Aging of orientation fluctuations in stripe phases

Direct Imaging of Interfacial Fluctuations in Confined Block Copolymer with in Situ Slow-Scan-Disabled Atomic Force Microscopy

Using environmentally controlled, high-speed atomic force microscopy (AFM), we examine dynamic fluctuations of topographically confined poly(styrene-block-methyl methacrylate) (PS-b-PMMA) cylinders. During thermal annealing, fluctuations drive perturbations of the block copolymer (BCP) interface between polymer domains, leading to pattern roughness. Whereas previous investigations have examined roughness in room-temperature and kinetically quenched samples, we directly visualize the dynamics of PS/PMMA interfaces in real space and time at in situ temperatures above the glass transition temperature, T_g. Imaging under these experimentally challenging thermal annealing conditions is critical to understanding the inherent connection between thermal fluctuations and BCP pattern assembly. Through the use of slow-scan-disabled AFM, we dramatically improve the imaging time resolution for tracking polymer dynamics. Fluctuations increase in intensity with temperature and, at high temperatures, become spatially coherent across their confining potential. Additionally, we observe that topographic confinement suppresses fluctuations and correlations in the proximity of the guiding field. In situ imaging at annealing temperatures represents a significant step in capturing the dynamics of chain mobility at BCP interfaces.

Pathways to equilibrium orientation fluctuations in finite stripe-forming systems

Small-angle orientation fluctuations in ordered stripe-forming systems free of topological defects can exhibit aging and anisotropic growth of two length scales. In infinitely extended systems, the stripe orientation field develops a dominant modulation length λ_{∥}^{*}(t) in the direction parallel to the stripes, which increases with time t as λ_{∥}^{*}(t)∼t^{1/4}. Simultaneously, the orientation correlation length ξ_{⊥}(t) in the direction perpendicular to the stripes increases as ξ_{⊥}(t)∼t^{1/2} [Riesch et al., Interface Focus 7, 20160146 (2017)2042-889810.1098/rsfs.2016.0146]. Here we show that finite systems of size L_{⊥}×L_{∥} with periodic boundary conditions reach equilibrium when the dominant modulation length λ_{∥}^{*}(t) reaches the system size L_{∥} in the stripe direction. The equilibration time τ_{eq}^{∥} is solely determined by L_{∥}, with τ_{eq}^{∥}∼L_{∥}^{4}. In systems with L_{⊥}<L_{∥}^{2}/2πλ_{p}, where λ_{p} is the undulation penetration length, the initial aging and coarsening dynamics changes at the crossover time τ_{C}^{⊥}∼L_{⊥}^{2} to an aging and coarsening dynamics described by the one-dimensional Mullins-Herring equation, before reaching equilibrium at τ_{∥}^{eq}. Our work reveals the two pathways to equilibrium in stripe phases with periodic boundary conditions, the finite-size scaling behavior of equilibrium orientation fluctuations, and the characteristic exponents associated with the influence of a finite system size.

Scaling properties of ageing orientation fluctuations in stripe phases

We investigate the non-equilibrium dynamics of an ordered stripe-forming system free of topological defects. In particular, we study the ageing and the coarsening of orientation fluctuations parallel and perpendicular to the stripes via computer simulations based on a minimal phase-field model (model B with Coulomb interactions). Under the influence of noise, the stripe orientation field develops fluctuations parallel to the stripes, with the dominant modulation length λ*∥ increasing with time t as λ*∥ ∼ t1/4 and the correlation length perpendicular to the stripes ξ⊥θ increasing as ξ⊥θ ∼ t1/2. We explain these anisotropic coarsening dynamics with an analytic theory based on the linear elastic model for stripe displacements first introduced by Landau and Peierls. We thus obtain the scaling forms and the scaling exponents characterizing the correlation functions and the structure factor of the stripe orientation field. Our results reveal how the coarsening of orientation fluctuations prevents a periodically modulated phase free of topological defects from reaching equilibrium.