Universality of the Entanglement Plateau Modulus of Comb and Bottlebrush Polymer Melts

A combination of scaling analysis and rheological experiments was used to study correlations between the entanglement plateau modulus and grafting density of graft polymers in a melt. Using the crowding parameter Phi, which describes overlap of side chains belonging to neighboring macromolecules, we identified two classes of graft polymers combs and bottlebrushes that demonstrate distinct conformational and rheological behaviors. In comb systems, both the backbones and sparsely grafted side chains are coiled that allow side chains of neighboring macromolecules to overlap (Phi< 1). In bottlebrush systems, steric repulsion between densely grafted side chains causes chain extension and inhibits side chain interpenetration (Phi >= 1). The ratio G(e,gr)/G(e,lin) similar or equal to phi(3)(1 + (Phi/0.7)(3)) of the plateau modulus of a graft polymer melt, G(e,gr), to that of a melt of linear chains, G(e,lin), is a universal function of the crowding parameter Phi congruent to phi(-1)n(sc)(-1/2) and graft polymer composition phi = n(g)/(n(g) + n(sc)), where n(sc) and n(g) are the degrees of polymerization of side chains and a spacer separating two consecutive side chains along the polymer backbone, respectively. This dependence of the plateau modulus is verified for poly(n-butyl acrylate) combs and other graft polymer systems reported in the literature. In a special case of graft polymers with long entangled side chains, the G(e,gr)/G(e,lin) ratio is proportional to phi(2).