Effect of Polymer Architectures on the Entanglement of Combs and Bottlebrushes

We study correlations between the entanglement plateau modulus and architecture of graft polymers in a melt. To distinguish between two types of graft polymers – combs and bottlebrushes – we introduce the crowding parameter Φ, which describes mutual interpenetration of the neighboring macromolecules. In comb systems, both the backbones and sparsely grafted side chains are coiled and allow side chains of neighboring macromolecules to overlap (Φ<1). In bottlebrush systems, however, the steric repulsion between densely grafted side chains results in chain extension and inhibits side chain interpenetration (Φ≥1).The ratio G_e,gr/G_e,lin≈φ³(1+(Φ/0.7)³) of the plateau modulus of a graft polymer melt, G_e,gr, to that of a linear polymer melt, G_e,lin, is a universal function of the crowding parameter Φ≈φ^-1n_sc^-1/2 and graft polymer composition φ=n_g/(n_g + n_sc), where n_sc and n_g are the degrees of polymerization of side chains and a spacer separating consecutive side chains along the backbone, respectively. Such universal behavior is verified by poly(n-butyl acrylate) combs and other graft polymers reported in literatures. For graft polymers with entangled side chains, the G_e,gr/G_e,lin ratio is proportional to φ².