Elongational/shear viscosities ratio

This simple approach was found to be more accurate than models that used the shear viscosity ratio for blends with a viscosity ratio not far from unity [55], as the torque reflects to some extent also the effects of melt elasticity and elongational flow component. Lyngaae-Jorgensen and Utracki developed a model which predicts the range of cocontinuity, and not only the phase inversion point [56,57]. This model, which is based on the percolation theory, relates the degree of cocontinuity 0A. which can be determined experimentally from extraction experiments, to the volume fraction of the given blend component (pp (Eq. (3.27)). 

In manufacturing and processing polymer blends, it is thus important that the viscosity ratio be within the optimal range in the actual processing conditions. Not only the polymers to be blended but also the temperature and processing conditions (shear, elongation) should be carefully selected. Other factors, such as interfacial tension [46,47] and elasticity of the blended polymers, may also influence the blend morphology. 

In contrast to rotational shear flow, deformation and breakage occurs over the whole range of viscosity ratio in an irrotational (extension) flow produced, for example, in a 4-roll apparatus (Fig. 23) from which the data shown in Fig. 21 were obtained [76]. Comparing the critical conditions for breakage by shear and by elongation. Fig. 23 shows that for equal deformation rates, irrotational flow tends to be more damaging than rotational flow. 

Elongational flow is more effective than simple shear flow for a given viscosity ratio. 

Fig. 7.23 Critical capillary number for droplet breakup as a function of viscosity ratio p in simple shear and planar elongational flow. [Reprinted by permission from H. P. Grace, Chem. Eng. Commun., 14, 2225 (1971).]...

Figure 19. The ratio of elongational to shear viscosities The theoretical dependence of the ratio of elongational to shear viscosity coefficients on the invariant of the additional stress tensor is calculated according to equation (9.71) and depicted by the dashed curve. The solid curves represent experimental data for systems listed in Table 3. Adapted from the paper of Pokrovskii and Kruchinin (1980).

By eliminating the velocity gradient from relations (9.69) and (9.70) we can obtain an expression for the ratio of the coefficients of elongational and shear viscosity... 

An elongational or extensional viscosity (%) develops as a result of a conformational transition when disperse systems are forced through constrictions, or compressed or stretched (Kulicke and Haas, 1984 Rinaudo, 1988 Barnes et al., 1989 Odell et al., 1989 Clark, 1992). The intuitive logic is that the random coils resist the initial distortion. % is believed to elicit the human sensation of stringiness (Clark, 1995). If shear viscosity is denoted iq, rheologists define a Trouton ratio as %/ti, wherein % > T) by a factor approximating 3 for uniaxial extension and 6 for biaxial extension. Alternatively stated, the Newtonian ly calculates to one-third to one-sixth % (Steffe, 1992). 

As a conclusion, if the viscosity ratio p between the internal and external phases lies between 0.01 and 2, the shear applied on a polydisperse emulsion made of large drops leads to a monodisperse one with a mean diameter governed by the stress. This fragmentation occurs through elongation of the drops and the development of a Rayleigh instability with a characteristic time of the order of one second. The obtained monodispersity probably results from the fact that the Rayleigh instability develops under shear for a critical diameter of the deformed drops. 

This formula is crude, and it does not account for differences in shear rates between the droplet and the medium (which are large when the viscosity ratio differs greatly from unity). Nevertheless, because of the shear-rate-dependence of, Eq. (9-22) can predict a.minimum in droplet size as a function of shear rate that is observed in some cases (Sundararaj and Macosko 1995 Plochocki et al. 1990 Favis and Chalifoux 1987). Viscoelastic forces have indeed been shown to suppress the breakup of thin liquid filaments that would otherwise rapidly occur via Rayleigh s instability (Goldin et al. 1969 Hoyt and Taylor 1977 Bousfield et al. 1986). Elongated filaments, for example, are observed in polymer blends (Sondergaard... 

The Trouton ratio (Tr) may also be defined as the ratio of the elongation viscosity (rj ) to the steady-shear viscosity (rj),... 

FIGURE 11.8 The effect of the viscosity ratio, drop over continuous phase inn/tlc), on the critical Weber number for drop breakup in various types of laminar flow. The parameter a is a measure of the amount of elongation occurring in the flow for a — 0, the flow is simple shear for a — 1, it is purely (plane) hyperbolic. 

The flow pattern can be intermediate between simple shear and pure elongation, and such a situation is very common in stirred vessels and comparable apparatus at Re < Recr. The extent to which elongation contributes to the velocity gradient can be expressed in a simple parameter, here called a, that can vary between 0 and 1. As shown in Figure 11.8, a little elongation suffices to reduce markedly the magnitude of Wecr and allows breakup at a higher viscosity ratio. 

The mechanisms governing deformation and breakup of drops in Newtonian liquid systems are well understood. The viscosity ratio, X, critical capillary number, and the reduced time, t, are the controlling parameters. Within the entire range of X, it was found that elongational flow is more efficient than shear flow for breaking the drops. 

Han and Funatsu [1978] studied droplet deformation and breakup for viscoelastic liquid systems in extensional and non-uniform shear flow. The authors found that viscoelastic droplets are more stable than the Newtonian ones in both Newtonian and viscoelastic media they require higher shear stress for breaking. The critical shear rate for droplet breakup was found to depend on the viscosity ratio it was lower for < 1 than for A, > 1. In a steady extensional flow field, the viscoelastic droplets were also found less deformable than the Newtonian ones. In the viscoelastic matrix, elongation led to large deformation of droplets [Chin and Han, 1979]. 

The microrheology makes it possible to expect that (i) The drop size is influenced by the following variables viscosity and elasticity ratios, dynamic interfacial tension coefficient, critical capillarity number, composition, flow field type, and flow field intensity (ii) In Newtonian liquid systems subjected to a simple shear field, the drop breaks the easiest when the viscosity ratio falls within the range 0.3 < A- < 1.5, while drops having A- > 3.8 can not be broken in shear (iii) The droplet breakup is easier in elongational flow fields than in shear flow fields the relative efficiency of the elongational field dramatically increases for large values of A, > 1 (iv) Drop deformation and breakup in viscoelastic systems seems to be more difficult than that observed for Newtonian systems (v) When the concentration of the minor phase exceeds a critical value, ( ) >( ) = 0.005, the effect of coalescence must be taken into account (vi) Even when the theoretical predictions of droplet deformation and breakup... 

The shear flow is inefficient for dispersing one polymer in another if they significantly differ in viscosity, especially when the viscosity ratio X 3.8. The elongational field is more proficient and rapid. On all accounts, viz. ... 

The earliest determinations of elongational viscosity were made for the simplest case of uniaxial extension, the stretching of a fibre or filament of liquid. Trouton [1906] and many later investigators found that, at low strain (or elongation) rates, the elongational viscosity he was three times the shear viscosity n [Barnes et al., 1989], The ratio Mb/M is referred to as the Trouton ratio, Tr and thus ... 

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