolefin FK A15 phases that further have the benefit of possessing intrinsic chirality and chemical reactivity that are associated with the sugar domain.…”