Self-Consistent Field Theory Predicts Universal Phase Behavior for Linear, Comb, and Bottlebrush Diblock Copolymers

An Efficient and Accurate SCF Algorithm for Block Copolymer Films and Brushes Using Adaptive Discretizations

Self-consistent field (SCF) theory serves as a robust tool for unraveling the intricate behavior exhibited by soft polymeric materials. However, the accuracy and efficiency of SCF calculations are crucially dependent on the numerical methods employed for system discretization and equation-solving. Here, we introduce a simple three dimensional SCF algorithm that uses real-space methods and adaptive discretization, offering improved accuracy and efficiency for simulating polymeric systems at surfaces. Our algorithm's efficacy is demonstrated through simulations of two distinct polymeric systems, namely, block copolymer (BCP) films and polymer brushes. By enhancing spatial resolution in regions influenced by external forces and employing finer contour discretization at grafting chain ends, we achieve significantly more accurate results at very little additional cost, enabling the study of 3D polymeric systems that were previously computationally challenging. To facilitate the widespread use of the algorithm, we have made our 1D-3D SCF code publicly available.

Phase Behavior and Conformational Asymmetry near the Comb-to-Bottlebrush Transition in Linear-Brush Block Copolymers

This study explores how conformational asymmetry influences the bulk phase behavior of linear-brush block copolymers. We synthesized 60 diblock copolymers composed of poly(trifluoroethyl methacrylate) as the linear block and poly[oligo(ethylene glycol) methyl ether methacrylate] as the brush block, varying the molecular weight, composition, and side-chain length to introduce different degrees of conformational asymmetry. Using small-angle X-ray scattering, we determined the morphology and phase diagrams for three different side-chain length systems, mainly observing lamellar and cylindrical phases. Increasing the side-chain length of the brush block from three to nine ethylene oxide units introduces sufficient asymmetry between the blocks to alter the phase behavior, shifting the lamellar-to-cylindrical transitions toward lower brush block compositions and transitioning the brush block from the dense comb-like regime to the bottlebrush regime. Coarse-grained simulations support our experimental observations and provide a mapping between the composition and conformational asymmetry. A comparison of our findings to strong stretching theory across multiple phase boundary predictions confirms the transition between the dense comb-like regime and the bottlebrush regime.

Confinement-induced self-assembly of a diblock copolymer within a non-uniform cylindrical nanopore

The self-assembly of a diblock copolymer melt confined within a non-uniform cylindrical nanopore is studied using the self-consistent field theory. The non-uniformity manifests in the form of a converging-diverging cylindrical nanopore. The axial variation in pore diameter presents a range of curvatures within the same confinement pore as opposed to a single curvature in a uniform-diameter cylindrical pore. The introduction of multiple curvatures leads to the formation of novel microstructures not accessible in uniform cylindrical confinement. The well-known equilibrium structures like a single helix, double helices, and concentric lamella under cylindrical confinement transition into new morphologies such as hyperboloidal phases, microstructures containing rings with a bead, rings with spheres, and a squeezed helical phase as the pore diameter varies axially. The converging-diverging geometry of the confining pore renders the helical phases seen in the cylindrical pore less favorable. A phase diagram in the parametric space of the block fraction and the ratio of the smallest and largest pore radii has been constructed to depict the order-order transition of various microstructures. The ratio of radii, a measure of the non-uniformity of the pore, along with the pore length brings out some interesting morphologies. The mechanism of these structural transitions is understood as the interplay between the variation in pore curvature attributed to the non-uniformity, the spontaneous curvature of the block copolymer interface, and the enthalpic interaction between the segregated blocks.

Frank-Kasper Phases of Diblock Copolymer Melts: Self-Consistent Field Results of Two Commonly Used Models

We constructed phase diagrams of conformationally asymmetric diblock copolymer A-B melts using the polymer self-consistent field (SCF) calculations of both the dissipative particle dynamics chain (DPDC) model (i.e., compressible melts of discrete Gaussian chains with the DPD non-bonded potential) and the "standard" model (i.e., incompressible melts of continuous Gaussian chains with the Dirac δ-function non-bonded potential) in the χN-ε plane, where χN and ε characterize, respectively, the repulsion and conformational asymmetry between the A and B blocks, at the A-block volume fraction f = 0.2 and 0.3. Consistent with previous SCF calculations of the "standard" model, σ and A15 are the only stable Frank-Kasper (FK) phases among the five FK (i.e., σ, A15, C14, C15 and Z) phases considered. The stability of σ and A15 is due to their delicate balance between the energetic and entropic contributions to the Helmholtz free energy per chain of the system, which, within our parameter range, increases in the order of σ/A15, Z, and C14/C15. While in general the SCF phase diagrams of these two models are qualitatively consistent, A15 is not stable for the DPDC model at the copolymer chain length N = 10 and f = 0.3; any differences in the SCF phase diagrams are solely due to the differences between these two models.

The impact of intramolecular polydispersity on the self-assembly of ABn miktoarm star copolymers

The self-assembly behaviors of AB_n miktoarm star copolymers as one typical type of asymmetric architecture have been studied well in the past few decades due to their deflected phase boundaries. In particular, recently, they have attracted renewed theoretical interest due to their expanded spherical phase region that stabilizes complex Frank-Kasper spherical phases. However, previous theoretical studies have never considered AB_n copolymers with unequal arm lengths, which is more or less the case for synthesized copolymers. In this work, we investigate the self-assembly behaviors of AB_n miktoarm star copolymers with unequal B-arms using self-consistent field theory. We propose an intramolecular polydispersity index (iĐ) to quantify the distribution of unequal B-blocks. Accordingly, we further propose a simple quantity of an effective arm number n_equ = n/iĐ for quantitatively comparing the phase boundaries between various AB_n copolymer samples with different arm numbers or different distributions of B-blocks. Our results indicate that different AB_n copolymers with equal n_equ exhibit similar phase diagrams. On the other hand, we also found that the phase boundaries of two different samples with same n_equ are not exactly overlapped. We speculate that the effect of spontaneous curvature may be mainly controlled by n_equ, but the packing frustration of B-blocks may also be dependent on the other quantities that are closely related to the shape of the distribution of B-arms, such as higher order polydispersity indexes.