Frank–Kasper Phases in Block Polymers

Dorfman, Kevin D.

doi:10.1021/acs.macromol.1c01650

Cited by 78 publications

(76 citation statements)

References 168 publications

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“…Beyond TPN phases, it can be expected that medial packing motifs play a key role in other complex BCP morphologies, notably the Frank-Kasper (FK) phases of sphere-like domains that are stabilized at high [45]. It is well appreciated that phases "self-alloy" into multiple symmetry-and volume-inequivalent domains.…”

Section: Discussionmentioning

confidence: 99%

“…We anticipate that properly accounting for terminal spreading via an extension of mSST may be critical for understanding open questions in FK formation. For example, what controls the sequence of stable sphere phases (BCC→ σ → A15) found when increasing elastic asymmetry and core block fraction [43,45]?…”

Section: Discussionmentioning

confidence: 99%

“…( 1), κ B /κ A ∝ , so that when > 1, the B matrix domains are stiffer compared to the A tubular domains [16,41]. Notably, elastic asymmetry in sphere-forming BCP has been understood to unlock a host of "self-alloying" Frank-Kasper crystal structures through the amplification of packing frustration in the coronal blocks [16,[43][44][45].…”

Section: Elastic Asymmetry and Double-gyroid Stabilitymentioning

confidence: 99%

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Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

Reddy,

Dimitriyev,

Grason

2021

Preprint

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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 -TPN assemblies require molecules to occupy variable local domain shapes, confounding attempts to understand their formation. Here, we examine the double-gyroid (DG) network phase of block copolymer (BCP) melts, a prototypical soft self-assembly system, by using a geometric formulation of the strong stretching theory (SST) of BCP melts. The theory establishes the direct link between molecular BCP packing, thermodynamics of melt assembly and the medial map, a generic geometric measure of the center of complex shapes. We show that "medial packing" is essential for thermodynamic stability of DG in strongly-segregated melts, reconciling a long-standing contradiction between infinite-and finitesegregation theories, corroborating our SST predictions at finite-segregation via self-consistent field calculations. Additionally, we find a previously unrecognized non-monotonic dependence of DG stability on the elastic asymmetry, the comparative entropic stiffness of matrix-forming to tubularnetwork forming blocks. The composition window of stable DG -intermediate to competitor lamellar and columnar phases -widens both for large and small elastic asymmetry, seemingly overturning the heuristic view that packing frustration is localized to the tubular domains. This study demonstrates utility of geometric optimization of medial tesselations for understanding in soft-molecular assembly. As such, the particular medial-based approach deployed here is readily generalizable to study packing frustration far beyond the case of DG morphologies in neat BCP assemblies.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Elastic Asymmetry and Double-gyroid Stabilitymentioning

confidence: 99%

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Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

Reddy,

Dimitriyev,

Grason

2021

Preprint

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“…The cubic Pm

\overset{3}{true}

n phase belongs to the Frank–Kasper (FK) phases, [12] i.e. a class of tetrahedrally close packed (TCP) structures which were firstly introduced for intermetallic compounds and more recently observed in self‐assembled soft‐matter structures [5a, 7d] …”

Section: Figurementioning

confidence: 99%

Tetrahedral Liquid‐Crystalline Networks: An A15‐Like Frank–Kasper Phase Based on Rod‐Packing

Chen

Poppe

et al. 2022

Angewandte Chemie

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The Pmtrue3‾ n cubic and other low‐symmetry Frank–Kasper phases are known to be formed by soft spheres, ranging from metals to block copolymer micelles and colloidal nanoparticles. Here, we report a series of X‐shaped polyphiles composed of sticky rods and two non‐symmetric branched side‐chains, which self‐assemble into the first example of a cubic liquid‐crystalline phase representing a tetrahedral network of rods with a Pmtrue3‾ n lattice. It is the topological dual to the Weaire–Phelan foam, being the Voronoi tessellation of the A15 sphere packing, from which this network is obtained by Delaunay triangulation.

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“…In the space-filling arrangement the micelles form dodecagonal quasicrystals, and the authors attributed the disorder to frustration in the macromolecular packing. This was followed by work examining the roles of symmetry breaking and exchange of mass, [14], issues of stability [9], effective descriptions from Voronoi type models [17], and identification of periodic Laves phases with large unit cells, [6]. The study of complex packing phases in diblock copolymers is complicated by the possibility of micelles exchanging mass via transport of the diblock polymer chains.…”

Section: Introductionmentioning

confidence: 99%

Defects and Frustration in the Packing of Soft Balls

Jao¹,

Promislow²,

Sottile³

2022

Preprint

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Motivated by the recent studies of quasi-periodic equilibria of dense packing of diblock and star polymers, this work introduces the Hookean-Voronoi energy, a minimal model for the packing of soft, deformable balls. Considering the planar case, we investigate the equilibrium packings of N deformable circles in a periodic square, deriving a reduced formulation of the system and establish that it has a large, O(N 2 ), family of ordered "single-string" minimizers. Through numerical investigation of gradient flows from random initial data, we show that for modest values of N the system frequently equilibrates to quasi-ordered states with low energy and large basins of attraction. For larger N the energy of equilibria approach a limiting distribution, shaped by two mechanisms: a proliferation of moderate-energy disordered equilibria that block access of the gradient flow to lower energy quasi-ordered states and a rigid threshold on the maximum energy of stable states that precludes high-energy equilibria. The system frustration, measuring gap between the average equilibrium energy of the gradient flow and the system ground state, increases with N but saturates.

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Frank–Kasper Phases in Block Polymers

Cited by 78 publications

References 168 publications

Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

Tetrahedral Liquid‐Crystalline Networks: An A15‐Like Frank–Kasper Phase Based on Rod‐Packing

Defects and Frustration in the Packing of Soft Balls

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