Generalized Ohta-Kawasaki Model
Open Access DepositedTheory and Numerics
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In this doctoral dissertation, we introduce a generalized Ohta-Kawasaki (GOK)model to investigate the nonlocal effect on the pattern formation of some binary systems with general long-range interactions. In the one-dimensional case, the GOK model exhibits bubble patterns similar to those of the standard Ohta-Kawasaki (OK) model. However, through Fourier analysis, we discover that the generalized model may impose an upper limit on the optimal number of bubbles, regardless of the magnitude of the repulsive strength. The existence of such an upper bound is determined by the eigenvalues of the nonlocal ker- nels. Furthermore, we explore the conditions under which the nonlocal horizon parameter either promotes or demotes bubble splitting, and apply the analytical framework to several case studies involving different nonlocal operators. Next, we study the asymptotic compatibility of the Fourier spectral method for the GOK model in multidimensional spaces. Using Fourier collocation discretization for the spatial variables, we show that asymptotic compatibility holds in both two- and three- dimensional periodic domains. For temporal discretization, we adopt the second-order backward differentiation formula (BDF) method and prove that the proposed time dis- cretization schemes preserve the energy dissipation law for certain nonlocal kernels. In numerical experiments, we validate the asymptotic compatibility, confirm the second-order convergence rate in time, and verify the energy stability of the proposed schemes. Notably, we discover a novel square lattice pattern that emerges when specific nonlocal kernels are applied to the model. Furthermore, the numerical results confirm the existence of an upper bound for the optimal number of bubbles in two dimensions for some specific nonlocal kernels. Finally, we explore the effects of the nonlocal horizon parameter δ, observing promotion or demotion of bubble splitting. These findings consistent with the theoretical studies in 1D case. Next, we present spectral methods for the standard Ohta-Kawasaki (OK) and Nakazawa- Ohta (NO) models in disk and spherical domains, representing diblock and triblock copoly- mer systems, respectively. These methods focus on studying coarsening dynamics and equilibrium pattern formations for the OK and NO models. For spatial discretization, we apply the ultraspherical spectral method in the disk domain and the Double Fourier Sphere (DFS) method in the spherical domain. For temporal discretization, we adopt the second- order backward differentiation formula (BDF) method and show the energy stability of the numerical schemes in both semi-discrete and fully discrete discretizations. To the best of our knowledge, this study is the first to develop numerical methods for diblock and triblock copolymer systems with long-range interactions in the disk domain. In our numerical experiments, we validate the second-order temporal convergence rate and confirm the energy stability of the proposed scheme. Our results show that coarsening dynamics in diblock copolymers lead to bubble assemblies both inside the interior and along the boundary of the disk. For the triblock copolymer system, we observe several novel pattern formations, including single- and double-bubble assemblies in the unit disk. These findings are detailed through extensive numerical experiments. For the spherical domain, the numerical experiments show several pattern forma- tions for both the OK and NO models. These include single-bubble assemblies in binary systems and different type of bubble assemblies in ternary systems, such as double-bubble and mixed-bubble assemblies. Importantly, these numerical results successfully reproduce experimental biomembrane patterns, highlighting the predictive accuracy of the OK model for applications in biology. Furthermore, we numerically explore the relationship between the repulsive strength and the number of bubbles in the assemblies, verifying the two-thirds law in the OK and NO models. This analysis provides quantitative insights into the param- eter dependencies in copolymer systems.
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