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

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.

Entropy production in thermal phase separation: a kinetic-theory approach

Entropy production during the process of thermal phase-separation of multiphase flows is investigated by means of a discrete Boltzmann kinetic model. The entropy production rate is found to increase during the spinodal decomposition stage and to decrease during the domain growth stage, attaining its maximum at the crossover between the two. Such behaviour provides a natural criterion to identify and discriminate between the two regimes. Furthermore, the effects of heat conductivity, viscosity and surface tension on the entropy production rate are investigated by systematically probing the interplay between non-equilibrium energy and momentum fluxes. It is found that the entropy production rate due to energy fluxes is an increasing function of the Prandtl number, while the momentum fluxes exhibit an opposite trend. On the other hand, both contributions show an increasing trend with surface tension. The present analysis inscribes within the general framework of non-equilibrium thermodynamics and consequently it is expected to be relevant to a broad class of soft-flowing systems far from mechanical and thermal equilibrium.

Grain coarsening in two-dimensional phase-field models with an orientation field

In the literature, contradictory results have been published regarding the form of the limiting (long-time) grain size distribution (LGSD) that characterizes the late stage grain coarsening in two-dimensional and quasi-two-dimensional polycrystalline systems. While experiments and the phase-field crystal (PFC) model (a simple dynamical density functional theory) indicate a log-normal distribution, other works including theoretical studies based on conventional phase-field simulations that rely on coarse grained fields, like the multi-phase-field (MPF) and orientation field (OF) models, yield significantly different distributions. In a recent work, we have shown that the coarse grained phase-field models (whether MPF or OF) yield very similar limiting size distributions that seem to differ from the theoretical predictions. Herein, we revisit this problem, and demonstrate in the case of OF models [R. Kobayashi, J. A. Warren, and W. C. Carter, Physica D 140, 141 (2000)PDNPDT0167-278910.1016/S0167-2789(00)00023-3; H. Henry, J. Mellenthin, and M. Plapp, Phys. Rev. B 86, 054117 (2012)PRBMDO1098-012110.1103/PhysRevB.86.054117] that an insufficient resolution of the small angle grain boundaries leads to a log-normal distribution close to those seen in the experiments and the molecular scale PFC simulations. Our paper indicates, furthermore, that the LGSD is critically sensitive to the details of the evaluation process, and raises the possibility that the differences among the LGSD results from different sources may originate from differences in the detection of small angle grain boundaries.

Hexagonal phase ordering in strongly segregated copolymer films

Strongly segregated copolymer mixtures with uneven composition ratio can form hexagonally ordered thin films. A simplified model describing the size and position of micellelike clusters is derived, allowing for investigation of much larger domain sizes than in previous studies. Simulations of this model are performed to study the generation of large scale order and evolution of pattern defects. We find three temporal regimes exhibiting different scaling laws for orientational correlation length and defect number. In the early stage, topological defects are rapidly eliminated by pairwise annihilation. A slower intermediate stage is characterized by the migration of grain boundaries and the elimination of small grains. In the final stage, grain boundaries become pinned and the evolution halts. A scaling law for defect interaction is proposed which is consistent with the crossover between the first and second stages.

Time-dependent Ginzburg-Landau model for nonfrustrated linear ABC triblock terpolymers

A time-dependent Ginzburg-Landau (TDGL) model is proposed to simulate the ordering of linear ABC triblock terpolymers. The model, in its current form, is applicable to nonfrustrated triblock systems, with the specific condition that χAC≫χAB≈χBC. Simulations are presented that demonstrate the model's ability to evolve a wide variety of morphologies throughout time, including tetragonal, core-shell hexagonal, three-phase lamellar, and beads-in-lamellar phases. The model also incorporates an interaction term to study templated substrates for directed self-assembly. The efficiency of the TDGL model enables large-scale simulations that allow investigation of self-assembly, and directed self-assembly, processes that may exhibit very small defect concentrations.

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.

Theory and application of shapelets to the analysis of surface self-assembly imaging

A method for quantitative analysis of local pattern strength and defects in surface self-assembly imaging is presented and applied to images of stripe and hexagonal ordered domains. The presented method uses "shapelet" functions which were originally developed for quantitative analysis of images of galaxies (∝10(20)m). In this work, they are used instead to quantify the presence of translational order in surface self-assembled films (∝10(-9)m) through reformulation into "steerable" filters. The resulting method is computationally efficient (with respect to the number of filter evaluations), robust to variation in pattern feature shape, and, unlike previous approaches, is applicable to a wide variety of pattern types. An application of the method is presented which uses a nearest-neighbor analysis to distinguish between uniform (defect-free) and nonuniform (strained, defect-containing) regions within imaged self-assembled domains, both with striped and hexagonal patterns.

Defect structures and ordering behaviours of diblock copolymers self-assembling on spherical substrates

One of the main differences of ordered structures constrained on curved surfaces is the nature of topological defects. We here explore the defect structures and ordering behaviours of both lamellar and cylindrical phases of block copolymers confined on spherical substrates by the Landau-Brazovskii theory, which is numerically solved by a highly accurate spectral method with a spherical harmonic basis. For the cylindrical phase, isolated disclinations and scars are generated on the spherical substrates. The number of excess dislocations in a scar depends linearly on the sphere radius. The defect fraction characterizing the ordering dynamics decays exponentially. The scars are formed from the isolated disclinations via mini-scars. For the lamellar phase, three types of defect structures (hedgehog, spiral and quasi-baseball) are identified. The disclination annihilation is the primary ordering mechanism of the lamellar phase.

Evolution of lateral ordering in symmetric block copolymer thin films upon rapid thermal processing

This work reports experimental findings about the evolution of lateral ordering of lamellar microdomains in symmetric PS-b-PMMA thin films on featureless substrates. Phase separation and microdomain evolution are explored in a rather wide range of temperatures (190-340 °C) using a rapid thermal processing (RTP) system. The maximum processing temperature that enables the ordering of block copolymers without introducing any significant degradation of macromolecules is identified. The reported results clearly indicate that the range of accessible temperatures in the processing of these self-assembling materials is mainly limited by the thermal instability of the grafted random copolymer layer, which starts to degrade at T > 300 °C, inducing detachment of the block copolymer thin film. For T ⩽ 290 °C, clear dependence of correlation length (ξ) values on temperature is observed. The highest level of lateral order achievable in the current system in a quasi-equilibrium condition was obtained at the upper processing temperature limit after an annealing time as short as 60 s.

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.

Nanoscale surface pattern evolution in heteroepitaxial bimetallic films

Nanoscale self-assembly dynamics of submonolayer bimetallic films was studied through simulation of a coarse-grained mesoscopic model. Simulations predict a phase transition sequence (hexagonal→stripe→inverse hexagonal) consistent with experimental observations of Pb/Cu(111) heteroepitaxial growth. Post-transition ordering dynamics of hexagonal and inverse hexagonal patterns was simulated and quantified in order to predict pattern quality and evolution mechanisms. Correlation length scaling laws and nanoscale evolution mechanisms were predicted through simulation of experimentally relevant length (≈1 μm(2)) and time scales, with findings supporting evidence of universal pattern behavior with other hexagonal systems. Results provide detailed dynamics and structure of this novel self-assembly process applicable to the design and optimization of functional bimetallic materials, such as bimetallic catalysts.

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.

Stripe domain coarsening in geographical small-world networks on a Euclidean lattice

We study phase ordering dynamics of spatially periodic striped patterns on the small-world network that is derived from a two-dimensional regular lattice with distance-dependent random connections. It is demonstrated numerically that addition of spatial disorder in the form of shortcuts makes the growth of domains much slower or even frozen at late times.

Coarsening in potential and nonpotential models of oblique stripe patterns

We study the coarsening of two-dimensional oblique stripe patterns by numerically solving potential and nonpotential anisotropic Swift-Hohenberg equations. Close to onset, all models exhibit isotropic coarsening with a single characteristic length scale growing in time as t1/2. Further from onset, the characteristic lengths along the preferred directions x and ŷ grow with different exponents, close to 1/3 and 1/2, respectively. In this regime, one-dimensional dynamical scaling relations hold. We draw an analogy between this problem and model A in a stationary, modulated external field. For deep quenches, nonpotential effects produce a complicated dislocation dynamics that can lead to either arrested or faster-than-power-law growth, depending on the model considered. In the arrested case, small isolated domains shrink down to a finite size and fail to disappear. A comparison with available experimental results for electroconvection in nematics is presented.

Morphologies and flow patterns in quenching of lamellar systems with shear

We study the behavior of a fluid quenched from the disordered into the lamellar phase under the action of a shear flow. The dynamics of the system is described by Navier-Stokes and convection-diffusion equations with the pressure tensor and the chemical potential derived by the Brazovskii free energy. Our simulations are based on a mixed numerical method with the lattice Boltzmann equation and a finite difference scheme for Navier-Stokes and order parameter equations, respectively. We focus on cases where banded flows are observed with two different slopes for the component of velocity in the direction of the applied flow. Close to the walls the system reaches a lamellar order with very few defects, and the slope of the horizontal velocity is higher than the imposed shear rate. In the middle of the system the local shear rate is lower than the imposed one, and the system looks like a mixture of tilted lamellae, droplets, and small elongated domains. We refer to this as a region with a shear-induced structures (SIS) configuration. The local behavior of the stress shows that the system with the coexisting lamellar and SIS regions is in mechanical equilibrium. This phenomenon occurs, at fixed viscosity, for shear rates under a certain threshold; when the imposed shear rate is sufficiently large, lamellar order develops in the whole system. Effects of different viscosities have been also considered. The SIS region is observed only at low enough viscosity. We compare the above scenario with the usual one of shear banding. In particular, we do not find evidence for a plateau of the stress at varying imposed shear rates in the region with banded flow. We interpret our results as due to a tendency of the lamellar system to oppose the presence of the applied flow.

Hydrodynamic coarsening in striped pattern formation with a conservation law

We observed in numerical simulations that the interaction of striped-pattern-forming instability and a neutrally stable zero mode induces patterns of domains of upflow hexagons coexisting with domains of downflow hexagons. They appear only when hydrodynamic flow is present.

Ordering mechanisms in two-dimensional sphere-forming block copolymers

We study the coarsening dynamics of two-dimensional hexagonal patterns formed by single microdomain layers of block copolymers, using numerical simulations. Our study is focused on the temporal evolution of the orientational correlation length, the interactions between topological defects, and the mechanisms of coarsening. We find no free disclinations in the system; rather, they are located on large-angle grain boundaries, commonly where such boundaries bifurcate. The correlation lengths determined from the scattering function, from the density of dislocations, and from the density of disclinations exhibit similar behavior and grow with time according to a power law. The orientational correlation length also grows following a power law, but with a higher exponent than the other correlation lengths. The orientational correlation length grows via annihilation of dislocations, through preferential annihilation of small-angle grain boundaries due to poor screening of the strain field around dislocations located on small-angle grain boundaries. Consequently, the patterns are characterized by large-angle grain boundaries. The most commonly observed mechanism of coarsening is the collapse of smaller grains residing on the boundary of two larger grains delimited by large-angle grain boundaries. Simulations agree remarkably well with experimental results recently obtained.

Shear-induced grain boundary motion for lamellar phases in the weakly nonlinear regime

We study the effect of an externally imposed oscillatory shear on the motion of a grain boundary that separates differently oriented domains of the lamellar phase of a diblock copolymer. A direct numerical solution of the Swift-Hohenberg equation in shear flow is used for the case of a transverse/parallel grain boundary in the limits of weak nonlinearity and low shear frequency. We focus on the region of parameters in which both transverse and parallel lamellae are linearly stable. Shearing leads to excess free energy in the transverse region relative to the parallel region, which is in turn dissipated by net motion of the boundary toward the transverse region. The observed boundary motion is a combination of rigid advection by the flow and order parameter diffusion. The latter includes break up and reconnection of lamellae, as well as a weak Eckhaus instability in the boundary region for sufficiently large strain amplitude that leads to slow wave number readjustment. The net average velocity is seen to increase with frequency and strain amplitude, and can be obtained by a multiple scale expansion of the governing equations.