Suggested Rheological Models

In order to simplify the final step in generating the required material parameter versus shear rate curves from by the above procedure, appropriate rheological models have been suggested for fitting the unified curves. 

In the master rheograms of viscosity versus shear rate given in Chapter 4, the following four models have been suggested [10,11] covering a varied range of shear rates  

The modified Carreau model and the modified Ellis model are limited to relatively low values of shear rates and shear stresses, reflectively, whereas the modified Ostwald-de Waele power-law model is applicable to the higher-shear-rate region where the data points fall in a straight line on the log-log plot of -q X MFI versus y/MFL 

The plot of I) X MFI versus y/MFI on the log-log scale in the low-shear-late region yields the limiting values i)o X MFI by mere readout as shown in Fig. 6.1. 

The functional behavior at large shear rates on the same log-log plot in Fig. 6.1 being linear defines K and n direcQy. The slope of the strai t line defines rt — 1, whereas is the value of i X MFI when y/MFI = 1 novided the point satisfies the power-law equation). The best values of K and R — 1 can be easily computed by regressional analysis of the data at high shear rates. 

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It is suggested that these results could change significantly the way mechanisms of synovial joint lubrication are examined. Effects of biochemistry of the system on wear of articular cartilage are likely to be important such effects may not be related to physical/rheological models of joint lubrication. 

The utility of the master curves presented in Chapters 4 and 5 is obvious in the sense that the individual curves for any grade of polymer at any temperature of interest can be merely readout and replotted by knowing the appropriate MFI value. However, the easier method is to use equations which fit the master curve and regenerate data by mere substitution of the MFI value. In order to do this, rheological models have been suggested in Chapter 6. 

For a given suspension rheology and flow rate there is a critical permeability of the filter, below which no cake will be formed. The model also suggests that the equilibrium cake thickness can be precisely controlled by an appropriate choice of suspension flow rate and filter permeability. 

The rheological properties change behavior, relative to more dilute solutions, above cp = 0.2, where non-Newtonian behavior is then exhibited. The power law dependence of rj on cp is in harmony with the Zimm rather than the Rouse model, which suggests that hydrodynamic interactions between these polymers, in a mean field sense, are important. Electrical properties also begin to deviate for Nafion solutions above cp = 0.2, and mechanical percolation is essentially the same for the sodium and acid forms. 

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

I would also like to list some of the challenges that will provide the foundation for where the profession has to go (Fig. 2). This is not meant to be comprehensive, but to suggest some of what we should be doing. This wish list derives from work Bob Brown and I have done on modeling flows of polymer fluids. The first item has to do with the need to understand the effects of polymer structure and rheology on flow transitions in polymeric liquids and on polymer processing operations. In the past, we ve studied extensively the behavior of Newtonian fluids and how Newtonian flows evolve as, say, the Reynolds number is varied. We have tools available to... 

All the above formulas are one-parameter equations, i.e. they relate the dispersion viscosity only to the volume fraction of particles contained in it. This limits the range of applicability of the equations to not very high dispersion concentrations. To take account of the influence of the structure of concentrated dispersions on their rheological behavior, Robinson [12] suggested that the viscosity of dispersions is not only propertional to the volume fraction of solid phase, but is also inversely proportional to the fraction of voids in it. (At about the same time Mooney [40], who proceeded from a hydrodynamic model, arrived, using theoretical methods, at the same conclusion). Robinson s equation contains the relative sedimentation volume value — S, which depends on the particle size distribution of the dispersion... 

In summary, the QCM-D technique has successfully demonstrated the adsorption of pectin on the BSA surface as well as determined the viscoelastic properties of the pectin layer. As pectin concentrations increase, the adsorbed mass of pectin estimated from the Voigt model show higher values than those estimated from the Sau-erbrey equation because the former takes into account the hydrated layer. But the similar increase of thickness of pectin suggests that the pectin chains form a multilayer structure. In agreement with our previous rheology results, the main elastic character of the pectin layer in terms of Q-tool software tells us the network structure of the pectin layer on the BSA surface. In summary, QCM-D cannot only help to better understand the polysaccharide/protein interactions at the interface, but also to gain information of the nanoscale structure of polysaccharide multilayers on protein surface. 

The homogeneous non-Newtonian capillary tube-power law model has a number of limitations. The models assume a power law relationship for the emulsion, and any deviations from this rheological behavior will lead to errors. The power law constants n and K are obtained by using viscometry, and their validity in porous media is questionable. No transient permeability reduction (assumption 4) is predicted, even though experimental evidence suggests otherwise. This model is seen to have validity only for high-quality emulsions that approach steady state quickly and have small droplet-size to pore-size ratios. 

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