Second Newtonian viscosity

Finally a phenomenon should be mentioned which polymer solutions show more often than polymer melts viz. a second Newtonian region. This means that with increasing shear rate the viscosity at first decreases, but finally approaches to another constant value. As the first Newtonian viscosity is denoted by rjor the symbol // x, is generally used for the second Newtonian viscosity. Empirical equations as those presented in Chap. 15 now need an extra term r oo to account for this second Newtonian region. This leads to ... 

Continuing in the pseudoplastic region it is often found that an upper threshold can be reached beyond which no further reduction in viscosity occurs. The curve then enters a second linear region of proportionality the slope of which is the second Newtonian viscosity. 

Figure 5, pertadning to the reduced temperature = 2, shows the behavior typical for the isotropic phase, viz. a first Newtonian viscosity, r/ = 1.1 in reduced units for smadl shear rates, a strong shear thinning for intermediate ones and the approach to the second Newtonian viscosity r/iao = 0.1 for high shear rates. Calculations with imposed sheaur rate and imposed shear stress give equivalent results. 

Bueche-Ferry theory describes a very special second order fluid, the above statement means that a validity of this theory can only be expected at shear rates much lower than those, at which the measurements shown in Fig. 4.6 were possible. In fact, the course of the given experimental curves at low shear rates and frequencies is not known precisely enough. It is imaginable that the initial slope of these curves is, at extremely low shear rates or frequencies, still a factor two higher than the one estimated from the present measurements. This would be sufficient to explain the shift factor of Fig. 4.5, where has been calculated with the aid of the measured non-Newtonian viscosity of the melt. A similar argumentation may perhaps be valid with respect to the "too low /efi-values of the high molecular weight polystyrenes (Fig. 4.4). 

Figure H1.1.4 A complete flow curve for a time-independent non-Newtonian fluid. r 0 and i , are the viscosities associated with the first and second Newtonian plateaus, respectively. Regions (1) and (2) correspond to viscosities relative to low shear rates induced by sedimentation and leveling, respectively. Regions (3) and (4) correspond to viscosities relative to the medium shear rates induced by pouring and pumping, respectively. Regions (5) and (6) correspond to viscosities relative to high shear rates by rubbing and spraying, respectively.

In 77) the authors give dependencies of the maximum Newtonian viscosity upon amplitude of periodic strain velocity q0 = f(e) for polyethylene and polystyrene. It has been also revealed that the dependency of normalized viscosity upon the velocity of stationary shear T /r 0 = f(y q0) obtained at r. = 0 coincides with a similar dependency when acoustic treatment is employed, i.e., at e 0. In other words, the effect of shear vibrations and velocity of stationary shear upon valuer] can be divided, representing the role of the first factor in form of dependency q0(s0) and that of the second in form of dependency (n/q0) upon (y r 0) invariant in relation to e. 

Second, the properties of such liquids are well determined by the stationary dependence of effective viscosity r eff on shear rate y (at large y, the lowest Newtonian viscosity is attained at T <, 40-60 °C). At higher temperatures, together with t eff(y), one has to known the dependence of t x on time as well. This dependence is linear, and its coefficients are exponentially dependent on temperature. 

Illy = 0, and therefore the non-Newtonian viscosity can only be a function of the second invariant rj(y) =/(//y). In practice, this functionality is expressed via the magnitude of y, and is given by... 

There is a reasonable correspondence between the experimental and calculated viscosity values at a shear rate of 10 s-1. The calculated 77 value at y 103 s-1 is too low, however, because at this shear rate the second Newtonian region is approached. To estimate rj Eq. (16.51) can be applied with... 

The impact of non-Newtonian viscosity behavior on effective film thickness has been successfully modeled with an equation developed by Bair and Khonsari21 that incorporates the second Newtonian obtained from the Car-reau viscosity equation. 

Results. In Figure 1 are shown the viscosity versus shear stress data for xanthan solutions (.1 to 1 mg/ml) in 0.5M NaCl, 0.04M phosphate buffer, pH 7, containing 0.02% NaN3 as a preservative. The data show a Newtonian plateau between 0.001 and 0.08 dyne/cm for 0.1, 0.2, and 0.3 mg/ml. As the shear stress increases beyond 0.1 dyne/cm, a sharp drop occurs in the viscosity. The viscosity decreases until a second Newtonian plateau is reached at 2-20 dyne/cm. For higher xanthan concentrations the low-shear stress Newtonian plateau occurs at lower shear stresses and the transition between the two plateaus is broadened. Whitcomb and Macosko (2) have reported similar data except that their data did not extend into the low-shear Newtonian range at low concentrations. 

The second Newtonian range occurring at very high shear stresses is often difficult to measure many rheologists even doubt its existence. Since a general flow law has not been discovered to date, the flow behavior cannot, of course, be characterized by one or two parameters. The difficulty is compounded in that the apparent viscosity of some liquids is time dependent. 

The phenomena described above are the basis of the structural viscosity and viscoelastic behavior mentioned in Sec. il. The region of elastic response and the region of viscous flow depicted in Fig. 33 correspond to the lirsi and to the second Newtonian plateau of Fig. 5, respectively. Further, the floe destruction region shown in Fig. 34 corresponds to the shear-thinning portion between both plateaus of Fig. 

Solution Rheology. Solutions of polyacrylamides tend to behave as pseudoplastic fluids in viscometric flows. Dilute solutions are Newtonian (viscosity is independent of shear rate) at low shear rates and transition to pseudoplastic, shear thinning behavior above a critical value of the shear rate. This critical shear rate decreases with the polymer molecular weight, polymer concentration, and the thermodynamic quality of the solvent. A second Newtonian plateau at high shear rates is not readily seen, probably because of mechanical degradation of the chains... 

The mechanism(s) of a particulate fluid electroviscous effect is still not fully resolved and quantified. It is not strictly relevant to this work and is therefore not dealt with in detail. At this stage it can only be said that it is a very multi-parameter and multidisciplinary event and, secondly, it should be understood that there is little change in the viscosity p of the fluid as it is normally defined in its continuum context save for a derived effective or non Newtonian viscosity sense. The term electroviscous, which has often been used to describe the present class of fluids, is misleading in this case. Rather, the held imposes a yield stress type of property on the fluid which is similar to, but not the same as, that which is a feature of the ideal Bingham plastic. This can readily be seen by referring to Figs. 6.63 to 6.66 inclusive. It is alternatively possible to claim that either the plastic viscosity changes with shear rate or the electrode surface yield stress does. 

The flow behavior of a polymer melt (logarithm apparent viscosity vs logarithm shear rate) is illustrated in Fig. 3-8. Basically, the melt has flow behavior very similar to a Newtonian (relatively constant apparent viscosity) at low shear rates. Similarly, at very high shear rates, a second Newtonian type of behavior is encountered. As such, therefore, the constitutive equation should represent, as closely as possible, the behavior shown in Fig. 3-8. The goodness of this representation is directly proportional to the number of constants in the constitutive equation The more constants, the truer the representation. 

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