Aspect ratio, shear viscosity

According to the Einstein theory, the intrinsic viscosity of a spherical particle suspension is 2.5. However, for a colloidal suspension of nonspherical particles, [r ] > 2.5. Jeffery [112] obtained the viscosity of an ellipsoidal particle suspension under shear. Incorporating Jeffery s results of velocity fields around the particle, Simha [113] obtained expressions for two explicit limiting cases of ellipsoids. Kuhn and Kuhn [114] also obtained an expression for intrinsic viscosity for the full range of particle aspect ratio (p) by taking an approach similar to Simha s method. 

Brenner (1974) has presented numerical results for the suspension stresses in various flows. Figure 6-14 plots the intrinsic viscosity [defined in Eq. (6-6)] for oblate and prolate spheroids of various aspect ratios as functions of the Peclet number. Note that as the aspect ratio of the spheroid increases, the zero-shear viscosity increases, and the suspension shows more shear thinning. The suspension also becomes more elastic when the aspect ratio p for prolate or 1/ for oblate spheroids) is large see Fig. 6-15, which plots Ni N2 versus Pe for prolate spheroids of various aspect ratios p. Typically, N2 is roughly an order of magnitude less than Ni, so this plot of Nj, mainly reflects the behavior of V,. 

In fact, the fiber contribution to the shear viscosity of a fiber suspension at steady state is modest, at most. The reason is that, without Brownian motion, the fibers quickly rotate in a shear flow until they come to the flow direction in this orientation they contribute little to the viscosity. Of course, the finite aspect ratio of a fiber causes it to occasionally flip through an angle of n in its Jeffery orbit, during which it dissipates energy and contributes more substantially to the viscosity. The contribution of these rotations to the shear viscosity is proportional to the ensemble- or time-averaged quantity (u u ), where is the component of fiber orientation in the flow direction and Uy is the component in the shear gradient direction. Figure 6-21 shows as a function of vL for rods of aspect... 

In transient shear flows starting from an isotropic distribution of fiber orientations, considerably higher viscosities will be initially observed, until the fibers become oriented. In Bibbo s experiments, t]r for isotropically oriented fibers is around 3.5 for v = 75. These viscosities can also be predicted reasonably well by semidilute theory and by simulations (Mackaplow and Shaqfeh 1996). Figure 6-25 shows the shear stress as a function of strain for a polyamide 6 melt with 30% by weight glass fibers of various aspect ratios, where the fibers were initially oriented in the flow-gradient direction. Notice the occurrence of a stress overshoot (presumably due to polymer viscoelasticity), followed by a decrease in viscosity, as the fibers are reoriented into the flow direction. 

At low enough shear rates, polymeric nematics ought to obey the same Leslie-Ericksen continuum theory that describes so well the behavior of small-molecule nematics. The main difference is that polymers have a much higher molecular aspect ratio than do small molecules, which leads to greater inequalities in the the numerical values of the various viscosities and Frank constants and to much higher viscosities. 

Bicerano et al. (1999) provide a simplified scaling viscosity model for particle dispersions that states the importance of the shear conditions, the viscosity profile of the dispersing fluid, the particle volume fraction and the morphology of the filler in terms of its aspect ratio, the length of the longest axis and the minimum radius of curvature induced by flexibility. 

Figure 13.15. Effects of volume fraction O and aspect ratio Af on the zero-shear viscosity rio(relative) for dispersions of infinitely rigid anisotropic filler particles. The curve labels denote Af. At any given d> and Af, T)o increases more with fibers than it does with platelets.

Figure 13.18. Comparison between model and measurements for clays dispersed in polymers. Viscosities observed as a function of shear rate by Krishnamoorti et al [46] for dispersions of silicate platelets (weight fractions of 0.06 and 0.13) in poly(dimethyl siloxane) at T=301K are indicated with symbols. Calculated results, assuming platelets to be monodisperse flexible cylinders with aspect ratio Af=(thickness/diameter)=0.01, are indicated as lines, (a) Relative viscosity=r)(dispersion)/r (polymer). (b) Dispersion viscosity, r)(dispersion).

Figure 17. Low shear limit viscosity variation with solid volume fraction and aspect ratio for suspensions of spheroids and cylinders (118).

Figure 7.12 illustrates the effect of shear rate, initial drop diameter, and the viscosity ratio on the droplet aspect ratio, p. For low and high values of X, pseudoplastic dependence has been observed [Talstoguzov et al., 1974]. 

Figure 7.12. Effect of the shear rate (top), the initial drop diameter (middle), and the viscosity ratio (bottom) on the drop aspect ratio for the systems water-gelatin-dextran (W-G-D, circles) and water-gelatin-polyvinyl alcohol (W-G-P, squares) [Talstoguzov et al., 1974].

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