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

Abstract

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 – networks require molecules to occupy variable local environments, confounding attempts to understand their formation. Here, we examine the double-gyroid network phase by using a geometric formulation of the strong stretching theory of block copolymer melts, a prototypical soft self-assembly system. The theory establishes the direct link between molecular packing, assembly thermodynamics and the medial map, a generic measure of the geometric center of complex shapes. We show that “medial packing” is essential for stability of double-gyroid in strongly-segregated melts, reconciling a long-standing contradiction between infinite- and finite-segregation theories. Additionally, we find a previously unrecognized non-monotonic dependence of network stability on the relative entropic elastic stiffness of matrix-forming to tubular-network forming blocks. The composition window of stable double-gyroid widens for both large and small elastic asymmetry, contradicting intuitive notions that packing frustration is localized to the tubular domains. This study demonstrates the utility of optimized medial tessellations for understanding soft-molecular assembly and packing frustration via an approach that is readily generalizable far beyond gyroids in neat block copolymers.

Medial Packing, Frustration, and Competing Network Phases in Strongly Segregated Block Copolymers

“…In Figure B,C, we show the free energy comparison of these same competing phases for elastically asymmetry diblocks: ϵ = 0.3, relatively stiffer tubular A block; and ϵ = 3.0 relatively stiffer matrix B block. We observe windows of mDG stability over Lam and Hex phases for both limits of elastic asymmetry, consistent with our previously reported results …”

“…In comparison, SCFT at these finite χ N predicts a lower free energy for DG than Hex or Lam for the full range of ϵ, indicating that it retains an equilibrium stability window up through χ N = 75. In comparison, mSST predicts that the free energy of DG exceeds that of Lam and Hex at intermediate 1.0 ≲ ϵ ≲ 2.0, corresponding to the vanishing equilibrium stability window, as previously reported . While we note some basic consistency in the non-monotonic variation of the DG to Lam free energy with ϵ, mSST shows greater range in relative free energy variation.…”

“…The geometric cost of chain stretching is well-approximated by the results of parabolic brush theory , and encoded in the total second moments of volume I α ≡ prefix∫ V α normald V 0.25em z 2 for each subdomain α, where z is a brush height coordinate extending from the IMDS at z = 0. While parabolic brush theory accounts for the statistics of chains whose free ends are allowed to exist anywhere within a brush in a manner consistent with melt conditions, it fails to be self-consistent for brushes on surfaces whose curvatures force the free chain ends to splay apart. , However, we have previously shown , that the necessary “end-exclusion zone” corrections to parabolic brush theory are negligible for network phases over the full range of parameters where network phases compete for stability, increasing the free energy per chain by ≲0.01%, with appreciable increases in free energy predicted for highly curved sphere phases, justifying our use of parabolic brush theory for the results presented in this manuscript.…”

“…Our choice of minimizing the free energy over a set of four basis functions rather than the two of our previous study 31 was in order to assess (i) whether the addition of further Fourier modes to the generating surfaces had a significant impact on the results of the calculation and (ii) the generalizability of the variational calculation to further degrees of freedom. As shown in SI Figure S2, these additional modes lead to slight reductions (on the order of 10 −5 for DG, 10 −3 for DD and DP) in the calculated free energy (within the constraints of the convergence criteria supplied to the Nelder−Mead algorithm) that seemingly diminish with each added mode.…”

“…We observe windows of mDG stability over Lam and Hex phases for both limits of elastic asymmetry, consistent with our previously reported results. 31 However, while both mDD and mDP structures are far from stable for ϵ = 0.3 (stiffer A block), mDD is in close competition with Lam and Hex phases for ϵ = 3.0 (stiffer B block) in the window where DG is stable. This indicates an asymmetry how medial packing relaxes entropic penalties in the two blocks: the cost of stretching for mDD and mDP is much larger than mDG when the A block is stiffer, whereas the three phases have comparable stretching costs when the B block is stiffer.…”