Scaling theory for radial distributions of star polymers in dilute solution in the bulk and at a surface, and scaling of polymer networks near the adsorption transition

Abstract

Monomer density profiles ρ(r) and center–end distribution functions g(rCE) of star polymers are analyzed by using a scaling theory in arbitrary dimensions d, considering dilute solutions and the good solvent limit. 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 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. For center-adsorbed stars at attractive walls, ρ(r∥,z) behaves as ρ(r∥,0)∼r−(d−1)+1/ν(d−1)∥, ν(d−1) being the exponent describing (d−1)-dimensional stars, while ρ(0,z) decays exponentially. On the other hand, the center–end distribution function at short distances is described by nontrivial exponents. For free stars with f arms, g(rCE)∼(rCE)θ( f ) for small rCE, where θ( f ) is expressed in terms of the configuration-number exponent γ( f ) and the exponent γ of linear polymers as θ( f ) =[γ−γ( f+1) +γ( f )−1]/ν. For center-adsorbed stars, at repulsive or marginal walls gs(rCE∥,ze) behaves as gs(rCE∥,0) ∼(rCE∥)θ∥( f ), gs(0,zE) ∼(zE)θ⊥( f ) with θ∥( f ) =[γ1−γs( f+1) +γs( f )−1]/ν and θ⊥( f ) =[γ−γs( f+1) +γs( f )−1]/ν, γ1 being the exponent of a linear polymer with one end at the surface. The scaling theory of general polymer networks at the adsorption transition is also presented. The configuration-number exponent γ𝒢 for a polymer network 𝒢 with nh h functional units in the bulk, n′h h-functional units at the surface and totally composed of f linear polymers with the same length is given by γSB𝒢 =α−1−f+ν +∑∞h=1[nhΔh +nhΔSBh]. Δh and ΔSBh are related, respectively, to the exponents of star polymers as γ( f )=α−1+(γ−α)f/2+Δf and γSBs( f ) =α−1+ν+(γ−α)f/2 +ΔSBf, with α given by α=2−νd. The exponent γSBs( f ) is evaluated by means of the renormalization-group ε=4−d expansion to the first order.