Microphase separation in thin block copolymer films: a weak segregation   mean-field approach
  In this paper we consider thin films of AB block copolymer melts confined between two parallel plates. The plates are identical and may have a preference for one of the monomer types over the other. The system is characterized by four parameters: the Flory-Huggins chi-parameter, the fraction f of A-monomers in the block copolymer molecules, the film thickness d, and a parameter h quantifying the preference of the plates for the monomers of type A. In certain regions of parameter space, the film will be microphase separated. Various structures have been observed experimentally, each of them characterized by a certain symmetry, orientation, and periodicity. We study the system theoretically using the weak segregation approximation to mean field theory. We restrict our analysis to the region of the parameter space where the film thickness d is close to a small multiple of the natural periodicity. We will present our results in the form of phase diagrams in which the absolute value of the deviation of the film thickness from a multiple of the bulk periodicity is placed along the horizontal axis, and the chi-parameter is placed along the vertical axis; both axes are rescaled with a factor which depends on the A-monomer fraction f. We present a series of such phase diagrams for increasing values of the surface affinity for the A-monomers. We find that if the film thickness is almost commensurate with the bulk periodicity, parallel orientations of the structures are favoured over perpendicular orientations. We also predict that on increasing the surface affinity, the region of stability of the bcc phase shrinks. 
Hindrik Jan Angerman, Albert Johner, Alexander N. Semenov