Models Shear viscosity data

Fitting a Bird-Carreau model to viscosity data. Table 7.5 shows the measurements of Ballenger et. al [5] of the viscosity as a function of shear rate for polystyrene at 453 K. 

Figure 6-29 shows that this simple model gives a good fit to the concentration-dependence of the shear viscosity data for polystyrene latices in 5 x 10" M NaCl. 

Correlation Between Steady-Shear and Oscillatory Data. The viscosity function is by far the most widely used and the easiest viscometric function determined experimentally. For dilute polymer solutions dynamic measurements are often preferred over steady-shear normal stress measurements for the determination of fluid elasticity at low deformation rates. The relationship between viscous and elastic properties of polymer liquids is of great interest to polymer rheologists. In recent years, several models have been proposed to predict fluid elasticity from shear viscosity data. 

Figure 4 illustrates how the Carreau-Yasuda model meets the shear viscosity data of Fig. 2. A non-linear fitting algorithm (i.e. Marquardt-Levenberg) was used to obtain the parameters given in the inset. As can be seen the fit curve provides a shear viscosity function that corresponds reasonably well with experimental data so that the high shear behavior is asymptotic to a power law and the very low shear behavior corresponds to the pseudo-Newtonian viscosity po- The characteristic time X (56.55 s) can be considered as the reverse of a critical shear rate (i.e. = Yc = 0.0177 s ) that corresponds to the intersection between the high shear power...

The entire discussion above is concerned with getting accurate shear viscosity data. It is also possible to use slit geometry to obtain information on the normal stress differences in shear. As with a capillary, extrudate swell occurs as liquid leaves a slit. Again, starting fi-om an integral model, a relation similar to eq. 6.2.27 can be derived... 

Fitting experimental data with the above model is not straightforward and requires both an extanded set of data and the appropriate fitting strategy, as demonstrated below with shear viscosity data on a series of Ti02 filled polystyrene at 180°C, as published by Minagawa and White [/. Appl. Polym. Sci, 20,501-523,1976]. 

A6.4.9 Fitting the Filled Polystyrene Shear Viscosity Data Model equation ... 

Given here are shear viscosity data at 20°C for a 500-ppm solution of a high-molecular-weight polyacrylamide in distilled water. Obtain the best-fit parameters for the power-law and Carreau models. ... 

Fig. 1 Shear viscosity data (symbols) and Cross-WLF model fit (curves) to the viscosity data for the ABS resin at 190,230,and270°C.

Figure 5 Comparison between experimentally determined transient shear viscosity data (5a-5c)/first normal stress coefficient (5d-5f) and predictions of the XPP, PTT-XPP and mLeonov models for HDPE Tipelin at 180 °C.

Modeling of the melt viscosity of polyethylene and random copolymers of ethylene and a-olefins has been extensively dealt with in the past. Empirical viscosity models of the form of the Generalized Cross/Carreau models can and have been fitted to viscosity data for INSITE Technology Polymers. The shear viscosity data is usually measured at 190 °C in the molten regime from 0.1 - 100 rad/s. Of the several models available, the Cross model provides a good fit to the data with a minimum number of fitting parameters. The Cross model is of the form ... 

Other schemes have been proposed in which data are fit to a lower, even order polynomial [19] or to specific rheological models and the parameters in those models calculated [29]. This second approach can be justified in those cases when the range of behavior expected for the shear viscosity is limited. For example, if it is clear that power-law fluid behavior is expected over the shear rate range of interest, then it would be possible to calculate the power-law parameters directly from the velocity profile and pressure drop measurement using the theoretical velocity profile... 

The plot of viscosity versus shear rate is shown in Fig. 3-3b, in which the line represents Eq. (3-24), with n = 0.77 and m = 1.01 dyn s /cm2 (or poise ). In this case the power law model represents the data quite well over the entire range of shear rate, so that n = n is the same for each data point. If this were not the case, the local slope of log T versus log rpm would determine a different value of n for each data point, and the power law model would not give the best fit over the entire range of shear rate. The shear rate and viscosity would still be determined as above (using the local value of n for each data point), but the viscosity curve could probably be best fit by some other model, depending upon the trend of the data (see Section III). 

Although the analysis here was performed using a power law viscosity model, other models could be used. For other viscosity models, the power law value n would be calculated using two reference shear rates, one higher and one lower than the shear rate calculated using Eq. 7.41. These high and low shear rates and viscosity data would be used to determine a local n value as follows ... 

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. 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

Basic Protocol 2 is for time-dependent non-Newtonian fluids. This type of test is typically only compatible with rheometers that have steady-state conditions built into the control software. This test is known as an equilibrium flow test and may be performed as a function of shear rate or shear stress. If controlled shear stress is used, the zero-shear viscosity may be seen as a clear plateau in the data. If controlled shear rate is used, this zone may not be clearly delineated. Logarithmic plots of viscosity versus shear rate are typically presented, and the Cross or Carreau-Yasuda models are used to fit the data. If a partial flow curve is generated, then subset models such as the Williamson, Sisko, or Power Law models are used (unithi.i). 

To calculate the shear rate constant, k, a relationship must be established between shear rate and viscosity of a non-Newtonian calibration fluid. A cone-and-plate viscometer is used to determine a correlation between shear rate and viscosity that can be fit to a power law model. The power law correlation is then applied to viscosity data calculated from the impeller viscometer and Eq. 4. The shear rate constant can be calculated as follows ... 

Sewell and co workers [145-148] have performed molecular dynamics simulations using the HMX model developed by Smith and Bharadwaj [142] to predict thermophysical and mechanical properties of HMX for use in mesoscale simulations of HMX-containing plastic-bonded explosives. Since much of the information needed for the mesoscale models cannot readily be obtained through experimental measurement, Menikoff and Sewell [145] demonstrate how information on HMX generated through molecular dynamics simulation supplement the available experimental information to provide the necessary data for the mesoscale models. The information generated from molecular dynamics simulations of HMX using the Smith and Bharadwaj model [142] includes shear viscosity, self-diffusion [146] and thermal conductivity [147] of liquid HMX. Sewell et al. have also assessed the validity of the HMX flexible model proposed by Smith and Bharadwaj in molecular dynamics studies of HMX crystalline polymorphs. 

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