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.

We analyze the energetics of sphere-like micellar phases in diblock copolymers in terms of wellstudied, geometric quantities for their lattices. We argue that the A15 lattice with P m3n symmetry should be favored as the blocks become more symmetric and corroborate this through a self-consistent field theory. Because phases with columnar or bicontinuous topologies intervene, the A15 phase, though metastable, is not an equilibrium phase of symmetric diblocks. We investigate the phase diagram of branched diblocks and find that the A15 phase is stable.The ability to control the self-assembly of complex lattices by manipulating molecular architecture remains an essential aspect in the creation of new, functional materials. With only a few tunable parameters, diblock copolymer melts exhibit a wide variety of equilibrium phases which can be understood via the mean-field Gaussian chain model of "AB" diblock copolymers composed of immiscible A and B blocks [1,2]. Indeed, in a system where the A and B-blocks are otherwise identical, there are only two thermodynamic variables, φ, the volume fraction of A type monomers, and χN , where χ is the Flory-Huggins parameter characterizing the repulsive interactions between the A-and B-type monomers and N is the degree of polymerization [3]. In this letter, we present a model which predicts that the the A15 (shown in Fig. 1a) lattice of diblocks is stable relative to other sphere-like phases for sufficiently large φ or, in other words, sufficiently symmetric diblocks. We corroborate this prediction by recalculating the phase diagram for symmetric diblocks (Fig. 1b) via a self-consistent field theory (SCFT) for diblock copolymer melts [4].The "classical" diblock phases are well-understood: near the order-disorder transition (ODT), Leibler developed a Landau-like theory in the weak-segregation regime to establish the stability of a body-centered cubic (BCC) phase, a hexagonal phase of columns and a lamellar phase [5]. Moreover, Semenov's picture of spherical micelles interacting through a disordered copolymer background when φ ≪ 1 accounts for the appearance of the facecentered cubic (FCC) lattice near the ODT in the meanfield phase diagram [6]. The more exotic gyroid phase was discovered [7,8,9] and was explained successfully by Matsen and Schick via SCFT [4]. In our study of the A15 lattice, we find that the hexagonal and gyroid phases intervene and thus there should be no stable A15 lattice for simple diblocks. However, sphere-like topologies are favored by branched diblock copolymers [10,11,12] and dendritic polymers [13,14]; with this in mind we predict that sphere-like phases are stabilized and that the A15 phase is a ground state for this class of structures. By implementing, to our knowledge, the first full SCFT treatment of branched molecules (shown in Fig. 1c) we have verified our theory.In the dilute regime, Semenov's picture treats each micelle as an undistorted sphere so that the outer block extends to a spherical unit cell of radius R S . This unit-cell approximation pr...

The Materials Genome Initiative (MGI) advanced a new paradigm for materials discovery and design, namely that the pace of new materials deployment could be accelerated through complementary efforts in theory, computation, and experiment. Along with numerous successes, new challenges are inviting researchers to refocus the efforts and approaches that were originally inspired by the MGI. In May 2017, the National Science Foundation sponsored the workshop "Advancing and Accelerating Materials Innovation Through the Synergistic Interaction among Computation, Experiment, and Theory: Opening New Frontiers" to review accomplishments that emerged from investments in science and infrastructure under the MGI, identify scientific opportunities in this new environment, examine how to effectively utilize new materials innovation infrastructure, and discuss challenges in achieving accelerated materials research through the seamless integration of experiment, computation, and theory. This article summarizes key findings from the workshop and provides perspectives that aim to guide the direction of future materials research and its translation into societal impacts.

We study ABn miktoarm star block copolymers in the strong segregation limit, focusing on the role that the AB interface plays in determining the phase behavior. We develop an extension of the kinked-path approach which allows us to explore the energetic dependence on interfacial shape. We consider a one-parameter family of interfaces to study the columnar to lamellar transition in asymmetric stars. We compare with recent experimental results. We discuss the stability of the A15 lattice of spherelike micelles in the context of interfacial energy minimization. We corroborate our theory by implementing a numerically exact self-consistent-field theory to probe the phase diagram and the shape of the AB interface.

We use both mean-field methods and numerical simulation to study the phase diagram of classical particles interacting with a hard-core and repulsive, soft shoulder. Despite the purely repulsive interaction, this system displays a remarkable array of aggregate phases arising from the competition between the hard-core and shoulder length scales. In the limit of large shoulder width to core size, we argue that this phase diagram has a number of universal features, and classify the set of repulsive shoulders that lead to aggregation at high density. Surprisingly, the phase sequence and aggregate size adjusts so as to keep almost constant inter-aggregate separation.PACS numbers: 61.30. Dk, 61.30.St,61.46.Bc Entropy is a potent force in the theory of selfassembly. It can be argued, through entropic considerations alone, that hard spheres will self-assemble into the face-centered-cubic (fcc) lattice or any of its many variants related through stacking faults. As a result, when a material exhibits an fcc phase, it is often attributed to the optimality of the close-packed lattice. When less common or less dense lattices are formed, all sundry of explanations are invoked, ranging from quantum mechanics [1], lattice effects [2], partially filled Landau levels [3], and even soft interactions [4,5]. While there has been concerted effort to tailor the pair interaction to achieve a desired periodic arrangement [6] this must be done in the context of those packing motifs that arise from generic interactions. For instance, it would be no trick to tailor a potential to make an fcc lattice.With this in mind, here we consider a seemingly simple extension of the hard sphere model, namely a hard core interaction of radius σ with a soft shoulder of radius σ s and height ǫ (HCSS):When σ s /σ 1 this potential models hard spheres with a soft pair repulsion and was used to study isostructural transitions in Cs and Ce [7]. In generic repulsive potentials, it has been shown that as the range of the soft repulsion grows (corresponding to σ s /σ ≈ 2) a rich variety of density-modulated ground states appear [8, 9, 10] which can be characterized as periodic arrangements of regular sized clusters of the original spheres. In this Letter we establish a sufficient condition on the pair potential for clustering and the subsequent ordering of the clusters which generalizes results on soft potentials without hard cores [11]. We develop a self-consistent field theory for soft repulsion and use it to study the formation of striped phases. We corroborate our analytic treatment with numerical solutions that also predict the existence of hexagonal and inverted hexagonal phases with both fluid and crystalline order in the clusters, as shown in Fig. 1. We also present results from Monte Carlo simulations of the HCSS potential which both stimulate and support the more general results. In all cases, we find that over the range of stable aggregate structures the lattice constant remains fixed while the clusters change their size and morphology so as to maint...