First Normal Stress Function

At the limit of low shear rates J(y) approaches J° according to the Coleman-Markovitz relation [Eq. (3.21)] and the experimental results in Section 5. In 

Such behavior is qualitatively understandable in terms of partial disentanglement in steady shear flow. In highly entangled systems (cM gM )Je0 is of the form (Section 5)  

Since g is a monotonically decreasing function of shear rate, departing from unity at the onset of shear rate dependence on tj,J(y) should show positive curvature (a montonic increase with shear rate) for systems of narrow molecular weight distribution. The shear rate dependence shown by Nagasawa s data (Fig. 8.16) is qualitatively similar but actually somewhat greater than that predicted with values of g from Eq. (8.24). 

Such qualitative agreement should not be interpreted too broadly in favor of disentanglement however. Sakai et al. (199) have also observed that J(y) increases with y even in narrow distribution systems of lower concentration, that is, 

Although the FENE model has been shown earlier to be inappropriate for polymer systems on other grounds, it nevertheless illustrates an instance in which J(y) increases with shear rate naturally and without the need for postulating structural changes in the system. 

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First normal stress function, pt t — p22 at steady state in steady simple shear flow. 

Equation (10) cannot be applied until A, the equivalent relaxation time for the fluid, is known. However, A is defined by the linear Maxwell model, and actual polymer solutions exhibit marked nonlinear viscoelastic properties [5,6,7]. For both fresh and shear degraded solutions of Separan AP 30 polyacrylamide, which exhibit pronounced drag reduction in turbulent flow, Chang and Darby [8] have measured the nonlinear viscosity and first normal stress functions, and Tsai and Darby [6] have reported transient elastic properties of similar solutions, A nonlinear hereditary integral function containing six parameters has been proposed to represent the measured properties [8], The apparent viscosity function predicted by this model is ... 

Fig. 19. Shear stress and first normal stress difference plotted as a function of shear rate for different molar masses, and b at different concentrations of polystyrene in toluene... 

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

The symbols Nt and N2 denote the normal stress functions in steady state shear flow. Symmetry arguments show that the viscosity function t](y) and the first and second normal stress coefficients P1(y) and W2(y) are even functions of y. In the... 

Fig. 2.1. First normal stress difference (pu—p22) as a function of shear rate q and doubled storage modulus 2 G as a function of angular frequency for a poly-dimethyl siloxane (M = 536,000) at a measurement temperature of 20° C. (o) (A n/C) cos 2y,...

The material functions, k i and k2, are called the primary and secondary normal stress coefficients, and are also functions of the magnitude of the strain rate tensor and temperature. The first and second normal stress differences do not change in sign when the direction of the strain rate changes. This is reflected in eqns. (2.51) and (2.52). Figure 2.31 [41] presents the first normal stress difference coefficient for the low density polyethylene melt of Fig. 2.30 at a reference temperature of 150°C. 

For example, Figs. 2.43 and 2.44 present the measured [55] viscosity and first normal stress difference data, respectively, for three blow molding grade high density polyethylenes along with a fit obtained from the Papanastasiou-Scriven-Macosko [59] form of the K-BKZ equation. A memory function with a relaxation spectrum of 8 relaxation times was used. 

Equations 12.2-6 and 12.2-7 seem to imply that the entrance (or ends ) pressure losses are simply related to the first normal stress difference function at the capillary wall. Indeed they find that... 

In the case of elastic fluids and for simple shear flow, the first normal stress difference is N[ =on — o22- When shearing a fluid between two plates (x, direction), the first normal stress difference N( forces the plates apart (x2 direction). The first normal stress difference N i is shown together with the measured shear stress x as a function of the shear rate in Fig. 3.9. In the range of shear rates investigated, the shear stress in the case of silicone oil is substantially greater than the normal stress difference and we see substantially greater normal stress differences for viscoelastic PEO solution than for viscous silicone oil. 

Figure 3.9 First normal stress difference N, and shear stress a as a function of shear rate for Baysilone M 1000 silicone oil and aqueous PEO solution (c=1wt.-%. Polyox WSR 301)...

As before, one may establish that small changes in the deformation rate are accompanied by dramatic changes in the phase behavior. Furthermore, the first normal stress difference was reported in Ref. [112] as a function of polymer concentration at constant temperature and shearing stress. For the discussion below, it is important to keep in mind that the plots of the normal stress... 

An important conclusion is that it is clear that Lodge s constitutive equation is not able to describe non-Newtonian behaviour in steady shear, because both the viscosity and the first normal stress coefficient appear to be no functions of the shear rate. 

In order to elucidate the correlation method it may be recalled that the viscosity 77 approaches asymptotically to the constant value r c with decreasing shear rate q. Similarly, the characteristic time t approaches a constant value xQ and the shear modulus G has a limiting value G0 at low shear rates. Bueche already proposed that the relationship between 77 and q be expressed in a dimensionless form by plotting 77/r]0 as a function of qx. According to Vinogradov, also the ratio t/tq is a function of qxQ. If the zero shear rate viscosity and first normal stress are determined, then a time constant x0 may be calculated with the aid of Eqs. (15.60). This time constant is sometimes used as relaxation time, in order to be able to produce general correlations between viscosity, shear modulus and recoverable shear strain as functions of shear rate. 

FIG. 15.46 Viscosity, 77, and first normal stress difference, Nh of Vectra 900 at 310 °C as functions of shear rate, according to Langelaan and Gotsis (1996). The first normal stress coefficient, Yi, is estimated from N, by the present author. ( ) Capillary rheometer ( ) and ( ) cone and plate rheometer ( ) complex viscosity rj (A) non-steady state values of the cone and plate rheometer. Courtesy Society of Rheology. 

Figure 5 Viscosity (t ) and first normal stress difference N] as a function of shear rate (.i) a-Variation ofT) with if b-Variation of Ni with for EC 40X in acetic acid...

Figure 6 Steady state functions for LD at 160°C (experimental and calculated), (o) elongational viscosity, ( ) shear viscosity, (A) first normal stress difference.

Figure 1-1 First Normal Stress Coefficient Data of Starch Dispersions with Different Concentrations as a Function of Shear Rate (Genovese and Rao, 2003). Abbreviations cwm, cross-linked waxy maize tap, tapioca gran, granule.

Figure 3.34 Shear stress (open symbols) and first normal stress difference (closed symbols) as functions of shear rate for two solutions of very-high-molecular-weight poly--4nethylmethacrylate (M = 23.8 x 10 )...

Figure 3.35 Steady-state values of the reduced shear stress <712/ and first normal stress difference N / as functions of dimensionless shear rate y Zr predicted by the equations of a constraint-release reptation theory (see Problem 3.10) for Xd/Zr — (a) 50, (b) 150, and (c) 500, where Zd is the reptation time and Zr is the Rouse retraction time. See also Marracci and lanniruberto (1997). (From Larson et al. 1998, with permission.)...

The Doi-Edwards equation predicts an overshoot in shear stress as a function of time after inception of steady shearing, but no overshoot in the first normal stress difference (Doi and Edwards 1978a). Typical overshoots in these quantities for a polydisperse melt are shown in Fig. 1-10. For monodisperse melts, the Doi-Edwards model predicts that the shear-stress maximum should occur at a shear strain yt = Yp, of about 2, roughly independently of... 

Figure 6.38 Viscosity (open symbols) and first normal stress difference (closed symbols) as a function of shear stress for neat polypropylene melt (Q. )> the same melt filled with 50% by weight CaC03 particles of size 2.5 fx, (A, A), and the filled melt with a titanate coupling agent ( , ). (From Han et al. 1981, reprinted with permission from the Society of Plastics Engineers.)...

Figure 11.11 Shear viscosity rj and first normal stress difference as functions of shear rate y for Vectra B-950 at 300°C. The different symbols are for data acquired on different cone-and-plate and capillary rheometers. (From De Neve et al. 1993a, with permission of the Journal of Rheology.)...

Figure 11.17 Dimensionless shear stress and first normal stress difference as functions of dimensionless shear rate predicted by the Doi with the closure approximation for the fourth moment, (uuun) — (uu) (uu) and U — 2U = 6.

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