Emergent symmetries in block copolymer epitaxy

The directed self-assembly (DSA) of block copolymers (BCPs) has shown promise in fabricating customized two-dimensional (2D) geometries at the nano- and meso-scale. Here, we discover spontaneous symmetry breaking and superlattice formation in DSA of BCP. We observe the emergence of low symmetry phases in high symmetry templates for BCPs that would otherwise not exhibit these phases in the bulk or thin films. The emergence phenomena are found to be a general behavior of BCP in various template layouts with square local geometry, such as 44 and 32434 Archimedean tilings and octagonal quasicrystals. To elucidate the origin of this phenomenon and confirm the stability of the emergent phases, we implement self-consistent field theory (SCFT) simulations and a strong-stretching theory (SST)-based analytical model. Our work demonstrates an emergent behavior of soft matter and draws an intriguing connection between 2-dimensional soft matter self-assembly at the mesoscale and inorganic epitaxy at the atomic scale.

) while red and green indicate the degenerate phase of the other orientation (the green domains in Fig. 3a).

Supplementary Discussion
SCFT Simulations. SCFT was employed to simulate post templated DSA of BCP. Details of the formalism can be found in a previous study (S2). For the purpose of this work, we chose f = 0.7 where f is the volume fraction of polymer A, making A the majority block. Different values of χN (15 < χN < 23) and different thickness of brush layers were used with no direct effect on the BCP self-assembled pattern (Fig. S6).
The results of square grid post template were implemented using χN = 15 and the 3 2 434 post pattern using χN = 17. Here, N refers to the degree of polymerization and χ is the Flory-Huggins parameter. Both blocks have equal statistical segment length. Simulations are conducted in 2D assuming no variation taking place across film thickness. The computational grid has a pixel size of 0.12Rg where Rg is the polymer radius of gyration. The overall size (height and depth) of the computational cell was modified by changing the total number of pixels.
A masking method was employed to mimic the effect of the post pattern in directed self-assembly. A pressure field w+ = (wB + wA)/2 was imposed as a mask where posts are located to create polymer inaccessible areas. A magnitude of w+ = 8 was applied to a circular post of diameter 16 pixels. Pattern chemical affinity towards block B was achieved using an exchange potential w-= (wB -wA)/2 = -8, surrounding the posts with a thickness of 10 pixels.
For the initial study of 3 2 434 template, we used squares and triangles of equal side length. This fixed the angle of the rhombus to 60 o . The simulation results reproduced the essential phases observed experimentally: 2,3,5 domains inside a rhombus in the range of Lp = 2.10 to 2.50 L0 (denoted as II, III, and V in Fig. S7a, S7b, and S7d respectively). Nonetheless, defective mixed phases were frequently observed with a probability that depends on the computational cell size. Hence, additional computational variables were explored, namely the rhombus angle θ (Fig. 3d and Fig S7). Fig. S7h shows the free energy landscape as a function of Lp and θ. It is evident that in the range studied in this work, the landscape is dominated by three plain phases of 2, 4, and 5 ( Fig.  S7a, S7c, and S7d, respectively). Qualitatively, phase II effectively occupies the small Lp < 2.25 L0 independent of θ. Large Lp > 2.3 L0 shows phase V for θ > 60 o while phase VI is located where θ < 60 o . Plain phase III is confined to a small stability window of 2.3L0 < Lp < 2.4L0 and θ of 54 o . More interestingly, stable mixed phases II/V and III/V were observed for θ > 64 o and Lp ~ 2.3L0 as shown in Fig S7e and S7f. The mixed phases consisted of rows with an alternating number of domains inside the rhombus template. Indeed, the stable mixed phases show lower symmetry with a supercell size of twice that of the post. Clear BCP rhombus configuration is observed inside the square arrangement of the 3 2 434 template for mixed phases. The orientation of the rhombus is an indication of the communication across template regions as the short axis of the rhombus points towards the 5 phase, while the long axis points towards the 2 or 3 phases. The unequal number of BCP domains inside the rhombus template causes the four BCP domains inside the square template to deform into a rhombus shape. Conversely, a square configuration prevailed inside the square template due to equal tension of the surrounding domains for plain phases. The free energy landscape could explain the observed experimental results. For θ = 60 o and varying Lp, the polymer configurations are located in the vicinity of multiple plain phases and far from mixed phases. Hence, we see the ubiquitous presence of plain phase signatures with no long-range order.

Strong-stretching theory (SST) calculations.
In this work, we employ the strong-stretching theory (SST) calculations by Milner and Olmsted (S3, S4) for circular domains in the polyhedral interface limit (PIL). Here, a simple assumption for chain distribution (the straight-path ansatz) is employed, where the AB interface adopts the same shape as the lattice Voronoi cell.
For a linear AB diblock, the total segment number in a chain N = NA+ NB, where the fraction of A is f = NA/N. Segments of both blocks have equal volume and equal statistical segment length. The total free energy F [nkbT] consists of Fi representing the interfacial energy between blocks at the core/corona interface, and Fst representing the entropic cost of stretching of a Gaussian chain.
(1) F = Fi + Fst Where Ai is the total interfacial area for the core/corona contact and V is the total system volume. The entropic component of free energy is calculated using SST expression derived from the "parabolic brush" assumption Here the integral is taken over the entire block volume. Coordinate z is a radial distance from the AB interface, where junction points are localized. To map F[nkbT] as a function of domain arrangement inside the square template, the BCP micelles are assumed to be polyhedral unit cells that cover the entire space. These unit cells are the Voronoi cells of the lattice. For an asymmetric configuration (rhombus configuration), the BCP domains occupy unequal space where hexagonal and pentagonal unit cells are constructed. In the symmetric case, all BCP domains occupy pentagonal unit cells (Fig. 2e). In this regard, for every point in the F plot (Fig. 2a), the BCP domains are positioned and the space is divided into the corresponding Voronoi cells. The corresponding integrals were numerically calculated.