Energy transfer in the restricted geometry of lamellar block copolymer interfaces

We carried out simulations of energy transfer kinetics for lamellar block copolymer systems in which donor and acceptor dyes were attached to the block junctions. We considered blocks of homopolymers that were sufficiently immiscible and of sufficiently high molecular weight to employ the Helfand-Tagami distribution of block junctions. The morphology of such block copolymers has been frequently discussed in terms of an apparent dimension parameter, which is recovered from the analysis of fluorescence decay curves, using the Klafter--Blumen (KB) formalism. Here, we investigate how such apparent dimensions are influenced by the interface thickness between the two blocks (which is dependent on the Flory-Huggins chi parameter of the system). We also probe the dependence of this apparent dimension on the concentration of the dyes in labeled samples. This kind of dependence has been experimentally observed but never explained, perhaps because of the approximations inherent in using the KB model to analyze fluorescent decay curves for block copolymer systems. We have found that apparent dimensions approach three for reasonably broad interfaces, but decrease to near two for very narrow interfaces, in accordance with asymptotic formulas that we propose for strongly segregated, lamellar block copolymer melts. Global analysis of the decay curves, as well as weighted linear regression of the parameters obtained from individual analyses of the decays, suggest linear relationships between the apparent dimensions from KB analyses and acceptor concentrations. We discuss the dependence on interface thickness in terms of the basic (Forster) theory of direct energy transfer, and indicate why the KB model is a reasonable representation of lamellar block copolymers.