Partial localisation in a variational model for diblock copolymer-homopolymer blends

Patterns are everywhere in nature and the mathematical challenges in trying to describe and understand them are big. In the field of energy driven pattern formation diblock copolymers play a paradigmatic role. The combination of a huge experimental literature with theoretical approaches have led to the mathematically accessible Ohta- Kawasaki variational model (1986), which describes diblock copolymer melts on a mesoscopic scale. In recent years this model has flourished in the mathematical literature because it exhibits a rich pattern forming behaviour and is manageable by mathematical means, although it still has not revealed its deepest secrets. A diblock copolymer molecule consists of two different types of monomer molecules (U and V) which are covalently bonded together. At low temperatures both monomer types repel each other, but the chemical bond limits their movement away from each other. In melts consisting of many such molecules the diblock copolymers will arange themselves in such a way as to find a middle ground between these two competing influences—large scale repulsion and small scale tendency to mix. As a result patterns form, ranging from lamellar patterns if both monomer types are present in equal measure to little spherical regions of one monomer type embedded in a large sea of the second type if the types’ ratio is skewed. In this thesis we focus on a special type of patterns: partially localised patterns. These are extended in some directions and small in the others. One can think of structures resembling a curve or surface in three-dimensional space. The former are extended in one direction and small in the other two, while the latter are small in one direction and extended in the remaining two. These kind of patterns are not observed in diblock copolymer melts, nor in the Ohta-Kawasaki model for them, which is why we study slightly more complex systems: diblock copolymer-homopolymer blends. Such a blend contains next to diblock copolymer molecules also homopolymer molecules, which are built up out of a single type of monomer molecules. This extra ingredient increases the variety of patterns that can form and in particular it opens up the possibility of partially localised patterns forming. When there is much more homopolymer than diblock copolymer one might expect the latter to form partially localised structures in a background of homopolymer. We model these blends by the sharp interface limit of an extension of the Ohta- Kawasaki energy which incorporates the homopolymers as well as the diblock copolymers and consider the case where there are equal and specified amounts of U- and V-monomer in the diblock copolymer and the homopolymer fills up the remaining space. In one dimension we show that minimisers of the energy are generically concatenations of one-dimensional monolayers, i.e. sequences of alternating intervals containing either monomers of type U or type V, tucked in between two large intervals of homopolymer. A stability analysis for two-dimensional mono- and bilayers proves that for both there is a parameter regime in which they are stable, but that they can also be unstable. The bilayer is always stable, independent of other parameters, as long as the repulsion between the U- and V-monomers is strong compared to the repulsion between the homopolymer and either the U- or V-monomers. The preference of the U- and V-monomers to mix on small scales, physically caused by the covalent bond, is expressed in the model via the H-1-distance between the Uand V-rich regions. To study the effects of this term in greater detail we use the method of G - convergence. Our results show that when the effect of the H-1-norm is localised around curves in the plane, it prefers these curves to bend rather than break and to break rather than to stretch. The question what minimisers for the extended Ohta-Kawasaki energy in more than one dimension look like in detail is still open, as it is for the original Ohta-Kawasaki model, but our results indicate that in certain parameter regimes and on certain domains flat bilayers are good candidates, while in other situations curved structures are preferable.