High shear limit relative viscosity

A grafted layer of polymer of thickness L increases the effective size of a colloidal particle. In general, dispersions of these particles in good solvents behave as non-Newtonian fluids with low and high shear limiting relative viscosities (fj0 and rj ), and a dimensionless critical stress (a3aJkT) that depends on the effective volume fraction = (1 + L/a)3. The viscosities diverge at volume fractions m0 < for mo < fan < 4>moo> the dispersions yield and flow as pseudoplastic solids. 

Figure 12 shows the variation of the high shear limit relative viscosity variation with particle volume fraction. One can observe that large discrepancies are present in the experimental data among different studies. This indicates the difficulty in measuring the viscosity of suspensions. Many factors can affect the experimental measurements. For instance, the uniformity of the particles, properties of the suspending medium, the wall effects of the viscometer, and even the time of the experiment (92). 

Jones and co-workers (72, 88) found that the suspension viscosity variation with shear rate can be fitted fairly well by the Cross equation, equation 12, with m = 0.5 — 0.84. Both the low and high shear limit relative viscosities, Mro, Mroo> can be expressed by the Quemada s equation with 0max = 0.63 and 0.71, respectively. 

Figure 12. High shear limit relative viscosity variation with volume fraction.

Figure 9.3.1 High shear limit relative viscosity for monodisperse spherical particles as a function of solids volume fraction. Circles are data of Shapiro 8c Probstein (1992), squares are data of de Kruif et al. (1986), curves are semi-empirical equations.

Since the apparent viscosity 17—>00 as (f)— lk, it follows that l/k may be interpreted as the maximum packing fraction (f,- "f e high shear limit relative viscosity may therefore be written... 

The equation used to fit the data of the high shear limit relative viscosity giving (f), = 0.53 in Fig. 9.3.1 is Eq. (9.3.8) with C = 1.5 found to give the best fit to the viscosity data (Shapiro Probstein 1992). This equation was also used for determining d> , for the bidisperse data of Fig. 9.3.2. 

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 ... 

Figure 13 shows a typical plot of the steady shear relative viscosity versus the Peclet number for polystyrene spheres of various sizes suspended in various fluids. The success of the Peclet number scaling is well observed. One can also observe that the viscosity is higher when the shear rate is small, and at both high and low shear limits, the viscosity curve shows a plateau, corresponding to the high and low shear limit Newtonian behavior. The explanation for this behavior has been, in part, discussed earlier for the random packing limit of the particles. 

Figure 14 shows the variation of the steady shear relative viscosity at the high shear limit with the effective volume fraction as defined by equation 46 for poly(methyl methacrylate) (PMMA) suspensions of different sizes in decalin sterically stabilized by means of grafted poly (12-hydroxy stearic acid) chains with a degree of polymerization of 5. The stabilizing polymer layer thickness is 9 1 nm, in particular, A = 9 nm... 

Figure 9.4.1 Relative viscosity of a bidisperse coal slurry made up of a colloidal fine fraction of mean diameter 2.3 /j,m and a noncolloidal coarse fraction of 200—300 m particles of mean diameter about 250 fim as a function of shear rate. The volume fraction of the colloidal particles = 0.30 and of the coarse particles , = 0.52. The solid line is a mean curve through the measured viscosities of the colloidal fraction. The triangles are the experimental points for the measured viscosity for the fine plus coarse mixture. The dashed line is the fine relative viscosity experimental curve redrawn through the data points to illustrate the parallelism. The upward shift of this curve corresponds to a coarse relative viscosity log 77, = 2.13. [After Sengun, M.Z. Probstein, R.F. 1989. Bimodal model of slurry viscosity with application to coal-slurries. Part 2. High shear limit behavior. Rheol. Acta 28, 394-401. Steinkopff Darmstadt. With permission.]...

The viscosity curve of a typical non-Newtonian product such as polystyrene is plotted in Figure 8 as a function of the shear gradient. Because of the small gap between rotor and heating wall, together with the relatively high rotor speed, the thin-film machine works within the limits of shear gradient of 1000 to 10,000 sec 1. 

Now the technique provides the basis for simulating concentrated suspensions at conditions extending from the diffusion-dominated equilibrium state to highly nonequilibrium states produced by shear or external forces. The results to date, e.g., for structure and viscosity, are promising but limited to a relatively small number of particles in two dimensions by the demands of the hydrodynamic calculation. Nonetheless, at least one simplified analytical approximation has emerged [44], As supercomputers increase in power and availability, many important problems—addressing non-Newtonian rheology, consolidation via sedimentation and filtration, phase transitions, and flocculation—should yield to the approach. 

Figure 4.6 shows that below and above T. the apparent shear viscosity expressed by shear-stress/shear-rate takes on very hi values. Below the polymer could not be extruded within the limits of the apparatus and above T. FIFE of sufficient molecular mass is known to have mdt viscosities of above 10 Pa s Below polymorph II is a rigid crystal with practically only vibrational motion and above T. the melt viscosity is high because of entanglement of the relatively stiff long-chain molecules. 

The predicted drop size for a simple field is proportional to interfacial tension and inversely proportional to shear rate and matrix phase viscosity. Although Newtonian systems are relatively well understood, there are many limitations to this theory for predicting the morphology of a multiphase polymer system. Other difficulties in comparison with such ideal systems may include the complex shear fields applied in processing and the relatively high concentrations of the dispersed phase in most commercial polymer blends. 

A wide variety of plastics can be used although about 80 percent is PVC and 15 percent ABS. (Calendering consumes about 6 percent of total plastics consumption for all processes.) Other plastics used are HOPE, PP, and styrenes. The basic plastic limitation of the calendering process is the need to have a sufficiently broad melt index to allow a heat range for the process. This permits the material to have a relatively high viscosity in the banks of the calender (banks indicating where two rolls meet, or the nip of the rolls). As a result of the viscosity, a shear effect can be developed throughout the process, and especially between the calender rolls. Thus, the calender forms... 

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