The packing of soft materials: Molecular asymmetry, geometric frustration and optimal lattices in block copolymer melts

Grason, Gregory M.

doi:10.1016/j.physrep.2006.08.001

Cited by 127 publications

(187 citation statements)

References 115 publications

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“…[This line of reasoning also explains why BCC packing is favored over FCC, (IQ) FCC = 0.7405, in the SSL]. We hasten to add that this tradeoff involves yet another competition associated with optimally configuring the blocks within the particle core and corona (33), which presumably favors a single-core diameter.…”

Section: Discussionmentioning

confidence: 93%

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Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Lee

Leighton

Bates

2014

Proc. Natl. Acad. Sci. U.S.A.

230

362

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Frank-Kasper phases are tetrahedrally packed structures occurring in numerous materials, from elements to intermetallics to selfassembled soft materials. They exhibit complex manifolds of Wigner-Seitz cells with many-faceted polyhedra, forming an important bridge between the simple close-packed periodic and quasiperiodic crystals. The recent discovery of the Frank-Kasper σ-phase in diblock and tetrablock polymers stimulated the experiments reported here on a poly(isoprene-b-lactide) diblock copolymer melt. Analysis of small-angle X-ray scattering and mechanical spectroscopy exposes an undiscovered competition between the tendency to form self-assembled particles with spherical symmetry, and the necessity to fill space at uniform density within the framework imposed by the lattice. We thus deduce surprising analogies between the symmetry breaking at the body-centered cubic phase to σ-phase transition in diblock copolymers, mediated by exchange of mass, and the symmetry breaking in certain metals and alloys (such as the elements Mn and U), mediated by exchange of charge. Similar connections are made between the role of sphericity in real space for polymer systems, and the role of sphericity in reciprocal space for metallic systems such as intermetallic compounds and alloys. These findings establish new links between disparate materials classes, provide opportunities to improve the understanding of complex crystallization by building on synergies between hard and soft matter, and, perhaps most significantly, challenge the view that the symmetry breaking required to form reduced symmetry structures (possibly even quasiperiodic crystals) requires particles with multiple predetermined shapes and/or sizes.symmetry breaking | sphericity | Frank-Kasper phases | block polymers T he discovery of materials with aperiodic order, often referred to as "quasicrystals," 30 years ago (1, 2) heralded new and promising vistas for designing materials endowed with unique properties. In the 1950s Frank and Kasper (3, 4) recognized complex tetrahedral atomic-and molecular-packing geometries that bridge the familiar close-packed crystals [e.g., face-centered cubic (FCC), hexagonally close-packed (HCP), and body-centered cubic (BCC) structures] characterized by periodic order, and quasiperiodic crystals (QCs) that extend crystallography beyond the 230 space groups relevant to periodic crystals (5, 6). The scientific literature is replete with examples of Frank-Kasper phases in hard materials, particularly in the area of intermetallics (7-9), but also in a few complex elemental crystals, including manganese (10, 11) and uranium (12). Recently, this class of crystalline order has cropped up in a host of soft materials, including dendrimers (13), surfactant solutions (14), and block polymers (15, 16), often in close proximity to QC phases (17)(18)(19). To the best of our knowledge the principles underlying the formation of Frank-Kasper phases across both categories of materials have not been established, presenting enticing challenges to sc...

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Section: Discussionmentioning

confidence: 93%

“…This constraint mandates that the morphology created by block segregation must partition space at constant density. As a consequence the mesoscopic lattice symmetry imposes unavoidable restrictions on the size and shape of the particles that make up the ordered and disordered morphologies (32)(33)(34). We illustrate this feature in Fig.…”

Section: Discussionmentioning

confidence: 99%

Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Lee

Leighton

Bates

2014

Proc. Natl. Acad. Sci. U.S.A.

230

362

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“…The sphericity of a polyhedron may be quantified by the isoperimetric quotient IQ = 36πV 2 /S 3 , where V and S are its respective volume and surface area (10) As the water concentration (w 0 ) decreases in the ionic surfactant LLCs, the increased ion concentration leads to greater chemical incompatibility between the ion-rich aqueous domains and the hydrophobic regions, as well as enhanced charge screening between the negatively charged surfactant headgroups. Thus, the surfactant tails stretch to drive formation of larger and more deformable aggregates with lower interfacial curvatures, which pack into low-symmetry FK σ and A15 phases that maximize the ionic sphericities of the counterion atmospheres around the micelles.…”

Section: Resultsmentioning

confidence: 99%

“…Sphere-forming soft materials tend to prefer different packing symmetries from those of metallic solids (10). Although squishy spheres do form BCC and FCC crystals, they also form tetrahedrally closest-packed Frank-Kasper (FK) phases that contain combinations of 12-, 14-, 15-, and 16-coordinate lattice sites (11)(12)(13)(14).…”

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confidence: 99%

Low-symmetry sphere packings of simple surfactant micelles induced by ionic sphericity

Kim

Jeong

Yethiraj

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

118

137

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Supramolecular self-assembly enables access to designer soft materials that typically exhibit high-symmetry packing arrangements, which optimize the interactions between their mesoscopic constituents over multiple length scales. We report the discovery of an ionic small molecule surfactant that undergoes water-induced selfassembly into spherical micelles, which pack into a previously unknown, low-symmetry lyotropic liquid crystalline Frank-Kasper σ phase. Small-angle X-ray scattering studies reveal that this complex phase is characterized by a gigantic tetragonal unit cell, in which 30 sub-2-nm quasispherical micelles of five discrete sizes are arranged into a tetrahedral close packing, with exceptional translational order over length scales exceeding 100 nm. Varying the relative concentrations of water and surfactant in these lyotropic phases also triggers formation of the related Frank-Kasper A15 sphere packing as well as a common body-centered cubic structure. Molecular dynamics simulations reveal that the symmetry breaking that drives the formation of the σ and A15 phases arises from minimization of local deviations in surfactant headgroup and counterion solvation to maintain a nearly spherical counterion atmosphere around each micelle, while maximizing counterion-mediated electrostatic cohesion among the ensemble of charged particles.self-assembly | liquid crystals | surfactants | Frank-Kasper phases | lyotropic phase M olecular self-assembly provides a facile means of constructing a plethora of multifunctional soft materials, with mesoscopic structures that dictate their tailored properties and performance applications. Driven by noncovalent interactions between constituents, block polymers (1), giant shape amphiphiles (2), thermotropic liquid crystals (LCs) (3), lyotropic liquid crystals (LLCs) (4), and colloids (5) exemplify soft matter systems that spontaneously form periodic 1D lamellar phases, 2D columnar structures, and 3D packings of spherical particles. Columnar and spherical phases are useful as templates for mesoporous heterogeneous catalysts (6) and as microscale photonic bandgap materials (7). Manipulating supramolecular self-assembly to achieve specific materials morphologies and functions requires a fundamental understanding of the interplay between the structure and symmetry of the constituents and their multibody interactions.Although the packing of spherical objects (e.g., oranges and billiard balls) seems intuitively simple, point particles form a dizzying array of periodic crystals, quasicrystals (QCs), and structurally disordered glasses. Metallic elements typically form high-symmetry body-centered cubic (BCC), hexagonally closest-packed, and facecentered cubic (FCC) structures, due to the isotropy of metallic cohesion mediated by itinerant electrons (8). A few pure elements (e.g., Mn and U) form low-symmetry crystals with large and complex unit cells that maximize metallic cohesion against local constraints, such as maximization of Fermi surface sphericity (9).Sphere-forming soft mater...

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“…Microphase separation in block copolymer melts [1,2,3,4,5,6,7] has motivated the scientific community over the last couple of decades. Delicate balance [3] of chain conformational entropy and repulsive interaction energy between unlike monomers allows block copolymers to readily self-assemble into a number of ordered morphologies.…”

Section: Introductionmentioning

confidence: 99%

Morphology diagrams forA₂Bcopolymer melts: real-space self-consistent field theory

Kumar

Sides

et al. 2012

J. Phys.: Conf. Ser.

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The standard Gaussian model for block copolymermelts M W Matsen -Stability of the fcc structure in block copolymer systems Makiko Nonomura -SCFT simulation and SANS study on spatial distribution of solvents in microphase separation induced by a differentiating non-solvent in a semi-dilute solution of an ultra-high-molecular-weight block copolymer K Ando, T Yamanaka, S Okamoto et al. Abstract. Morphology diagrams for A2B copolymer melts are constructed using real-space self-consistent field theory (SCFT). In particular, the effect of architectural asymmetry on the morphology diagram is studied. It is shown that asymmetry in the lengths of A arms in the A2B copolymer melts aids in the microphase separation. As a result, the disorder-order transition boundaries for the A2B copolymer melts are shown to shift downward in terms of χN, χ and N being the Flory's chi parameter and the total number of the Kuhn segments,respectively, in comparison with the A2B copolymers containing symmetric A arms. Furthermore, perforated lamellar (PL) and a micelle-like (M) microphase segregated morphologies are found to compete with the classical morphologies namely, lamellar, cylinders, spheres and gyroid. The PL morphology is found to be stable for A2B copolymers containing asymmetric A arms and M is found to be metastable for the parameter range explored in this work. IntroductionMicrophase separation in block copolymer melts [1,2,3,4,5,6,7] has motivated the scientific community over the last couple of decades. Delicate balance[3] of chain conformational entropy and repulsive interaction energy between unlike monomers allows block copolymers to readily self-assemble into a number of ordered morphologies. Typical morphologies [3,8,9] include lamellar, cylinder, sphere, gyroid and Fddd network phases, which have been proven to be equilibrium states.Experimentally, the balance of the energy and entropy can be tuned by a number of molecular characteristics such as molecular weight [3], chain architecture [10,11], conformational asymmetry [12,13], polydispersity [14,15,16,17,18] and temperature. A number of studies [1,2,3,4,5,6,7] have been done to elucidate the effects of molecular weight, polydispersity, conformational asymmetry and temperature on the mircophase separation in linear di-block copolymer melts. However, the effects of chain architecture are not that well established. On the computational side, this is primarily due to a large parameter space for the simulations and resulting large number of morphologies, which need to be compared in terms of their free-energy

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The packing of soft materials: Molecular asymmetry, geometric frustration and optimal lattices in block copolymer melts

Cited by 127 publications

References 115 publications

Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Low-symmetry sphere packings of simple surfactant micelles induced by ionic sphericity

Morphology diagrams forA₂Bcopolymer melts: real-space self-consistent field theory

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The packing of soft materials: Molecular asymmetry, geometric frustration and optimal lattices in block copolymer melts

Cited by 127 publications

References 115 publications

Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Low-symmetry sphere packings of simple surfactant micelles induced by ionic sphericity

Morphology diagrams forA2Bcopolymer melts: real-space self-consistent field theory

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Product

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Morphology diagrams forA₂Bcopolymer melts: real-space self-consistent field theory