Mixed homopolymer brushes grafted onto a nanosphere

Microphase separation of mixed A∕B polymer brushes grafted onto a nanosphere with its radius comparable to the size of polymers is investigated by numerical implementation of the self-consistent field theory. The idea is to embed the sphere within a larger cubic computational cell and use a "masking" technique to treat the spherical boundary. The partial differential equations for the chain propagator on the sphere can thus be readily solved with an efficient and high-order accurate pseudospectral method involving fast Fourier transform on a cubic cell. This numerical technique can circumvent the "pole problem" due to the use of a spherical coordinate system in conventional finite difference or finite element grid. We systematically investigate the effect of the total grafting density, composition, chain length asymmetry between two grafted homopolymers as well as spherical radius, i.e., substrate curvature on the formation of island structure with specific arrangement in a regular lattice. A series of island structures with different island numbers representing specific structure symmetry ranging from 2 to 12 except for 11 are found, in contrast to conventional hexagonal arrangement for polymer brushes on a planar substrate. Among these parameters, the spherical radius plays a significant role in determining the type of island structures, i.e., the morphology formed on the sphere.