The Viscometric Functions

If a creep or start-up shearing test is continued until the stresses reach their steady-state values, the rheological response of the material is described completely by three functions of the shear rate. These are the viscosity and the first and second normal stress diiferences, which were defined by Eqs. 10.33 and 10.34. The three material functions of steady simple shear fliy) liy) and N2iy) are called the viscomern cywncrions, and they provide a complete description of the behavior in steady simple shear of an isotropic polymer, i.e., one that does not form a liquid crystal or another ordered phase at rest. 

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In example 2.2 we obtained that for steady shearing flows the viscometric functions for this constitutive equation are defined by... 

Comment how the viscometric functions for the shear flow of a Lodge rubber-liquid develop in Example 2.4 compare with experimental observations. 

The connection between the double value of the slip parameter obtained from the viscometric functions and the violation of the Lodge-Meissner rule becomes more evident when the time-strain separability of the model is considered. For this purpose, the Johnson-Segalman model can be rewritten under the form of a single integral equation, cancelling the Cauchy term, which gives the following form in simple shear flows ... 

The uniaxial extensional viscosity rj(s) and the viscometric functions rj(y) and ki(y), predicted by the Doi-Edwards model for monodisperse melts, are shown in Fig. 3-32. The Doi-Edwards model predicts extreme thinning in these functions the high-shear-rate asymptotes scale as 17 oc oc y , and4 i oc The second normal... 

We shall now examine the effects produced by the stresses generated during the reorientation process by calculating the viscometric functions that relate the shear and normal stress differences. For a planar geometry and using the convention in (Bird R. et al 1971) the first normal stress difference is defined by... 

From Eq. 83 we observe that the viscometric functions are insensitive to the direction of shear and that the primary normal stress coefficient is zero. Hence this is not a realistic model for most shear sensitive fluids. Eq. 82, with i = fi = constant and / 2 = 0 is the Newtonian fluid. If we keep ii shear rate dependent and set /t,2 = 0, we then have the GNF. Several special cases of the GNF are discussed below. 

The fact that both the shear stress and shear rate are independent of position in the gap is what makes the cone and plate arrangement so desirable for steady shear characterization. To get the normal stresses we note that, since cot0 0, Eq. 1672 indicates oee = ( ) Furthermore, since the viscometric functions... 

Of the viscometric functions, the viscosity is by far the easiest to measure and is thus the most often reported. As in the case of Newtonian fluids, the viscosity of a polymer depends on temperature and pressure, but for polymeric fluids it also depends on shear rate, and this dependency is quite sensitive to molecular structure. In particular, the curve of viscosity versus shear rate can be used to infer the molecular weight distribution of a linear polymer as is explained in Chapter 8. And in certain cases it can also tell us something about the level of long-chain branching. This curve is also of central importance in plastics processing, where it is directly related to the energy required to extrude a melt. 

Instead of carrying out equilibrium averaging, one can introduce a consistently averaged hydro-dynamic interaction , in which the averaging is not performed with the equilibrium distribution function, but rather with a distribution function that is consistent with the local flow field.This leads to shear-rate dependence for all the viscometric functions, and also to a nonzero (but positive) value of the second normal stress coefficient. 

Because of the fact that equation (118) does not give shear-rate dependence for the viscometric functions, many other postulates have been made for the segment creation and loss rates, and some other modifications have been made in the theory. If equation (113) is replaced by Q=l(/c — 2 y) Q] (c/ last paragraph of Section 8.5.1.3), then the constitutive equation becomes... 

The need to measure fluid properties at shear rates higher than those accessible with rotational viscometers arises because deformation rates can easily reach 10 -10 sec in polymer processing operations. To attain these high shear rates, we use flow through capillaries or slits and calculate the viscometric functions from a knowledge of the pressure drop-versus-flow rate relationship. 

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