StrAPS: Structural Angular Power Spectrum for Discovering Novel Morphologies in Block Copolymers

Abstract

The morphologies of phase separating systems have formal distinctions such as symmetry groups, but the analysis protocol for labeling a particular phase field with a morphology requires manual expertise, arbitrary thresholds, or established signatures. In this work, it is investigated if the angular power spectrum of the 3D structure factor can discriminate between morphologies. The 3D structure factor is computed on configurations of phase separating block copolymers generated by coarse-grained molecular dynamics simulations. The shell of structure factor values containing the primary peaks is isolated. This 2D field on a sphere is decomposed into spherical harmonic modes of even polynomial degree $\ell\le 12$, then further reduced to the rotationally invariant angular power spectrum. It is found that these few coefficients for low $\ell$ discriminate robustly between different morphologies. This analysis serves as an automatic tool for flagging novel structures, without a need to enumerate the plausible morphologies in advance.

Figures (5)

• Figure 1: Structural Angular Power Spectra (StrAPS). Black circles represent postprocessing of morphologies found in MD simulations. Open squares represent idealized morphologies imposed by assigning particle types based on (A) lamellar, (B) hexagonal, and (C) BCC spherical phase fields.
• Figure 2: Perspective renders. Three different morphologies of phase separated microstructures. (Left) Lamellae at $f=1/2$. (Middle) cylinders at $f=3/16$. (Right) spheres at $f=1/8$.
• Figure 3: Orthographic projections of particle centers. These highlight the real space lattices of (A) Lamellar (B) Hexagonal and (C) BCC structures.
• Figure 4: Samples from the 3D Structure. (A) lamellar, (B) Hexagonal, and (C) BCC morphologies. Values of $S(\mathbf{k})$ are presented for the shell of $\mathbf{k}$ vectors with magnitude $k^*$, as well as a plane in $k$ space. The darkness of each point indicates the value of $S(\mathbf{k})$. The gray and black wireframes in (C) indicate two competing BCC structures.
• Figure 5: Radially averaged structure factor. Small gray points indicate values for individual 3D k vectors. The Black line indicates the average over vectors with the identical magnitude. A gray band around the averaged line represents the standard error of the mean for each k. Data are shown for the (A) lamellar, (B) Hexagonal, and (C) BCC systems.