Normal stress differences in steady-state shear flow

In this section, we present another experimental method that also employs a capillary die. However, this method allows one to determine not only shear viscosity but also normal stress differences in steady-state shear flow by continuously supplying a polymeric... 

We now present the theory (Davis et al. 1973 Han 1974) that allows one to determine shear stress and first normal stress difference in steady-state shear flow using wall normal stress measurements along the axis of a slit die. Consider a fluid flowing through a slit die having the height h and the width w, and assume that flow has become fully developed. Then, for steady-state fully developed flow, the equations of motion... 

Temperature Dependence of Relaxation Time and First Normal Stress Difference in Steady-State Shear Flow of Linear Flexible Homopolymers... 

To obtain the expressions from Eq. (12.23) for bulk shear stress and bulk first normal stress difference in steady-state shear flow, for instance, one must evaluate the quantity (uuuu) defined by Eq. (12.20). For this, information on f(u, t) must be provided. Once /(u, t) is known, in principle one can calculate (uuuu) fromEq. (12.20). Several different ways of approximating (uuuu), referred to as closure (or decoupling) approximation, have been suggested in the literature (Advani and Tucker 1980, 1987 Doi 1981 Hand 1962 Hinch and Leal 1976 Marrucci and Grizzuti 1984) (see Chapter 9). The simplest form of approximation is... 

It is interesting to observe in Eq. (12.30) that an evaluation of the fourth-order tensor (uuuu) is not necessary to obtain expressions for, for instance, shear stress and first normal stress difference in steady-state shear flow (see Appendix 12B). 

In steady-state shear flow at very low rates of shear 7, the primary normal stress difference c — 022 is related to the dynamic storage modulus at very low... 

Figure 1.3 Plots of first normal stress difference (IVj) versns shear rate (y) for 4 wt% aqueous solution of polyacrylamide in steady-state shear flow at 25 C.

In this section, we present the rheological behavior of linear flexible homopolymers (i.e., without side-chain branching). We first present methods for obtaining temperature-independent plots for shear viscosity ( ) and first normal stress difference (N ) in steady-state shear flow and for dynamic moduli (G" and G") in oscillatory shear flow. We then discuss the effect of molecular weight and molecular weight distribution on the rheological behavior of linear flexible homopolymers. 

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

The steady-state shear flow properties in the low shear rate region and the dynamic functions were measured using a rotational viscometer (cone-plate type, RGM151-S, Nippen Rheology Kiki Co., Ltd., Japan). The cone radius R was 21.5mm, the gap between the central area of the cone and plate H was kept at 175p.m, and the cone angle 0 was 4°. The measurements were carried out at 200°C Steady state shear properties (shear viscosity //, and the first normal stress difference Ni) as well as dynamic functions (storage and loss moduli G, G", respectively. 

For steady-state shear flow of a spinodal blend, Onuki [218-220] postulated that the shear stress is the same in two phase-separating liquids, which leads to the effective viscosity and the first normal stress difference as ... 

The first manifestation of nonlinear behavior with increasing strain or strain rate is the appearance of normal stress differences in shearing deformation. For steady-state shear flow at small shear rates, several nonlinear models ° predict the relation for the primary normal stress difference given as equation 62 of Chapter 1, which, combined with equation 54, gives... 

Measurements of normal stress differences during steady shear flow, and of normal stress growth approaching steady-state flow and stress relaxation after cessation of flow, provide additional information about nonlinear viscoelastic properties. The conventional identifications of the normal stresses for simple shear have been shown in Fig. 1-16 their orientations in several examples of experimental geometry are sketched in Fig. 5-5. 

For steady-state shear flow the set consisting of Equation 9.30, Equation 9.31, and Equation 9.32 can be solved anal3d-ically for a = 0.5. This special value of the anisotropy factor was often observed in viscoelastic surfactant solutions. The shear stress and first and second normal stress difference can be expressed by the following equations... 

As we have demonstrated, an erroneous conclusion can be drawn on the sign of first normal stress difference in TLCP if the transient or steady-state shear flow experiments were carried out in the presence of residual normal force that was generated... 

The stress growth experiment of Figure 2(c) involves the study of the time evolution of the stresses when a fluid is brought instantaneously from a state of rest at t = 0 to a state of steady-state shear flow this is an idealized experiment which presumes that in the experiment one can effectively minimize inertial effects and achieve the linear velocity profile within an acceptably short time interval. One can then define the growth functions associated with the shear stress and the two normal-stress differences for t > 0 as follows... 

Figure 17 Shear viscosity /, coefficient of the first normal stress difference i, normalized integrated scattering intensity in the flow direction (f A axis) ll(y)/ ll(y = 0), and the neutral direction ( y axis) = 0) for PS548/D0P 6.0wt.% under steady-state shear flow at 27°C. The annotation...

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

Since at steady state the angular distribution of fiber orientations is predicted to be symmetric about the flow direction in a shearing flow, Eq. (6-50) implies that the normal stresses (e.g., a oc [u uy) will be identically zero. However, nonzero positive values of N have frequently been reported for fiber suspensions (Zimsak et al. 1994). Figure 6-24 shows normalized as discussed below, as a function of shear rate for various suspensions of high fiber aspect ratio. These normal stress differences are linear in the shear rate and can be quite large, as high as 0.4 times the shear stress, which is dominated by the contribution of the solvent medium, cr Fig- 6-24, the N] data are normalized... 

The stress in viscoelastic liquids at steady-state conditions is defined, in simple shear flow, by the shear rate and two normal stress differences. Chapter 13 reviews the evolution of both the normal stress differences and the viscosity with increasing shear rate for different geometries. Semiquantitative approaches are used in which the critical shear rate at which the viscosity starts to drop in non-Newtonian fluids is estimated. The effects of shear rate, concentration, and temperature on die swell are qualitatively analyzed, and some basic aspects of the elongational flow are discussed. This process is useful to understand, at least qualitatively, the rheological fundamentals of polymer processing. 

Starting with cell model of creeping flow, Choi and Schowalter [113] derived a constitutive equation for an emulsion of deformable Newtonian drops in a Newtonian matrix. The authors characterized the interphase with an ill-defined interfacial tension coefficient, Vu, affecting the capillarity number, k = (Judfvu. The analysis indicated that depending on magnitude of /cy the emulsion may be elastic, characterized by two relaxation times. For the steady-state shearing, the authors expressed the relative viscosity of emulsions and the first normal stress difference as ... 

Finally, it is also important to emphasize that the nonlinear rheology of viscoelastic surfactant solutions is characterized by the existence of normal stresses of nonnegligible magnitude. In steady-state flow, a nonzero first normal stress difference N has been detected once the first stable branch becomes shear-thinning. Ni was found to increase with 7 and a slight change of slope was observed at the... 

At very low shear rates, the normal stress coefficients 1,0 and 2,0 are also independent of 721 i.e., the normal stress differences are proportional to 721. At higher shear rates, 1 and 2 are observed to decrease. The course of stress relaxation after cessation of steady-state flow and the magnitude of the steady-state compliance J° are also strongly affected at high shear rates. In general, description of these phenomena requires more complicated constitutive equations than the single-integral models mentioned above. 

The normal stresses present during the steady-state flow also relax after its cessation, and the course of the primary normal stress difference, (oi j — shear rate which precedes it. However, the normal stress difference relaxes more slowly than the shear stress. 

---