Pattern Formation in Binary Systems with Inhibitory Long-range Interaction
Open AccessIn this thesis, we investigate pattern formation in a two-phase system based on the Otha-Kawasaki model for diblock copolymers. In the Ohta-Kawasaki model, the total energy of the systemincludes a short-range term - a Landau free energy and a long-range term - the Otha-Kawasakifunctional. The short-range term favors large domains with minimum perimeter and the long-rangeinhibitory term favors small domains. The balance of these terms leads to minimizers that possess avariety of patterns, including single droplets, droplet assemblies, stripes, wriggled stripes and combinationsthereof. We consider pattern formation in a two-dimensional domain with boundary (theinterior of an ellipse) and a two dimensional compact domain without boundary (the ellipsoid). Forboth domains, we consider a two-pronged approach. First, we consider the sharp interface model ofthe Ohta-Kawasaki system. This enables us to approximate the distribution of droplets in an equilibriumconfiguration. The sharp interface problem utilizes the Green’s function and a related functionthat we call the remnant function. The interaction of curvature with the nonlocal term influences theposition of a single droplet solution when an inhibitory parameter is small. The Green’s functionencodes the geometry of the domain into its construction. The two domains that we consider areconformal to domains (i.e., the unit disk in R2 and the unit sphere in R3) where the Green’s functionsare well-known. Using properties of conformal mappings, we are able to construct Green’sfunctions for the ellipse and the ellipsoid. For the ellipse, there exist stable assemblies consistingof boundary droplets, interior droplets and a combination of interior and boundary droplets. Sincethe boundary of the ellipsoid is empty, all droplets are interior. Sharp interface analysis typicallyassumes that the droplet radii are small. To analyze domains where the droplet size is moderateto large, we consider finite element solutions of the diffusive interface problem for both the ellipse domain and the ellipsoid domain. Using the estimates for droplet size and droplet location fromthe sharp interface model, we are able to develop a good initial guess for a solution of the diffusiveinterface model for certain disk assemblies. Moreover, we are able to consider equilibria wherethe droplet size is not necessarily small. Numerical solutions with stripes, wriggles, elongateddroplets, dog-bone shaped droplets are computed. These may be discovered by numerical investigationsof the diffusive model alone. Our method of solution for the diffusive interface problem isoptimization-based and utilizes an interior-point large-scale algorithm. We compute analytic expressionsfor the gradient and Hessian of the total energy of the system. Thus, stability of a numericalsolution can be assessed. We found this approach to be robust. In general, the solution space forthis class of problems is quite complicated. There are numerous local minima. While there may bea global minimum, it need not be unique. Nevertheless, using a combination of analysis and numericalcomputation we are able to assert conditions under which an equilibrium configuration existsand when it does not. The results on domains with boundary and compact domains leads to someinteresting questions regarding the ”efficient packing of droplets” in a domain that is topologically(as well as geometrically) constrained. While our analyses of the sharp interface problems reliedon computing the Green’s function, our analysis of the diffusive interface problem is not restrictedto domains that are conformal to simple domains such as the unit disk or unit sphere. Moreover,our numerical approach for the diffusive interface problem can be adapted to higher genus compactsurfaces.
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