Viscoelastic spectrum melts with

Some information concerning the intramolecular relaxation of the hyperbranched polymers can be obtained from an analysis of the viscoelastic characteristics within the range between the segmental and the terminal relaxation times. In contrast to the behavior of melts with linear chains, in the case of hyperbranched polymers, the range between the distinguished local and terminal relaxations can be characterized by the values of G and G" changing nearly in parallel and by the viscosity variation having a frequency with a considerably different exponent 0. This can be considered as an indication of the extremely broad spectrum of internal relaxations in these macromolecules. To illustrate this effect, the frequency dependences of the complex viscosities for both linear... 

Furthermore, in the line-shape analyses of these viscoelastic spectra, the molecular-weight distributions of P7, FIO, F35, and F80 included in the calculations are identical, respectively, to those extracted from the line-shape analyses of the spectra of the pure melt systems, some of which have been shown in Chapter 10. Thus, the close agreements between theory and experiment in the line-shape analyses have been achieved under the consistency of maintaining the same molecular-weight distributions of the samples between the pure-melt and blend-solution systems. Thus, in a quantitative way, the universality of viscoelastic spectrum is shown extending over the melt and blend-solution systems in accordance with Eq. (11.5) with the molecular weight normalized with respect to Mg for the melt and Mg for the blend solution. 

In the short-time or high-frequency region of the viscoelastic response (relaxation modulus G t), viscoelastic spectrum G (w) or creep compliance J(t)) of a polymer melt with modulus values in the range from lO to 10 °dynes/cm (or compliance values in the range from to... 

There are numerous other constitutive equations of both differential and integral type for polymer melts, and some do a better job of matching data from a variety of experiments than does the PTT equation. The overall structure of the differential equations is usually of the form employed here The total stress is a sum of individual stress modes, each associated with one term in the linear viscoelastic spectrum, and there is an invariant derivative similar in structure to the one in the PTT equation, but with different quadratic nonlinearities in t and Vv. The Giesekus model, for example, which is also widely used, has the following form ... 

The Phan-Thien/Tanner constitutive equation does not represent the state of the art in modeling melt flow at the time of this writing, but it is adequate to illustrate the response of melts of flexible polymers in complex flows and it has a mathematical structure that does not differ substantively from other equations with a firmer basis in molecular theory. Furthermore, it has been widely used in simulation studies to date. Hence, we will use it for illustrative purposes in this text, recognizing that it is likely to be replaced as the preferred constitutive equation for applications. The minimum rheological information required for simulations is thus the temperature-dependent linear viscoelastic spectrum and the temperature-dependent viscosity as a function of shear rate. Extensional data should be used, but they are often unavailable when the PTT equation is employed it is therefore common to select a reasonable value of to describe the extensional response. 

As mentioned in the beginning of this review (see Sect. 1), besides the theoretical importance of modelling and experiments in extension of molten polymers, there is an increasing interest in this field of rheology and mechanics of viscoelastic fluids from the technological point of view. This is connected with a wide spectrum of applied problems, the solution of which is based on data on melt extension. Below we shall discuss... 

In contrast to simple elastic solids and viscous liquids, the situation with polymeric fluids is somewhat more complicated. Polymer melts (and most adhesives are composed of polymers) display elements of both Newtonian fluid behavior and elastic solid behavior, depending on the temperature and the rate at which deformation takes place. One therefore characterizes polymers as viscoelastic materials. Furthermore, if either the total strain or the rate of strain is low, the behavior may be described as one of linear or infinitesimal viscoelasticity. In such a case, the stress-deformation relationship (the constitutive equation) involves not just a single time-independent constant but a set of constants called the relaxation spectrum,(2) and this, too, may be determined from a single stress relaxation experiment, or an experiment involving small-amplitude oscillatory motion. 

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