Linear viscoelastic models

Viscoelasticity and temperature dependences are explained using molecular dynamics and control theory. We have previously (Borg and Pääkkönen, 2009 [1-3]) applied control theory to model the relationship between the relaxation modulus, dynamic and shear viscosity, transient flow effects, power law and Cox-Merz rule related to the molecular weight distribution (MWD), and here these topics are discussed more generally. In this paper we show the direct simple relation to molecular dynamics using structural models comprising dumb-bells (Bird et al., 1987 [4]) with internal viscosity and elasticity in a statistical tube. The dumb-bell model is used to obtain the linear relation to the elasticity P′ value of function P′(ω) and the relation to the viscosity P″ value of function P″(ω) from chain friction. The applied principle is also valid for the relaxation modulus or shear viscosity. A new principle is presented for obtaining absolute values such as zero viscosity by modelling, which is first used to obtain absolute values for a target point at a high rate for unentangled chains (since close relaxed states of chain topology are much more complicated). An analytical model for the temperature dependency of viscoelastic flows is presented, which is many times more accurate than WLF or Arrhenius equations. Control theory and variations of tube diameter as a function of temperature gives linear relation between chain dynamics and viscoelastic properties. New compact formulas are presented to simultaneously model different polymer flows and temperatures. We have also found that the MWDs computed from the relaxation modulus or complex and the shear viscosity are not temperature sensitive, in contrast to what time-temperature superposition (TTS) suggests, although absolute viscoelastic values make them appear very temperature-dependent. TTS is verified for thermorheologically simple materials, and the reasons for it not holding are explained.