Onset of Shear Rate Dependence

Following the procedure used with JeR (Section 5), po data on these and other polymers were correlated in terms of cM/qMc, with results which are shown in Fig. 8.13. The parallel in behavior between the po and JeR master correlations is unmistakable. Even the relative positions of polymers on the master correlations are similar note for example the data on JeR and p0 for polybutadiene. Published data on relatively narrow distribution polyethylene (210,326) have not been included in Fig. 8.13 because departures from tj0 were too small to define P0 with accuracy. However, estimates of po from the data provided suggest that polyethylene may follow a different pattern than other polymers. Departures from rj0 seem to appear at anomalously low shear rates (326). Aside from tj0 values, viscoelastic data on well characterized crystallizable polymers in the melt state are rather scarce. Although not especially anticipated, it is certainly conceivable that erystallizability confers some unusual features to the flow behavior. 

The parallel between po and JeR has been noted elsewhere (208,213,328,329), and is not in fact fortuitous. It follows rather directly from the empirical observation that departures of t (y) and r/ (co) from t]0 are governed by the longest relaxation times of the system, combined with slight extensions of a reduced variables argument suggested by Markovitz for linear viscoelastic behavior (329). Suppose one wants to compare the forms of the dynamic moduli on 

differences among samples due to differences in Jc° and rj0 are removed in this reduced form. Since tj0 and Je° are sensitive characteristics of the long relaxation time processes, and since the initial departure of t] 0 — rjs from t]0 — s is governed by these same processes, one would espect significant departures from rj0 to occur near some roughly constant characteristic value of c% with a value of approximately unity. Finally, if the onset of shear rate dependence depends on the same processes, then Eq.(8.8) also gives an appropriate reduced form for f  

The onset of shear rate dependence therefore is governed by the product JeR/ and should occur near JeR/S = 1. The observed value of this product at departure will of course depend on the criterion chosen to define departure. It may also vary somewhat if the criterion requires a large departure and the form of rj(y) differs substantially from sample to sample. 

Aside from minor differences this equation is the same as that suggested recently by Prest et al. (208)9. 

---

Williams has derived the molecular weight and concentration dependence of a viscoelastic time constant t0 (actually the characteristic time governing the onset of shear rate dependence in the viscosity) from his theory (217-219). Employing a dimensional argument, he equates the parameters which control the shear rate dependence of chain configuration and the intermolecular correlation function. The result agrees with the observed form of characteristic relaxation time in concentrated systems [Eq.(6.62)] ... 

Fig. 8.11. Dimensionless shear rate / 0 locating the onset of shear rate dependence for viscosity in narrow distribution polystyrene systems. Symbols are O for solutions at 30° C in n-butyl benzene (155), 9 f°r solutions at 25° C in arochlor (177), and 4 for undiluted polymers at 159° and 183° C (324). Values for the intrinsic viscosity (cM=0) lie in the range /i0 = 1-2, varying somewhat with solvent-polymer interaction and molecular weight...

Fig. 8.13. Dimensionless shear rate /30 locating the onset of shear rate dependence in the viscosity in narrow distribution systems of linear polymers vs cM/qM. Symbols for data on additional polymers are A for undiluted 1,4 polybutadiene (322), for undiluted poly(dimethyl siloxane) (323), and O for solutions of polyvinyl acetate in diethyl phthalate (195). The dotted lines indicate the ranges of for the intrinsic viscosity...

A final piece of evidence against both finite extensibility and internal viscosity is provided by flow birefringence studies. One would expect each to produce variations in the stress optical coefficient with shear rate, beginning near the onset of shear rate dependence in the viscosity (307). Experimentally, the stress-optical coefficient remains constant well beyond the onset of shear rate dependence in r for all ranges of polymer concentration (18,340). 

Changes in excluded volume and in intramolecular hydrodynamic interaction appear at the present time to be the only acceptable explanations for the onset of shear rate dependence in systems without appreciable intermolecular interactions. It seems likely that both internal viscosity and finite extensibility would assume importance only at much higher shear rates. 

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). 

Characteristic reduced shear rate locating the onset of shear rate dependence in the viscosity. 

The use of cone-and-plate rheometers for polymer melts is limited to relatively low shear rates by the onset of flow instabilities, typically occurring not far beyond the onset of shear-rate dependence for t](y) and the a iVi crossing point. A capillary rheometer is sketched in Fig. 3.22. Stable operation at much higher shear rates is possible, but usually t]o cannot be determined because of instrumental limitations at low shear rates. The steady-state viscosity, however, can be obtained from measurements of the volumetric flow rate, Q, and the pressure drop, AP = P — Po, Po being the ambient pressure. For long tubes (L/D 1), the following equation applies for Newtonian liquids ... 

Master curves (7b = 25 °C) for the complex viscosity of two nearly monodisperse 1,4-polybutadiene melts [28] are shown in Fig. 3.24. One is linear rjo = 4.8 X 10 Pa s, J° = 2.1 X 10 Pa ), the other a three-arm star rjo = 2.8 x 10 Pa s, J° - 1. 4 X 10 Pa ). Their zero-shear viscosities are similar, but their recoverable compliances differ by a factor of seven and the shapes of their curves are obviously different, too. Figures 3.25(a) and (b) compare those results with steady-shear-viscosity data for nearly monodisperse polymers, showing master curves at 183 °C for five linear polystyrene samples [29] (48 500 < M < 242 000) in Fig. 3.25(a), and master curves at 106 °C for seven polybutadiene stars [30] (45 000 < M < 184000) in Fig. 3.25(b). Values of t]o were available for all samples, so knowledge of rj y)/r o was always available. Values of J° were not generally available, so Tq for the shear-rate reduction was estimated from the onset of shear-rate dependence. Agreement with the Cox-Merz rule is evident even in this rather severe test of using different samples and even different species. The... 

Figure 3.28 compares the viscosity function and De/D versus yw in a long capillary for a commercial sample of polystyrene [32] with = 220000, Mw/Mn = 3.1, t]o = 1.4 X 10 Pa s at 180 °C, 7° 6 x 10" At low shear rates, the viscosity levels off at t]o. Normal-stress differences are small in that region, as discussed before, and D /D is about 1.1, the computed value for slow flows of a Newtonian liquid [33]. The swell ratio then begins to rise near the shear rate at the onset of shear-rate dependence in the viscosity, a shear rate that, from yw based on Eq. (3.14), also locates the cr Ai crossover range, indicating... 

Viscosity of polymer systems approaches a limiting constant value at low shear rates. The onset of shear rate dependence is usually quite sharp for monodisperse polymers. However, the broader... 

---