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 accounting for the repulsion between the A monomers and the B monomers, 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. The theory is phenomenological in the sense that in the Landau expansion of the free energy the third and fourth order vertexes are approximated by constants, while the second order vertex is approximated by a quadratic function of the wave vector. As a consequence, the details of the architecture are not taken into account, and within the scope of the theory, our results apply equally well for multiblock copolymers and triblock copolymers, though the mapping from the rescaled parameters to the physical parameters would be a bit different. We restrict our analysis to the region of the parameter space where the interaction strength chi is close to its spinodal value Os, the composition f is close to the critical composition, the plate preference h is small, and the film thickness d is close to a small multiple of the natural periodicity L-0 = 2 pi/q(0) (the first three conditions define the weak segregation regime). We will present our results in the form of phase diagrams in which Delta similar to vertical bar(d/nL(0) - 1)/epsilon vertical bar is placed along the horizontal axis, and T similar to (chi - chi(s))/epsilon(2) is placed along the vertical axis, where epsilon is a measure for the composition asymmetry (f - 0.5). We present a series of such phase diagrams for increasing values of the rescaled parameter H = h epsilon(-3). We find that if Delta is small, corresponding to films whose thickness is commensurate with the bulk periodicity, parallel orientations of the structures are favored over perpendicular orientations. We also predict that on increasing the value of the parameter H, the region of stability of the bcc phase shrinks.