Linear viscoelastic range viscosity

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

In principle, the shear viscosity function in the linear viscoelastic range reduces to... 

In addition to the observations in shear that have just been discussed, there is considerable interest in the elongational response in polymeric fluids. The elon-gational or Trouton viscosity ije is three times the zero shear rate viscosity in the linear viscoelastic range. However, its behavior is more complex and, as shear rates get higher, the viscosity can even go through a maximum with increasing shear rate as illustrated in Figure 22 (77). 

Only a few non-linear viscoelastic properties have been studied with polymers of well-characterized structure. The most prominent of these is the shear-rate dependence of viscosity. Considerable data have now been accumulated for several polymers, extending over a wide range of molecular weights and concen-... 

At sufficiently low shear rates the viscosity is constant. If a constant shear rate in this range is imposed, the shear stress should grow monotonically to its steady state value c(oo) according to the equation from linear viscoelasticity ... 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on pol rmers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 

The Doi-Edwards theory of linear viscoelasticity predicts 1.2 for JeG, where is the plateau shear modulus. This value is significantly lower than typical experimental values found in the range 2—4 [3, 71]. This defect of the Doi-Edwards theory, along with its failure of predicting the 3.4 power law for viscosity, has been pointed out by Osaki and Doi [72]. It is associated with the fact that the Doi-Edwards theory yields a relaxation time distribution which is too narrow compared with observed ones [69]. Modifications of the Doi-Edwards theory have been made so as to bring JeG closer to measured values, but no remarkable success has as yet been achieved. 

Gagon and Denn estimated the linear viscoelastic parameters for two modes from Gregory s viscosity and relaxation time data by choosing a wedge spectrum, in which Gi = rjo/NXi, where N is the number of relaxation modes. They arbitrarily chose X2/X1 = 5. The parameter f was taken to be zero to reflect the shear insensitivity of the viscosity of PET, while e was taken to be 0.015. The results were insensitive to in the range from 0 to 0.015, so the simulation was in effect carried out for a two-mode Maxwell model. The results depended on the choice of the coefficient for the air drag coefficient, but the variation was small in the range 0.37 to 0.6. 

Most concentrated structured liquids shown strong viscoelastic effects at small deformations, and their measurement is very useful as a physical probe of the microstructure. However at large deformations such as steady-state flow, the manifestation of viscoelastic effects—even from those systems that show a large linear effects—can be quite different. Polymer melts show strong non-linear viscoelastic effects (see chap. 14), as do concentrated polymer solutions of linear coils, but other liquids ranging from a highly branched polymer such as Carbopol, through to flocculated suspensions, show no overt elastic effects such as normal forces, extrudate swell or an increase in extensional viscosity with extension rate [1]. 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

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... 

With this choice of j8, the theoretical picture presented in this section is consistent, for example, with the range in MJM (from about 2 to about 20) explored by the polyisoprene (PI) data set of Fetters et al. [5] using a single value of Mg of 5000 g mol" The viscosities of these melts cover five decades in magnitude yet the current theory, together with values for Gq and Tg consistent with data on linear PI predicts the entire range of viscoelastic spectra (see Fig. 9). 

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. To model this material behavior, viscoelasticity utilizes spring constants ( i and E2), dashpots (viscosity, v), and St. Venant sliders (a slider to account for nonrecoverable deformation) elements. The properties of these elements may be selected to cover a wide range of elastic and time-dependent viscous behavior. Viscoelastic models can be divided into both the number of elements employed and whether the elements are in a series or parallel arrangements. These elements may be linear or nonlinear and are combined as necessary for the model to describe the behavior of the sediment imder study. These models describe short-term behavior reasonably well, but tend to not yield reliable predictions of deformation for extended time periods. 

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