Shear rate limitations

Figure 5.12 The measured viscosity as a function of volume fraction at the low and high shear rate limits. This is the data for the system with the pair potential shown in Figure 5.9...

The slope between the low and high shear rate limits has a constantly changing gradient. In order to define the point of inflection on this slope and its tangent the second derivative is required ... 

The time ie represents the entanglement formation time at the current shear rate. The second term g is the average number of entanglement sites for a chain of fixed length in steady flow relative to the number in the zero shear rate limit ... 

Shear rate limits where the shear rate exponent changes value, becomes 1, or the consistence curve no longer follows a power law approximation. 

For polymer melts where the low shear rate limiting viscosity value is r ), r 3t]0 (14). Examples of extensional viscosity growth, either to a steady t](i ) value or to a strainhardening-like mode, are shown in Fig. 3.6 for the linear nonbranched polystyrene (PS), a high density polyethylene (HDPE) that is only slightly branched with short branches, and a long chain-branched low density polyethylene (LDPE) (15). 

The reformation of chiral nematic phase from nematic phase sometimes goes through the transient state of band texture. The formation of band texture is also a relaxation phenomenon. For HPC/H2O solutions, the band texture is only observed when the molecules have been well orientated in the shear direction.A critical lower shear rate limit exists because of the stability of chiral nematic textures. 

An upper shear rate limit also exists above which no bands are formed. The upper critical shear rate is associated with the flow-aligning region of molecular dynamics. Band texture formation is driven by the release of energy, which has been stored in the mesophase during shear. The evolution of the band texture depends on the previous shear rate applied to mesophase. The rate of evolution of the band texture... 

Smah sample size Uniform shear rate Limitations... 

Shear Rate Limitations. Extruders are run at relatively low screw speeds. Small extruders have variable drives to adjust screw speeds from about 20 to 150 rpra for extruders up to about 2 in. in diameter. As the screw diameter increases the screw speed must be reduced. The peripheral screw speed is given by Equation k... 

Figure 18.6 Viscosity of particle dispersions the solid lines correspond to the shear rate limits calculated from the Krieger-Dougheity equation (see text) using the values of and dimensionless shear rate or Peclet number (Pe) given in the inset table, of Chou et aL [76] on acrylic latex particles (O) 7S0 nm spheres, ( ) 960 run spheres, (A) multilob particles...

The tern) is denoted as fU the steady state creep compliance, and is the inverse of i)o. the viscosity in the low-shear rate limit. Note that an elustic solid, after the initial defonnatioii. has a slope e )ual lu /eru (y.. — 0 t)u meaning that the solid does not flow. 

The disadvantage of the low shear rate limitation can be negated by applying an oscillatory rotary motion to the moving plate. Thus, one can measure complex viscosity as a function of frequency see also Section 6.2.6.5 on dynamic analysis. In this fashion, frequency levels up to about 500 radians/s are possible this corresponds to shear rates of about 500 s. ... 

At low values of 7, the coefficients 1 and 2 approach the zero-shear-rate limits of j,o and 2,0 respectively. All the single-integral equations mentioned above predict for the limiting primary normal stress coefficient ... 

Newtonian regimes are nevertheless widely observed in polymer melts in the high and low shear rate limits, where the viscosities are designated by and respectively. This is reflected by the empirical expressions widely used in engineering practice to describe the response to steady shear flow, an example being the Cross model [Eq. (55)], which reduces to the well known power law of Eq. (56) when tj tj , with n = l/(m+l) between 10 and 20 for most polymer melts. 

Performance in static load shear tests (see Fig. 5) can be related to the viscosity of the adhesive when dealing with uncrosslinked or lightly crosslinked (below the gel point) PSAs. If we assume that this material will creep governed by the zero shear rate limiting viscosity of the adhesive, rjo, and that the area determining the shear stress, r, is the constantly decreasing overlap area w /(/), then... 

Stressing viscosity (i, for uniaxial, biaxial, and planar extension, stressing viscosity for planar extension, and shear viscosity as functions of time ter inception of steady straining for polyisobutylene. The solid line is the low shear rate limit of. Extension and shear rates are 0.08s except the biaxial which is 0.02s-. From Retting and Lawn, 1991. 

To test this result Beavers and Joseph (1975) measured h versus for a polyisobutylene solution. Some of their results are reproduced in Figure 5.3.6. We see that there is an approximately constant static climb, hs(0), with different radii and a fairly large region of linearity in f2. This linearity appears to justify dropping the higher order terms in eq. 5.3.36. From the slopes in Figure 5.3.6, Beavers and Joseph calculate p = 0.97 0.5 g/cm at 26°C. Davis and Macosko (1973) estimated ir, o and V 2,o from the low shear rate limits of N from cone and plate thrust data and N] — N2 from parallel disk data and obtained p = 0.98 0.5 g/cm on the same material. Joseph etal. (1984) have made similar comparisons on a number of polymers. 

Some experimental viscosity results on dilute suspensions of rigid spheres x, glass 5 fim in zinc iodide glycerin (Manley and Mason, 1954) O. polystyrene aqueous latices, 0.42, 0.87 /xm (Saunders, 1961) v. low shear rate, and A, high shear rate limits for nonaqueous polystyrene latices, 0.16-0.43 urn (Krieger, 1972). 

There are limited measurements of the viscoelastic properties of hard-sphere colloids at elevated shear rates, stresses, and frequencies, de Kruif, et al. report rir of silica spheres in the small and large shear-rate limits, as functions of (53). At smaller concentrations, rjr depends but weakly on /c. Above

shear thinning becomes apparent, de Kruif, et al. propose that r]r diverges 2 — 4>/4>m), with 4>m being 0.71 or 0.63 in the large and small shear limits. Jones, et al. also find weak shear thinning for (p > 0.395, the extent of shear thinning increases quite substantially for volume fractions between 0.59 and 0.60(55). 

We can also determine the viscosity in the zero shear rate limit by application of Eq. (6.107) as... 

Several quantities are used to describe the low-shear-rate limiting viscosity of a solution tj in terms of the viscosity of the solvent, rj, and the concentration of polymer, c. These are defined as follows. 

The intrinsic viscosity is defined in terms of the viscosity in the limit of zero shear-rate, but the viscosity of a polymer solution depends on the shear-rate. Factors that enhance the shear-rate dependency of [77] are high solvent power, large molecular weight and broad molecular weight distribution [66,67]. To avoid consideration of this parameter, measurements must be made at sufficiently low shear rates that the viscosity is essentially equal to its low-shear rate limiting viscosity. 

The subscript zero indicates that this is the viscosity exhibited in the regime of linear viscoelasticity, ie., the low-shear-rate limiting viscosity. It is thus defined as shown by Eq. 4.10 and is called the zero-shear viscosity. 

This is a curious result, as it indicates that a nonlinear property can be calculated from linear data, but it has been found to describe accurately the response of polymeric liquids at sufficiently low shear rates. The low-shear-rate limiting behavior indicated by the above equations, which involves monotone increasing functions, is always shown in plots of nonlinear data the nonlinear responses involve overshoots in the material functions, but should always start out at short times, when the strain is still small, by following the low-shear-rate, LVE curve. Then, as the shear rate increases, the nonlinear data fall below the linear envelopes at shorter and shorter times [44,45]. These features can be seen in Fig. 10.9, which shows the data of Menezes and Graessley [46] for shear and first normal stress difference in start-up of simple shear. The dashed lines are calculated from the linear spectrum using Eqs.4.8 and 10.49. As expected, the... 

The ratio N y) 2 cr y), ie., SR/2, is often called the recoverable shear. However, it is only equal to the actual ultimate recoil y, i.e., the strain recovered after sudden release of the shear stress during steady shear, in the low shear rate limit, as indicated by Eq. 10.69. This is the behavior predicted by the rubberKke liquid model. 

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