Review on tube model based constitutive equations for polydisperse linear and long-chain branched polymer melts

Rheological equations of state are of great importance for characterization of polymer melts and for simulation of polymer processing. This concise review considers tube model based constitutive equations developed in the last 40 years since the original publication of Doi and Edwards in 1978. The emphasis is on the concepts, assumptions, and material parameters introduced to model nonlinear viscoelasticity of polydisperse linear and long-chain branched polymer melts. Linear viscoelasticity is assumed to be known, either from linear-viscoelastic modeling or by experimental characterization. The scope is limited to constitutive equations which are based on the linear-viscoelastic relaxation modulus and can be expressed in terms of integral, differential, or integro-differential equations. Multimode models based on the concept of preaveraged stretch require a large number of nonlinear model parameters. Relaxing the assumption of a constant tube diameter, the number of nonlinear model parameters can be drastically reduced to 2 or 3, independent of the number of Maxwell modes needed to represent the linear viscoelasticity.