Self-Assembly of Lamellar Microphases in Linear Gradient Copolymer Melts

The ability to create `designer copolymers' with tunable properties by tailoring their monomer composition has garnered recent interest in their molecular self-assembly. Here we investigate lamellar microphases in linear gradient binary copolymer melts using a variety of techniques, including solutions of self-consistent field equations, scaling theory, and analysis of the strong-segregation limit. The Flory scaling theory predicts the scaling of the equilibrium lamellar width $L_{eq}$ as a function of comonomer incompatibility as characterized by \textit{$\chi $}. From the strongly segregated limit there are conformational fluctuations, and it is the tradeoff between the entropic effect of these relative to repulsive comonomer interactions that determines $L_{eq}$. We discover that $L_{eq}$ /$R_{g} \quad \sim $ (\textit{$\chi $N})$^{1/6}$; remarkably, this is the same result as for symmetric diblock copolymers, although for quite different physical reasons.