Vanishing Shear Rate

A complete discussion of the dependence of the steady-flow viscosity at very small shear rates, iio, on temperature, molecular weight, and molecular weight distribution is outside the scope of this book it is available in other sources. - The theoretical basis has been given in Chapters 10 and 11, and the relation to chemical structure through the monomeric friction coefficient has been pointed out in Section B4 of Chapter 12. A few additional comments are pertinent. 

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We can also calculate other viscoelastic properties in the limit of low shear rate (linear viscoelastic limit) near the LST. The above simple spectrum can be integrated to obtain the zero shear viscosity 0, the first normal stress coefficient if/1 at vanishing shear rate, and the equilibrium compliance J... 

The term director is usually only defined in the limit of vanishing shear rate where the order parameter tensor has uniaxial symmetry. Even at finite shear rates, however, the order parameter tensor S remains well-defined it can be represented pictorially by an ellipsoidal shape, whose major axis is the largest eigenvalue of S. If, for example, the major axis of S lies in the x y plane, then its angle 9 measured clockwise with respect to the x direction is given by... 

Predictions of the (idealized) MCT equations for the potential part of the equilibrium, time-dependent shear modulus ( ) of hard spheres for various packing fractions 0 are shown in Fig. 3 and calculated from the limit of (lid) for vanishing shear rate ... 

The normalized density fluctuation functions are calculated self-consistently within MCT from (13, 14) at vanishing shear rate [2, 38], which turn into quiescent MCT equations ... 

For vanishing shear rate, 7 = 0, an ideal glass state exists in the ITT approach for steady shearing. All density correlators arrest at a long time limit, which from (25) close to the transition is given by fpq(t °°, > Q,y = Q) = fi = -2,) + C ( ). Consequently the modulus remains elastic at long... 

Yield stresses can be measured with a variety of techniques. These include measuring the shear stress at vanishing shear rates, extrapolation of data using rheological models that include yield stresses and stress relaxation experiments, and others (Barbosa-Canovas and Peleg, 1983 Lang and Rha, 1981). 

The viscosity of a dilute polymer solution depends on the nature of polymer and solvent, the concentration of the polymer, its average molecular mass and molecular mass distribution, the temperature, and the shear rate. The most important characteristic quantity in a very dilute solution, at vanishing shear rate, is the limiting viscosity number, which is defined as [1]... 

There is another kind of stress relaxation experiment that is applicable only to viscoelastic liquids, which are capable of steady-state shearing deformation at a constant strain rate y. The stress during steady flow (sufficiently slow to ensure linear behavior) is <7 = yrio, where tjo is the viscosity corresponding to vanishing shear rate. If the flow is abruptly stopped, the stress will gradually decay (Fig. 1 -5). Its time dependence can be shown to be given by... 

At finite concentrations, the frequency dependence of G and G" — (aris can be examined directly without scaling the coordinates as in preceding figures. However, measurements at different temperatures may be combined by reduction to a standard temperature To G and G" — coris are multiplied by Toco/Tc and u> is multiplied by a shift factor aj which is given by rjo Vs)ocT/ t]o — Vs)coTq. Here r]o is the steady-flow (vanishing shear rate) viscosity and the subscript 0 otherwise refers to the reference temperature. This is the method of reduced variables which will be discussed fully in Chapter 11 with explanation of its rationale, and affords an extension of the effective frequency range. 

In discussing the effects of dilution in the terminal zone, we review first the viscosity at vanishing shear rate, the terminal relaxation time., and the steady-state-compliance. 

Vo 1 31 steady-state shear viscosity at vanishing shear rate... 

With respect to the number of variables, it is fairly obvious that material functions are necessarily nonlinear but, of course, in well-selected asymptotic conditions of one of the parameters, with all the others constant, rnie may recover a linear behavior. For instance, at constant temperature, the shear viscosity function at vanishing shear rate of a pure, tmfilled polymer is the so-called pseudo-Newtonian viscosity, i.e. rio = limri(Y), and for (infinitesimally) low strain ampli-... 

We noted in Section 10.7.2 that the second-order fluid approximation for flows only marginally removed from the rest state indicates that the first and second normal stress differences are second order in the shear rate, so that the first and second normal stress coefficients Pj q and T z 0 approach non-zero limiting values at vanishing shear rate. The second-order approximation also predicts that the net stretching stress in uniaxial extension is second order in the Hencky strain rate, and this implies that the extensional viscosity approaches its limiting zero-strain-rate value 3t7o with a non-zero slope ... 

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