Scaling theory for radial distributions of star polymers in dilute solution in the bulk and at a surface. II. ε expansion for monomer densities

Abstract

The behavior of monomer density profiles of a star polymer in a d-dimensional good solvent, which was predicted in an earlier paper using scaling arguments, is now derived by using the renormalization group ε=4−d expansion method. Both the case of a free star in the bulk and of a center-adsorbed star at a free surface are considered. In the latter case of a semi-infinite problem, a distinction is made between repulsive walls, attractive walls—where for large arm length l, the configuration of the star is quasi-(d−1)-dimensional—and ‘‘marginal walls,’’ where for l→∞ the transition from d-dimensional to (d−1)-dimensional structure occurs. For free stars, ρ(r) behaves as r−d+1/ν for small r, where ν is the exponent describing the linear dimensions of the star, e.g., the gyration radius Rgyr∼lν. For center-adsorbed stars at repulsive or marginal walls, ρ(r∥,z) behaves as ρ(r∥,0)∼r−d+λ( f )∥ and ρ(0,z)∼z−d+1/ν, where r∥ and z denote the distances parallel and perpendicular to the surface, respectively; the new exponent λ( f ) depends explicitly on the number of arms f in general. We calculate this exponent λ( f ) to first order in ε=4−d; then λ( f ) is obtained to be (f−1)ε/4+𝒪(ε2) for repulsive walls and 2−ε/4+𝒪(ε2) for marginal walls.