Zero-shear rate viscosity from

Fig. 37. Effect of the diluent molecular weight in blends with 1,800,000 molecular weight PS (X = 0.3) on craze fibril stability (e — ej. The samples were unfiltered and the strain rate used was 5 X 10 s Mc(as2Me) is the critical molecular weight for entanglement effects on the zero-shear-rate viscosity (From Ref. courtesy Macromolecules (ACS))...

These relationships mean that we can extend any oscillatory plot if steady-state data such as zero-shear-rate viscosity from a creep experiment is available (or vice versa). This then dictates the G curve, at least to the shear rate where the viscosity departs from r o, which in turn prescribes the frequency at which G" departs from t o< . 

The intimate contact data shown in Figure 7.16 were obtained from three-ply, APC-2, [0°/90o/0o]7- cross-ply laminates that were compression molded in a 76.2 mm (3 in.) square steel mold. The degree of intimate contact of the ply interfaces was measured using scanning acoustic microscopy and image analysis software (Section 7.4). The surface characterization parameters for APC-2 Batch II prepreg in Table 7.2 and the zero-shear-rate viscosity for PEEK resin were input into the intimate contact model for the cross-ply interface. Additional details of the experimental procedures and the viscosity data for PEEK resin are given in Reference 22. 

The distance between the L, and peaks, Alogx, decreases as the concentration decreases (Ref 1, Chap. 4) (see Fig. 9.13). This distance is important because it provides an objective measure of the length of the rubbery plateau. The results at hand seem to suggest that the plot of A log Xyn against log M, where is the volume fraction of polymer, gives a line of slope 3.4. The line intercepts the abscissa at a value of log((j)M) that is identical with that found from the break in the logpo vs log(( )M) plot, where Po is the zero shear rate viscosity. 

It is expected that the same picture that gives a good account of the linear viscoelastic behavior of polymer melts should also hold for semidilute and concentrated solutions. In the case of semidilute solutions some conclusions can be drawn from sealing arguments (19,3, p. 235). In this way, concentration dependence of the maximum relaxation time tmax the zero shear rate viscosity r Q, and the plateau modulus G% can be obtained, where t is the viscosity of the solvent. The relevant parameters needed to obtain Xmax as a function of concentration are b, c, N, kgT, and Dimensional analysis shows that... 

The above-described thickeners satisfy the criteria for obtaining very high viscosities at low stresses or shear rates. This can be illustrated from plots of shear stress a and viscosity tj versus shear rate y (or shear stress), as shown in Figure 10.25. These systems are described as pseudoplastic or shear thinning. The low shear (residual or zero shear rate) viscosity tj(0) can reach several thousand Pa s, and such high values prevent creaming or sedimentation [24, 25]. 

For colloidal particles, the dimensionless parameters are generally small and non-Newtonian effects dominate. Considering the same example as above, but with particles of radius a = 1 /xm, the parameters take on the values Pe = y, N y = 10 y, and N = 10 y so that for shear rates of 0.1 s or less they are all small compared to unity. The limit where the values of the dimensionless forces groups are very small compared to unity is termed the low shear limit. Here the applied shear forces are unimportant and the structure of the suspension results from a competition between viscous forces. Brownian forces, and interparticle surface forces (Russel et al. 1989). If only equilibrium viscous forces and Brownian forces are important, then there is well defined stationary asymptotic limit. In this case, there is an analogue between suspensions and polymers which is similar to that for the high shear limit, wherein the low shear limit for suspensions is analogous to the zero-shear-rate viscosity limit for polymers. 

The cone-and-plate and parallel-plate rheometers are rotational devices used to characterize the viscosity of molten polymers. The type of information obtained from these two types of rheometers is very similar. Both types of rheometers can be used to evaluate the shear rate-viscosity behavior at relatively low vales of shear rate therefore, allowing the experimental determination of the first region of the curve shown in Figure 22.6 and thus the determination of the zero-shear-rate viscosity. The rheological behavior observed in this region of the shear rate-viscosity curve cannot be described by the power-law model. On the other hand, besides describing the polymer viscosity at low shear rates, the cone-and-plate and parallel-plate rheometers are also useful as dynamic rheometers and they can yield more information about the stmcture/flow behavior of liquid polymeric materials, especially molten polymers. 

A master curve can be constructed as indicated in Figure 22.8, where the zero-shear-rate viscosity t]q has to be evaluated for each one of the indicated viscosity curves. Both, the effect of temperature and pressure on the viscosity versus shear rate curve can be addressed by considering a shift factor that may be related, for instance, to the free volume of the system by means of the Williams, Landel, and Ferry (WLF) equation [9, 15, 23, 24]. With the aid of this shift factor, the new viscosity curve can be constructed from known viscosity values and the reference curve at the prescribed values of temperature and a pressure. The use of shift factors to take into account the temperature dependence on the viscosity curve was also used by Shenoy et al. [19-21] in their methodology for producing viscosity curves from MFI measurements. 

De Gennes (1971) postulated that polymer molecules were constrained to move along a tube formed by neighbouring molecules. In a deformed melt, the ends of the molecules could escape from the tube by a reciprocating motion (reptation), whereas the centre of the molecule was trapped in the tube. When the chain end advanced, it chose from a number of different paths in the melt. This theory predicts that the zero-shear rate viscosity depends on the cube of the molecular weight. However, in the absence of techniques to image the motion of single polymer molecules in a melt, it is hard to confirm the theory. 

The frequency dependences of G (o)) and G"((o) for a typical polymer melt are reported in Figure 13.27. At low frequency G (ft)) < G"((0) and viscous behavior is dominant. This is the long-time or terminal region from which a measure of the zero shear rate viscosity can be derived. At intermediate frequencies, G ((o) > G"((0), and... 

Figure 4 shows the zero-shear-rate viscosity Tjo as a function of Ti of the silicone oils studied. The viscosity of a silicone oil changes with temperature as it varies from -20 through 3, 20, 50 to 70 °C. The average of the exponent of the studied samples is 1.196 0.203. There is obviously a correlation between Tio and T2. This means for a given, known silicone oil the viscosity can be determined by one NMR experiment. 

Fig. 5. The zero-shear-rate viscosity % as a function of T2 of the silicone oils studied, from the same data as Fig. 4.

A comparison of rheological NMR experiments and experiments for the determination of the molecular weight show that there are correlations between the zero-shear-rate viscosity, the T2 relaxation time and the molecular weight of silicone oils. The temperature was varied fiom -20 to 70 °C, the molecular weight from 650 to 250 (XX). 

A viscoelastic shift factor, as, can be found from the ratio of the experimentally measured zero shear rate viscosity (at test conditions of P, T, and SCF cOTicentratiOTi), to the experimentally measured zero shear rate viscosity (at some reference conditions of Fref> Iref, and reference SCF concentration). Once as is determined experimentally, a master curve can be constmcted by plotting ij/as vs. as 7 where tj is the measured viscosity and y is the measured shear rate. If the fractional free volume, /, is estimated from an equation of state as/ = 1 — pjp, where p is the mixture density and p is the mixture close-packed density, then the shift factor due to the presence of the SCF can be calculated from Eq. (18.3), provided the constant B is known. Experimentally, B for SCF-swoUen polymers has been found to be near unity [130,131], in agreement the universal constants of the WLF equation [132] for the temperature dependence of pure polymers. 

Table 18.7 lists viscosity reductions measured due to the presence of an SCF in the polymer. Because few researchers have reported the zero shear rate viscosity for both the pure polymer melt and the SCF-swollen melt, an experimental shift factor cannot usually be estimated from the data. Instead, Table 18.7 lists the ratio of viscosity measured in the presence of SCF diluent (rjoii) to the viscosity measured in the absence of diluent (rj), at some reference conditions. Unless otherwise specified, the reference state has no SCF, and has the same temperature as the experimental state. The pressure for the reference state, and dynamic characterization conditions (e.g., constant stress, cr, or constant y) are also specified in Table 18.7. 

Polymer solutions were characterized for conductivity, zero shear rate viscosity ( /o), and surface tension prior to extrusion. The extruded polymer was characterized for the presence of residual solvent and pyridine using Fourier Ti ansform Infrared Spectroscopy (FTIR). As a complimentary technique, CHN elemental analyzer was used to determine if any pyridinium acetate residue remained (based on detecting nitrogen) in the extruded polymer. For the latter, PCL films were cast from the 15% solution containing 0%, 0.5%, 1%, 2% and 5% pyridine. 

Interrelationships among the Viscoelastic Material Functions. There is a continuing disagreement within the molecular viscoelasticity commimity about which of the above methods should be used to characterize a material (20). In fact, if one can obtain the zero shear rate viscosity and any of the other functions, these methods are all equivalent. The issue, however, revolves aroimd the fact that some features that appear in the dynamic modulus disappear if the compliance is used as the function to represent the data and vice versa. Also, some measurements are more or less dominated by the viscosity contribution. As a result, some problems of misinterpretation of data could be averted if workers who prefer modulus representations would calculate the compliances. In addition, those who measure the compliance should calculate the moduli in order to provide the data in the format that is more common in the field because of the large number of commercial instruments that obtain dynamic moduli. The advent of modern software packages that make the interrelationships easily calculated makes this dispute seem to go away. The pathways to determine the different material functions, one from the other, are shown in Figure 6. 

The zero shear rate viscosity is calculated from (56)... 

FIG. 6 Zero-shear rate viscosity tjq vs. molecular weight M, obtained from the MD simulation and the Rouse model for small M (circles) or the reptation model for high M (squares). Also shown in the figure are experimentally obtained rjo values (triangles). 

The incorporation of the macromer was investigated and calculated by NMR measurements. The maximum incorporation rate was 0.52 mol%. This means that about 59.7 wt% of the polymer is composed of macromer units and, on average, every 400th carbon atom of the backbone chain is branched. The melting point of the long-chain branched polyethylene decreases from 136 to 12 PC, and the zero shear-rate viscosity increases from 142 to 280 Pa s. Such long-chain branched copolymer can be produced much more easily by metallocene/MAO catalysts than by Ziegler-Natta catalysts. 

The Rouse theory is clearly not applicable to polymer melts of a molar mass greater than (M ) for which chain entanglement plays an important role. This is obvious from a comparison of eqs (6.40)-(6.42) and experimental data (Figs 6.13 and 6.14) and from the basic assumptions made. However, for unentangled melts, i.e. melts of a molar mass less than (M ), both the zero-shear-rate viscosity and recoverable shear compliance have the same molar mass dependence as was found experimentally (Figs 6.13 and 6.14). The Rouse model does not predict any shear-rate dependence of the shear viscosity, in contradiction to experimental data. 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

FIG. 7 Dependence of zero shear rate viscosity for mixture of polyethylene and paraffin on number of C atoms in backbone. (Data from Ref. 21.)... 

The temperature dependence of D may be obtained from the zero shear-rate viscosity of the polymer in an indirect way. Both D and the zero shear-rate viscosity, Qq, are related through xd and hence the monomeric friction coefficient. One can obtain iJq from the stress relaxation modulus, G(t) > ... 

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