Limiting viscosity at zero shear rate

Limiting viscosity at zero shear rate, i.e., at the upper Newtonian plateau... 

The viscosities of most real shear-thinning fluids approach constant values both at very low shear rates and at very high shear rates that is, they tend to show Newtonian properties at the extremes of shear rates. The limiting viscosity at low shear rates mq is referred to as the lower-Newtonian (or zero-shear /x0) viscosity (see lines AB in Figures 3.28 and 3.29), and that at high shear rates Mo0 is the upper-Newtonian (or infinite-shear) viscosity (see lines EF in Figures 3.28 and 3.29). 

If the intrinsic viscosity is large (i.e., greater than about 4 deciliters per gram), the viscosity is likely to be appreciably dependent on the rate of shear in the range of operation of the usual capillary viscometer. Measurements in a viscometer permitting operation at a series of rates of shear extending to very low rates are then required in order to extrapolate nsp/c to its limiting value at a shear rate of zero. Extrapolation to infinite dilution does not eliminate the effect on this ratio of a dependence on shear rate. 

Et1oo is the activation energy for viscous flow at zero shear rate in the limit of T E can be estimated [7] in terms of the molar viscosity-temperature function HT1 by using Equation 13.12. Hp is expressed in units of g F1/3"mole"4/3 and normally estimated by using group contributions. For example, the use of group contributions gives HT)=4020 g J1/3 mole-4/3 for polystyrene, so that the use of Equation 13.12 with Hp=4020 and M=104.15 results in ET OO =57504 J/mole, in comparison with the experimental value [7] of 59000 J/mole. 

It is well known that at extremely low shear rates the slope of the r/y curve (Figure 3.26) is constant and that there exists some very low but finite threshold shear rate beyond which deviation from linearity commences. The slope of the initial linear portion of the curve is known as the limiting viscosity, the zero shear viscosity, or the Newtonian viscosity. Beyond this low shear rate region (initial Newtonian regime) the material is shear-softened (i.e., becomes pseudoplastic), a phenomenon which has its counterpart in the solid state where it is known as strain-softening. 

Figure 11.2 Viscosity vs. concentration for PBG M = 335 000) 4- m-cresol. (a) Zero shear/frequency limit of steady shear and dynamic viscosity, (b) Steady shear viscosity at various shear rates.

At higher strain rates even more complications arise. There the viscosity is no longer constant and shows a decrease with increasing rate, commonly called shear-thinning . We will discuss this effect and related phenomena in chapter 7, when dealing with non-linear behavior. In this section, the focus is on the limiting properties at low shear rates, as expressed by the zero shear rate viscosity , ryo, and the recoverable shear compliance at zero shear rate,... 

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 ... 

Which range should be considered The answer is the region near the origin of a plot like Fig. 2.2 for pseudoplastic materials. The slope of the tangent to a pseudoplastic curve at the origin is called the viscosity at zero rate of shear. Note that this is an extrapolation to a limit rather than an observation at zero shear (which corresponds to no flow). We shall use the symbol to indicate the viscosity of a polymer in the limit of zero shear, since the behavior is Newtonian (subscript N)in this region. 

When the fluid behaviour can be described by a power-law, the apparent viscosity for a shear-thinning fluid will be a minimum at the wall where the shear stress is a maximum, and will rise to a theoretical value of infinity at the pipe axis where the shear stress is zero. On the other hand, for a shear-thickening fluid the apparent viscosity will fall to zero at the pipe axis. It is apparent, therefore, that there will be some error in applying the power-law near the pipe axis since all real fluids have a limiting viscosity po at zero shear stress. The procedure is exactly analogous to that used for the Newtonian fluid, except that the power-law relation is used to relate shear stress to shear rate, as opposed to the simple Newtonian equation. 

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... 

First the temperature dependence of the limiting zero shear rate viscosity (Newtonian) is calculated at a shear rate of 0.01 1/s using the data in Table 3.7 ... 

Under shear flow, the minimum center-to-center separation, r, between the particles will be in the interval 2Rs < r < L i.e., at the high-shear limit r = 2R, and at quiescent conditions r = L. An analysis of the flow behavior in this case leads to the following expression for the zero shear rate (i.e., Pe < 1) limit of the relative viscosity ... 

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. 

The limiting viscosity number (LVN) of the polymers—i.e., the intrinsic viscosity at zero concentration and shear rate, was determined from the inherent viscosity measured at 0.05 to 0.2 grams/100 ml. concentration in decalin at 135° C. with a Schulken-Sparks viscometer. 

If values of rja are now plotted as a function of strain rate (Figure 13-62), it can be seen that although there is a dramatic decrease in apparent viscosity at high values of y, at low strain rates the apparent viscosity is essentially constant. Obviously, if you want to compare the rheological properties of different types of polymers, it is this strain rate-independent parameter that would be most useful, as it would presumably be a characteristic property of the polymer. This limiting value is called the zero shear-rate viscosity, rjm. 

For an unvulcanized polydimethylsiloxane, the biaxial viscosity was approximately six times the shear viscosity over the biaxial extensional rates from 0.003 to 1.0 s (Chatraei et al., 1981), a result expected for Newtonian fluids, that is, the relationship between the limiting value of biaxial extensional viscosity (j, ) at zero strain rate and the steady zero-shear viscosity (i o) of a non-Newtonian food is ... 

In this case, the system does not show a yield value rather, it shows a limiting viscosity ri 6) at low shear rates (that is referred to as residual or zero shear viscosity). The flow curve can be fitted to a power law fluid model (Ostwald de Waele)... 

The zero-shear viscosity rip is defined as the melt viscosity in the limit of y=0, and is a function of T and Mw. It is important to keep in mind, however, that rp) is very often not measured directly, but extrapolated from measurements at low shear rates. Such extrapolations can introduce an error in the value of r o if the range of shear rates used in the extrapolation is sufficiently high for non-Newtonian effects to begin manifesting themselves. 

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