Thermodynamic theory of viscoelasticity

The relaxation spectra in polymers arise from the existence of many possible modes for dissipating the strain energy raised by the imposed force. These modes are made up by coupling the simplest and fastest mode of relaxation involving the rotation of a conformer, typically represented by the picosecond rotation of the carbon to carbon bond. This fast relaxation process cannot take place easily in the condensed state crowded by the densely packed conformers, necessitating cooperativity among them. The domain of cooperativity grows at lower temperatures, toward the infinite size at the Kauzman zero entropy temperature. From the temperature dependence of the domain size, the well-known Vogel equation is derived, which is numerically equivalent to the empirical WLF and free volume equations. The molar volume is a crucial factor in determining the molar free volume and, therefore, in determining theT g of a material. The molar ΔC P is proportional to the logarithmic molar volume, and is greater for a polymer with a higherT g, but ΔC P per gram for it is smaller, as it is proportional to (logM) divided byM, whereM is the molecular weight of the conformer. From this theory, it is possible to predict the dependence of the characteristic relaxation time on temperature if eitherT g or the conformer size is known, since one can be derived from the other. From the Vogel equation with all parameters thus derived, it is possible to obtain a master relaxation curve and the spectrum from one set of dynamic mechanical data taken at one frequency over a range of temperatures. Whereas the linear viscoelastic principle is limited to small strains only, a real polymer is often deformed well beyond such a limit. Above a certain limit of strain energy level, linear viscoelastic deformation is no longer possible and the plastic deformation takes over. However, because a polymer typically manifests a spectrum of relaxation times, its behavior is a combination of viscoelastic and plastic behaviors. The ratio between the two behaviors depend on the rate of deformation, and can be precisely predicted from the linear viscoelastic relaxation spectrum. The combined behavior is termed viscoplasticity, and it applies to a wide range of practically important mechanical behaviors from the flow of the melt to the yield and fracture of glassy and crystalline solids.