Structural Relaxation and Anisotropic Elasticity of Ordered Block Copolymer Melts

TL;DR

The paper develops a SCFT-based framework to quantify terminal (long-time) elastic stiffness in ordered block copolymer melts across lamellar, columnar, and cubic morphologies for AB and ABA architectures. By computing the mixing free energy under quasistatic deformations and fitting to quadratic forms, it extracts the anisotropic moduli $C_{ijkl}$ and analyzes how domain spacing $D$, conformational asymmetry $\epsilon$, and chain architecture influence stiffness, including bridging/looping effects. It demonstrates a hierarchy where 3D cubic phases exhibit nonzero lower bounds while 1D and 2D phases have liquid-like directions, and shows that modulus predictions diverge from naive rule-of-mixtures due to interfacial enthalpy and domain-spacing adjustments. The work also explores polycrystalline averages (Voigt/Reuss) and reveals that domain topology and chain exchange across slip surfaces critically govern long-time relaxation, offering design rules for durable thermoplastics and guidance for experimental validation.

Abstract

Block copolymer (BCP) melts play a critical role in the design of thermoplastics, owing in large part to the creation of alternating nano-scale domains of soft and stiff components. Much of thermoplastic design has been focused on the short-time response associated with the dynamical rigidity of rubbery or glassy chains. However, less attention has been paid to the long-time relaxation and rigidity of microphase separated BCP melts or the role that domain morphology plays in modulating near-equilibrium response. We take advantage of the ability of self-consistent field theory (SCFT) to calculate equilibrium properties of BCP melts to explore the anisotropic elastic response of ordered ABA and AB copolymer melts as quasistatic deformation processes. This allows us to determine the anisotropic stiffness of the liquid crystal-like lamellar and columnar phases due to modulations in domain spacing, as well as the full stiffness tensor of the cubic BCC sphere and double gyroid phases. We explore elastic modulus landscapes for both single grain materials and random polycrystals over architectural parameters and segregation strengths, using AB diblock and ABA triblock melts as key examples. Finally, we re-examine basic assumptions of BCP melts as simple composites, dependence of melt stiffness on domain spacing, and the role of bridging chain conformations in altering melt relaxation, providing evidence for an interplay between mass transport and domain topology that remains to be understood.

Figures (8)

• Figure 1: (a) Schematic of relative modulus of different phases for BCPs, this work narrowing in on $\omega \to 0$. Anisotropic elasticity is highlighted, showing deformations yielding liquid and elastic responses. (b) The elastic responses under investigation in this work (uniaxial extension and shear). (c) The short and long time response of a shear deformation is shown, highlighting the liquid response for lamellae and cylinders.
• Figure 2: Modulus heat map for 1D lamellar phase for three elastic asymmetries, plotting full region of metastable lamellar phases overlaid by phase diagrams, where the lamellar stability window is between the yellow lines.
• Figure 3: Modulus heat map for 2D hexagonal cylinder phase for three elastic asymmetries for stable phase region, comparing Young's modulus (top) to shear modulus (bottom).
• Figure 4: Modulus heat map for 3D cubic phases for three elastic asymmetries for stable phase region, highlighting shear modulus for BCC (top) and gyroid (bottom).
• Figure 5: (a) Plot showing the scaling of longitudinal modulus with domain spacing for lamellae at $f_{\rm A}=0.5$, with domain spacing normalized by $\epsilon$ and $f_{\rm A}$. (b) Plot showing the percent difference between $C_{||}$ for ABA and AB, where $\Delta C_{||} = C_{||}^{\rm ABA}-C_{||}^{\rm AB}$.