Ohta-Kawasaki Model Research Articles

Asymptotically compatible schemes for nonlocal Ohta–Kawasaki model

AbstractWe study the asymptotical compatibility of the Fourier spectral method in multidimensional space for the nonlocal Ohta–Kawasaka model, a more generalized version of the model proposed in our previous work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989). By introducing the Fourier collocation discretization for the spatial variable, we show that the asymptotical compatibility holds in 2D and 3D over a periodic domain. For the temporal discretization, we adopt the second‐order backward differentiation formula method. We prove that for certain nonlocal kernels, the proposed time discretization schemes inherit the energy dissipation law. In the numerical experiments, we verify the asymptotical compatibility, the second‐order temporal convergence rate, and the energy stability of the proposed schemes. More importantly, we discover a novel square lattice pattern when certain nonlocal kernel are applied in the model. In addition, our numerical experiments confirm the existence of an upper bound for the optimal number of bubbles in 2D for some specific nonlocal kernels. Finally, we numerically explore the promotion/demotion effect induced by the nonlocal horizon , which is consistent with the theoretical studies presented in our earlier work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989).

Bayesian model calibration for block copolymer self-assembly: Likelihood-free inference and expected information gain computation via measure transport

We consider the Bayesian calibration of models describing the phenomenon of block copolymer (BCP) self-assembly, which is to infer model parameters and their uncertainty from noisy image data produced by microscopy or X-ray scattering characterization of BCP structures. To account for the long-range disorder in BCP structures, we introduce auxiliary variables in the model to represent this aleatory uncertainty. These variables, however, result in an integrated likelihood function for high-dimensional image data that is generally intractable to evaluate. We tackle this Bayesian inference problem using a likelihood-free inference (LFI) approach based on measure transport together with the construction of summary statistics for the image data. We show that expected information gains (EIG) from observed data about the model parameters can be computed with no significant additional cost. Lastly, we present a numerical case study using the Ohta–Kawasaki model for diblock copolymer thin film self-assembly and top-down microscopy characterization. We introduce several domain-specific energy and Fourier-based summary statistics for calibration and quantify their informativeness using EIG. The effect of various data corruptions, summary statistics, and experimental designs on the calibration results are studied using the proposed LFI method.

Linear energy-stable method with correction technique for the Ohta–Kawasaki–Navier–Stokes model of incompressible diblock copolymer melt

An efficiently linear time-marching method is proposed for the incompressible fluid flows coupled Ohta–Kawasaki model of diblock copolymer melt. Although this model satisfies the energy dissipation law under appropriate boundary conditions, the existences of nonlinear terms and coupling terms lead to difficulties of designing energy-stable numerical algorithms. To overcome the difficulties resulting from nonlinear and coupling terms, we herein construct a numerical method based on a highly efficient variant of scalar auxiliary variable approach. The stabilization technique is introduced to maintain the stability with large time steps. To improve the consistency, a simple and practical relaxation strategy is used to correct the difference between time-discretized modified energy and original energy. In each time step, all variables are totally decoupled and we only need to solve linear elliptic equations in a step-by-step manner. The unique solvability and energy stability are analytically proved. Extensive numerical experiments indicate that the proposed method not only achieves desired accuracy, energy stability, and consistency, but also has good capabilities on simulating various hydrodynamically coupled pattern formations.

Nonlocal effects on a 1D generalized Ohta–Kawasaki model

We propose a generalized Ohta–Kawasaki model to study the nonlocal effect on the pattern formation of some binary systems with general long-range interactions. While in the 1D case, the generalized Ohta–Kawasaki model displays similar bubble patterns as the standard Ohta–Kawasaki model, by performing Fourier analysis, we find that the optimal number of bubbles for the generalized model may have an upper bound no matter how large the repulsive strength is. The existence of such an upper bound is characterized by the eigenvalues of the nonlocal kernels. Additionally, we explore the conditions under which the nonlocal horizon parameter may promote or demote the bubble splitting, and apply the analysis framework to several case studies for various nonlocal operators.

Sharp-interface problem of the Ohta-Kawasaki model for symmetric diblock copolymers

The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.

Equilibrium validation in models for pattern formation based on Sobolev embeddings

<p style='text-indent:20px;'>In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.

Maximum Principle Preserving Schemes for Binary Systems with Long-Range Interactions

We study some maximum principle preserving and energy stable schemes for the Allen–Cahn–Ohta–Kawasaki model with fixed volume constraint. With the inclusion of a nonlinear term $$f(\phi )$$ in the Ohta–Kawasaki free energy functional, we show that the Allen–Cahn–Ohta–Kawasaki dynamics is maximum principle preserving. We further design some first order energy stable numerical schemes which inherit the maximum principle preservation in both semi-discrete and fully-discrete levels. Furthermore, we apply the maximum principle preserving schemes to a general framework for binary systems with long-range interactions. We also present some numerical results to support our theoretical findings.

Pathways connecting two opposed bilayers with a fusion pore: a molecularly-informed phase field approach.

A phase field model with two phase fields, representing the concentration and the head-tail separation of amphiphilic molecules, respectively, has been constructed using an extension of the Ohta-Kawasaki model (Macromolecules, 1986, 19, 2621-2632). It is shown that this molecularly-informed phase field model is capable of producing various self-assembled amphiphilic aggregates, such as bilayers, vesicles and micelles. Furthermore, pathways connecting two opposed bilayers with a fusion pore are obtained by using a combination of the phase field model and the string method. Multiple fusion pathways, including a classical pathway and a leaky pathway, have been obtained depending on the initial separation of the two bilayers. The study shed light on the understanding of the membrane fusion pathways and, more importantly, laid a foundation for further investigation of more complex membrane morphologies and transitions.

Dimension Reduction and Optimality of the Uniform State in a Phase-Field-Crystal Model Involving a Higher-Order Functional

We study a Phase-Field-Crystal model described by a free energy functional involving second order derivatives of the order parameter in a periodic setting and under a fixed mass constraint. We prove a $\Gamma$-convergence result in an asymptotic thin-film regime leading to a reduced 2-dimensional model. For the reduced model, we prove necessary and sufficient conditions for the global minimality of the uniform state. We also prove similar results for the Ohta-Kawasaki model.

Energy Stable Semi-implicit Schemes for Allen–Cahn–Ohta–Kawasaki Model in Binary System

In this paper, we propose a first order energy stable linear semi-implicit method for solving the Allen-Cahn-Ohta-Kawasaki equation. By introducing a new nonlinear term in the Ohta-Kawasaki free energy functional, all the system forces in the dynamics are localized near the interfaces which results in the desire hyperbolic tangent profile. In our numerical method, the time discretization is done by some stabilization technique in which some extra nonlocal but linear term is introduced and treated explicitly together with other linear terms, while other nonlinear and nonlocal terms are treated implicitly. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are proved for this method in both semi-discretization and full discretization levels. Numerical experiments indicate the force localization and desire hyperbolic tangent profile due to the new nonlinear term. We test the first order temporal convergence rate of the proposed scheme. We also present hexagonal bubble assembly as one type of equilibrium for the Ohta-Kawasaki model. Additionally, the two-third law between the number of bubbles and the strength of long-range interaction is verified which agrees with the theoretical studies.

How antagonistic salts cause nematic ordering and behave like diblock copolymers.

We present simulation results and an explanatory theory on how antagonistic salts affect the spinodal decomposition of binary fluid mixtures. We find that spinodal decomposition is arrested and complex structures form only when electrostatic ion-ion interactions are small. In this case, the fluid and ion concentrations couple and the charge field can be approximated as a polynomial function of the relative fluid concentrations alone. When the solvation energy associated with transferring an ion from one fluid phase to the other is of the order of a few kBT, the coupled fluid and charge fields evolve according to the Ohta-Kawasaki free energy functional. This allows us to accurately predict structure sizes and reduce the parameter space to two dimensionless numbers. The lamellar structures induced by the presence of the antagonistic salt in our simulations exhibit a high degree of nematic ordering and the growth of ordered domains over time follows a power law. This power law carries a time exponent proportional to the salt concentration. We qualitatively reproduce and interpret neutron scattering data from previous experiments of similar systems. The dissolution of structures at high salt concentrations observed in these experiments agrees with our simulations, and we explain it as the result of a vanishing surface tension due to electrostatic contributions. We conclude by presenting 3D results showing the same morphologies as predicted by the Ohta-Kawasaki model as a function of volume fraction and suggesting that our findings from 2D systems remain valid in 3D.

Low Density Phases in a Uniformly Charged Liquid

This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta-Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase. Under periodic boundary conditions, we prove that the considered energy $\Gamma$-converges to an energy functional of the limit "homogenized" measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplets per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

Effects of thermal fluctuations and block copolymers compositions on defects in directed self-assembly hole shrink process

We investigated the two critical defects on the directed self-assembly hole shrink process, i.e., polystyrene (PS) residue and placement error, by performing dynamic simulations with the Ohta–Kawasaki model. In the simulations, the thermal noise was added to generate stochastic variations in shape and location of the poly(methyl methacrylate) (PMMA) cylindrical domains. For the PS residue issue, we found that the volume fraction of the PMMA minor block, fPMMA, was an effective parameter, and that the PS residue could be minimized by increasing fPMMA from a conventional value of 0.30 to 0.40. On the other hand, the placement error of the PMMA cylindrical domain was affected little by the change in shape and size of the guide hole and by the connectivity to the guide bottom wall. It is speculated that the interfacial stiffness between the PMMA and PS domains would be essential to control the placement error of the PMMA cylindrical domains.

Optimization of directed self-assembly hole shrink process with simplified model

Application of the directed self-assembly of block copolymer to the hole shrink process has gained large attention because of the low cost and high potential for sublithographic patterning. In this study, we have employed a simplified model, called the Ohta-Kawasaki model to find the optimal process conditions, which minimize the morphological defects of the diblock copolymer, PS-b-PMMA. The model parameters were calibrated with cross-sectional transmission electron microscopy images. Our simulation results revealed that it is difficult to eliminate the morphological defects (i.e., PS residual layer) by only varying the shape of the guide hole. It turned out that changing the affinity of the bottom surface of the guide hole from “PMMA attractive” to “neutral” is a more effective way to obtain a reasonably wide, defect-free process window. Note that our simulations are not only computationally inexpensive, but are also comparable to the other detailed models such as the self-consistent field theory; they may also be feasible for large-scale simulations such as the hotspot analysis over a large area.

The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. Droplet Arrangement via the Renormalized Energy

This is the second in a series of papers in which we derive a Γ-expansion for the two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion known as the Ohta–Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. In our previous paper, we computed the Γ-limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order Γ-limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg–Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Thus, without appealing to the Euler–Lagrange equation, we establish for all configurations which have “almost minimal energy” the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of Γ-equivalence, the obtained results also yield an expansion of the minimal energy and a characterization of the zero super-level sets of the minimizers for the original Ohta–Kawasaki energy. This leads to the expectation of seeing triangular lattices of droplets as energy minimizers.

The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. I. Droplet Density

This is the first in a series of two papers in which we derive a $\Gamma$-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In this paper we show that an appropriate setting for $\Gamma$-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki energy functional $\Gamma$-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain a characterization of the limit charge density for the energy minimizers in the diffuse interface model.

Large-Scale Simulations of Directed Self-Assembly with Simplified Model

Directed self-assembly (DSA) is considered as a candidate for future patterning technology for semiconductor manufacturing. The DSA utilizes the phase separation of block copolymer and provides further resolution enhancement by the use of chemically and physically pre-patterned surface. In order for DSA to be a viable lithography solution, it is crucial to realize defect-free manufacturing process. In this study, we utilize the so-called Ohta-Kawasaki (OK) model to simulate the morphological defects of block copolymer formed on the chemically and physically pre-patterned surface. The OK model has advantages of the relatively low computational expense, scalability for large-scale simulations, and compatibility to the other simulation models such as self-consistent field theory (SCFT) through common physical properties of the materials. As test cases, we investigated the lamella defects of symmetric diblock copolymer formed on the chemically and physically pre-patterned surface. For the chemically pre-patterned surface, the two-dimensional (2D) dynamic simulations were performed including the thermal fluctuations, and the time evolution of the lamella defects was characterized as a function of the surface interactive parameter. In the three-dimensional (3D) dynamic simulations of the physically pre-patterned surface, effects of the trench width on the formation of the lamella defects was examined. Our preliminary results demonstrate that various types of the DSA lamella defects can be reasonably predicted by the OK model. It is expected that by calibrating the surface interactive parameters with experimental data, the OK model could be applied to various large-scale DSA simulations, e.g., hotspot analysis over a large area, and design/process optimizations with numerous parameters.

Dynamics of dislocations in a two-dimensional block copolymer system with hexagonal symmetry

Block copolymer thin films have attracted considerable attention for their ability to self-assemble into nanometre-scale architectures. Recent advances in the use of block copolymer thin films as nano-lithographic masks have driven research efforts in order to have better control of long-range ordering in the plane of the film. Irrespective of the method of sample preparation, different quasi-two-dimensional systems with hexagonal symmetry unavoidably contain translational defects, called dislocations. Dislocations control the process of coarsening in the nano/meso-scales and provide one of the most important mechanisms of length-scale selection in hexagonal patterns. Although in the last decade the nonlinear dynamics of topological defects in quasi-two-dimensional systems has witnessed significant progress, still little is known about the role of external fields on the creation and annihilation mechanisms involved in the relaxation process towards equilibrium states. In this paper, the dynamics of dislocations in non-optimal hexagonal patterns is studied in the framework of the Ohta-Kawasaki model for a diblock copolymer. Measurements of the climb and glide velocities as a function of the wave vector deformation reveal the main mechanisms of relaxation associated with the motion of dislocations.