Shear Dependence of Viscosity

Polymer fluids, including polymer solutions and melts, show deviations from ordinary Newtonian behavior, usually of the shear-thinning type. If the flow properties 

FIGURE 13.4 (a) Bohlin Gemini HRnano Rheometer by Malvern Instruments, and (b) parallel plate and cone-and-plate geometry. 

FIGURE 13.5 Idealized flow behavior of polymer melts and solutions indicating Newtonian 

Whereas a variety of constitutive equations have been proposed to describe non-Newtonian flow, one of the most useful relationships is the so-called power law 

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A basic theme throughout this book is that the long-chain character of polymers is what makes them different from their low molecular weight counterparts. Although this notion was implied in several aspects of the discussion of the shear dependence of viscosity, it never emerged explicitly as a variable to be investi-tated. It makes sense to us intuitively that longer chains should experience higher resistance to flow. Our next task is to examine this expectation quantitatively, first from an empirical viewpoint and then in terms of a model for molecular motion. 

Due to the increasing importance of biotechnology, which employs non-Newtonian fluids by far more frequently than chemical industry does, variable physical properties (e.g. temperature dependence, shear-dependence of viscosity) are treated in detail. It must be kept in mind that in scaling up such processes, apart from the geometrical and process-related similarity, the material similarity also has to be considered. 

The technological importance of xanthan gum rests principally on its unusual and distinctive properties25 28 29 49,116,251,257-260 in aqueous solution. Some of these properties are (1) remarkable emulsion-stabilizing and particle-suspension ability, (2) low concentrations yield high viscosities, (3) recoverable shear-thinning (extremely large shear dependence of viscosity), (4) little variation in viscosity with temperature under normal conditions of industrial utilization, and (5) gel formation when mixed with certain other, nongelling polysaccharides. 

The viscosity of polymer solutions is usually determined by measuring the flow time of a definite quantity of solution through a capillary. The driving force is the height of the fluid in the viscometer. A difficulty arises because polymer solutions are sufficiently oriented in ordinary capillary viscometers so that even at a low rate of shear the viscosity determined does not correspond to its real value at zero shear. In order to get values which are reproducible, regardless of the viscometer used, the viscosities, therefore, have to be determined at several rates of shear and extrapolated to zero shear rate as well as to zero concentration. This is particularly import t for high molecular-weight polymers where the shear dependence of viscosity is most pronounced. 

Hi) Shear Dependence of Viscosity, Non-Newtonian Behavior The presence of reinforcing fillers also increases the non-Newtonian behavior of elastomers. This effect is mainly due to the fact that the incorporation of fillers in elastomers decreases the volume of the deformable phase. As discussed in the following text, this decrease is not limited to the actual volume of the filler, but must also include the existence of occluded mbber. So, when filled mixes are submitted to shear forces, because of the lower deformable volume, the... 

Since the power-law and the Bingham plastic fluid models are usually adequate for modelling the shear dependence of viscosity in most engineering design calculations, the following discussion will therefore be restricted to cover just these two models where appropriate, reference, however, will also be made to the applications of other rheological models. Theoretical and experimental results will be presented separately. For more detailed accounts of work on heat transfer in non-Newtonian fluids in both circular and non-circular ducts, reference should be made to one of the detailed surveys [Cho and Hartnett, 1982 Irvine, Jr. and Kami, 1987 Shah and Joshi, 1987 Hartnett and Kostic, 1989 Hartnett and Cho, 1998]. 

Typically the viscosity of liquid crystalline fluids is highly dependent on shear rate over many orders of magnitude of y. This shear dependence of viscosity is illustrated by the data in Figure 7 for a 60 mole % PHB/PET copolyester. In this figure, data... 

In Figure 3 we have presented the stress growth curves at 275°C obtained at several different shear rates. The ppearance of the first peak occurs at y values of about 1.0 sec." whi]. e the second peak appears at values of y of about 5.0 sec.". In flexible chain systems stress overshoot occurs at values of y similar to the reciprocal of the loni gst relaxation time (t). Based on the shear dependence of viscosity, t should be at le st 100 sec. and hence for shear rates of the order of 0.01 sec.", overshoot should be observed. Howeyer, we observe that values of y must be of the order of 1.0 sec." before overshoot is observed. Hence, we cannot associate the- overshoot with the relaxation processes which occur in flexible chain polymers. 

Newtonian region, the latter may be complicated by unexpected relaxational contributions in the PCS decay function. Thus, based on the present results alone we are not able to unambiguously differentiate between the potential effects of hydrodynamic and weak associative interactions in congested solutions of PSM. Further work, utilizing techniques such as freeze-fracture electron microscopy and including a comprehensive analysis of the shear dependence of viscosity will be necessary to confirm the presence of secondary aggregation of PSM in solution. 

If a more complex shear dependence of viscosity, rjiy), than the power law fluid is now considered, then the expression for the shear rate at radius r is as follows ... 

In amoriDhous poiymers, tiiis reiation is vaiid for processes tiiat extend over very different iengtii scaies. Modes which invoived a few monomer units as weii as tenninai reiaxation processes, in which tire chains move as a whoie, obey tire superjDosition reiaxation. On tire basis of tiiis finding an empiricai expression for tire temperature dependence of viscosity at a zero shear rate and tiiat of tire mean reiaxation time of a. modes were derived ... 

Non-Newtonian fluids vary significantly in their properties that control flow and pressure loss during flow from the properties of Newtonian fluids. The key factors influencing non-Newtonian fluids are their shear thinning or thickening characteristics and time dependency of viscosity on the stress in the fluid. 

At least, in absolute majority of cases, where the concentration dependence of viscosity is discussed, the case at hand is a shear flow. At the same time, it is by no means obvious (to be more exact the reverse is valid) that the values of the viscosity of dispersions determined during shear, will correlate with the values of the viscosity measured at other types of stressed state, for example at extension. Then a concept on the viscosity of suspensions (except ultimately diluted) loses its unambiguousness, and correspondingly the coefficients cn cease to be characteristics of the system, because they become dependent on the type of flow. 

The simple result is attained when dependence of viscosity on shear rateq(Yapp)can be ignored in the interval of apparent shear rates realized along channel axis. Then,... 

Modify the dependence of viscosity on shear rate, producing low viscosity at high rates, and vice versa. 

Fig. 33. Dependence of viscosity on the shear rate of microgel solutions in C2H5OC2 H4OCOCCH3. EUP(MA+HD), c/t 70/30, EUP/S and EUP/EDMA(D), AIBN, P.-S. polystyrene [136].

The calculation method and equations presented in the previous sections are for Newtonian fluids such that the flow due to screw rotation and the downstream pressure gradient can be solved independently, that is, via the principle of superposition. Since most resins are highly non-Newtonian, the rotational flow and pressure-driven flow in principle cannot be separated using superposition. That is, the shear dependency of the viscosity couples the equations such that they cannot be solved independently. Potente [50] states that the flows and pressure gradients should only be calculated using three-dimensional (3-D) numerical methods because of the limitations of the Newtonian model. 

The viscosity of a thermosetting resin undergoing a curing reaction is a function of time, temperature, and degree of cure. In case a resin presents a non-Newtonian behavior, a dependence of viscosity on shear rate needs to be accounted for as well [111]. 

In steady shear flows, only the shear-rate dependence of viscosity is well documented in terms of molecular structure effects. Molecular theories are more... 

Viscoelastic behavior is classified as linear or non-linear according to the manner by which the stress depends upon the imposed deformation history (SO). Insteady shear flows, for example, the shear rate dependence of viscosity and the normal stress functions are non-linear properties. Linear viscoelastic behavior is obtained for simple fluids if the deformation is sufficiently small for all past times (infinitesimal deformations) or if it is imposed sufficiently slowly (infinitesimal rate of deformation) (80,83). In shear flow under these circumstances, the normal stress differences are small compared to the shear stress, and the expression for the shear stress reduces to a statement of the Boltzmann superposition principle (15,81) ... 

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