Molecular constitutive equation for unentangled branch copolymers

The dynamics and linear viscoelasticity of the heterogeneous polymeric system with complex molecular topology and semiflexibility are studied in this work. From the viewpoint of the bead-spring chain, the multi-intensity springs, multi-friction beads, and multi-site restrictions are systematically considered for the heterogeneity of the chain. These heterogeneities are incorporating into a general constitutive equation without entanglement effect based on graph theory, the maximum entropy principle, and Brownian dynamics. The general constitutive equation is validated for a series of bottlebrush copolymers. The predictions of the dynamic moduli, the longest relaxation time, and the zero-shear viscosity are well consistent with experimental observations. Besides, we provide a method to separate the modulus of local chain segments (such as arm or backbone) from the apparent modulus. The dependencies of the distribution of eigen relaxation modes and the dynamic moduli on the heterogeneities are also discussed.