Scaling Behavior of Block Copolymers in Spontaneous Growth of Lamellar Domains

Late Stage Domain Coarsening Dynamics of Lamellar Block Copolymers

Block copolymer (BCP) phase separation dynamics can be expected to differ significantly from that of the polymer blends due to the constraint of chain connectivity. BCP phase separation dynamics has been studied theoretically, but there has been little experimental evidence to confirm the BCP domain growth scaling laws put forward theoretically. Here, we investigate the dynamics of late-stage lamellar BCP domain coarsening and find that the scaling exponent for lamellar domain growth is ≈1/6 (0.17), irrespective of the annealing temperature, a value close to the scaling exponent of 0.2 predicted by theoretical studies. Furthermore, we show that the prefactors in the domain coarsening equation show Arrhenius dependence on temperature, indicating that the BCP domain growth dynamics is Arrhenius over the temperature range investigated.

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.

Defects in the Self-Assembly of Block Copolymers and Their Relevance for Directed Self-Assembly

Block copolymer self-assembly provides a platform for fabricating dense, ordered nanostructures by encoding information in the chemical architecture of multicomponent macromolecules. Depending on the volume fraction of the components and chain topology, these macromolecules form a variety of spatially periodic microphases in thermodynamic equilibrium. The kinetics of self-assembly, however, often results in initial morphologies with defects, and the subsequent ordering is protracted. Different strategies have been devised to direct the self-assembly of copolymer materials by external fields to align and perfect the self-assembled nanostructures. Understanding and controlling the thermodynamics of defects, their response to external fields, and their dynamics is important because applications in microelectronics either require extremely low defect densities or aim at generating specific defects at predetermined locations to fabricate irregular device-oriented structures for integrated circuits. In this review, we discuss defect morphologies of block copolymers in the bulk and thin films, highlighting (a) analogies to and differences from defects in other crystalline materials, (b) the stability of defects and their dynamics, and (c) the influence of external fields.

Aging of orientation fluctuations in stripe phases

Stripe patterns, observed in a large variety of physical systems, often exhibit a slow nonequilibrium dynamics because ordering is impeded by the presence of topological defects. Using computer simulations based on a well-established model for stripe formation, we show that a slow dynamics and aging occur also in stripe patterns free of topological defects. For a wide range of noise strengths, the two-time orientation correlation function follows a scaling form that is typical for systems exhibiting a growing length scale. In our case, the underlying mechanism is the coarsening of orientation fluctuations, ultimately leading to power-law spatial correlations perpendicular to the stripes. Our results show that even for the smallest amount of noise, stripe phases without topological defects do not reach equilibrium. This constitutes an important aspect of the dynamics of modulated phases.

Effect of noise on ordering of hexagonal grains in a phase-field-crystal model

We present a quantitative analysis of grain morphology of self-organizing hexagonal patterns based on the phase-field crystal model to examine the effect of stochastic noise on grain coarsening. We show that the grain size increases with increasing noise strength, resulting in enhanced hexagonal orientation due to noise up to some critical noise level above which the system becomes disordered.

Nucleation, growth, and coarsening of crystalline domains in order-order transitions between lamellar and hexagonal phases

Using the numerical solution of the time-dependent Ginzburg-Landau equation, we study the entire process of transformation between the lamellar and the hexagonal phases from the early-stage nucleation and growth to the late-stage coarsening regime. The metastable crystalline structure that nucleates first is identified in terms of the mean-field theory under the single-wave-number approximation. This has been borne out by the numerically efficient preparation of single-crystal structure developed via the noise-induced self-organization. We also present results for the scaling of the late-time domain growth, which is quantified by two measures: the structure factor and the orientational correlation function. In particular, the growth exponent is shown to be robust and indifferent to conservation of the order parameter.

Numerical study of domain coarsening in anisotropic stripe patterns

We study the coarsening of two-dimensional smectic polycrystals characterized by grains of oblique stripes with only two possible orientations. For this purpose, an anisotropic Swift-Hohenberg equation is solved. For quenches close enough to the onset of stripe formation, the average domain size increases with time as t(1/2). Further from onset, anisotropic pinning forces similar to Peierls stresses in solid crystals slow down defects, and growth becomes anisotropic. In a wide range of quench depths, dislocation arrays remain mobile and dislocation density roughly decays as t(-1/3), while chevron boundaries are totally pinned. We discuss some agreements and disagreements found with recent experimental results on the coarsening of anisotropic electroconvection patterns.

Dynamics of pattern coarsening in a two-dimensional smectic system

We have followed the coarsening dynamics of a single layer of cylindrical block copolymer microdomains in a thin film. This system has the symmetry of a two-dimensional smectic. The orientational correlation length of the microdomains was measured by scanning electron microscopy and found to grow with the average spacing between +/-1/2 disclinations, following a power law xi2(t) approximately t(1/4). By tracking disclinations during annealing with time-lapse atomic force microscopy, we observe dominant mechanisms of disclination annihilation involving tripoles and quadrupoles (three and four disclinations, respectively). We describe how annihilation events involving multiple disclinations result in similarly reduced kinetic exponents as observed here. These results map onto a wide variety of physical systems that exhibit similarly striped patterns.

Hydrodynamic interactions in ordering process of two-dimensional quenched block copolymers

The hydrodynamic coarsening of microphase separation in two-dimensional diblock copolymers is studied using numerical simulations. Results for symmetric and asymmetric block copolymers are compared. In contrast to the formation of the hexagonal phase where hydrodynamic flow appears not to be effective in enhancing domain coarsening, the late-time evolution of the lamellar phase proceeds faster, thus leading to a different power-law scaling with the addition of coupling of the velocity field to the order parameter.

Grain-boundary motion in layered phases

We study the motion of a grain boundary that separates two sets of mutually perpendicular rolls in Rayleigh-Bénard convection above onset. The problem is treated either analytically from the corresponding amplitude equations, or numerically by solving the Swift-Hohenberg equation. We find that if the rolls are curved by a slow transversal modulation, a net translation of the boundary follows. We show analytically that although this motion is a nonlinear effect, it occurs in a time scale much shorter than that of the linear relaxation of the curved rolls. The total distance traveled by the boundary scales as epsilon(-1/2), where epsilon is the reduced Rayleigh number. We obtain analytical expressions for the relaxation rate of the modulation and for the time-dependent traveling velocity of the boundary, and especially their dependence on wave number. The results agree well with direct numerical solutions of the Swift-Hohenberg equation. We finally discuss the implications of our results on the coarsening rate of an ensemble of differently oriented domains in which grain-boundary motion through curved rolls is the dominant coarsening mechanism.

Cell dynamics simulations of shear-induced alignment and defect annihilation in stripe patterns formed by block copolymers

The effect of large amplitude oscillatory shear on two-dimensional stripe patterns formed by block copolymers was investigated using cell dynamics simulations. A global orientational order parameter S and a correlation function for stripe normals G(r-r(')) were used to characterize the degree of stripe orientation under shear. The kinetics of stripe alignment, quantified by S, at various shear and quench conditions were studied as a function of strain amplitude, shear frequency, and temperature. The mechanisms of shear alignment and defect annihilation were investigated. A critical shear condition for complete alignment of stripes along the shear direction was also identified.

Ordering process in quenched block copolymers at low temperatures

We have studied domain growth of symmetric diblock copolymers undergoing microphase separation at low temperatures. We introduce a phenomenological nonlinear diffusion model with order-parameter-dependent mobility. Performing two-dimensional simulations, we find that the time-dependent scattering function exhibits dynamical scaling with a logarithmic growth law in the strong segregation limit where surface diffusion is the relevant mechanism for coarsening.