Newtonian behavior

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

The amount of curvature in plots like Fig. 2.2 is a measure of the deviation from Newtonian behavior. We can use the logic of calculus to argue that such curvature becomes less apparent as we examine progressively smaller segments of the line. This statement leads us to two important conclusions ... 

If our objective is to examine non-Newtonian behavior, we must design experiments which permit the relationship between and dv/dy to be studied over as wide a range as possible. This topic is taken up in the next section. 

If we wish to avoid the complication of non-Newtonian behavior, we must focus attention on a relatively narrow range of values for dv/dy. 

For the straight line in Fig. 2.5 where m = 1.0, this equation expresses direct proportionality between and y, the condition of Newtonian behavior. In the non-Newtonian region where m < 1, Eq. (2.11) may describe the data over an order of magnitude or so. Next we consider the relationship between the constant K and viscosity. If Eq. (2.11) is solved for K and the resulting expression multiplied and divided by 7 ", we obtain... 

Figure 2.5 reveals that polymer viscosity approaches Newtonian behavior for sufficiently low rates of shear. From an empirical point of view, this simply means that m 1 as 7 0. From a molecular point of view, in the region of... 

Newtonian behavior the rate of shear is small compared to the rate constant for the flow process. When molecular displacements occur very much faster than the rate of shear (7 < kj ), the molecules show maximum efficiency in dissipating the applied forces. When the molecules cannot move fast enough to keep pace with the external forces, they couple with and dissipate those forces to a lesser extent. Thus there is a decrease in viscosity from its upper, Newtonian limit with increasing 7/kj. The rate constant for the flow process is therefore seen to define a standard against which the rate of shear is to be judged large or small. In the next section we shall consider a molecular model in terms of which this rate constant can be analyzed. 

Polymers display Newtonian behavior = constant) at sufficiently low rates of shear [Eq. (2.29)]. 

Apply Eq. (2.27) to some of the data points to evaluate the apparent viscosity at different 7 s. The first section of Table 2.2 shows the results of such calculations. Note that the calculated 17 s are constant at low 7 values, indicating Newtonian behavior. Table 2.2 also expresses all 17 values relative to the Newtonian limiting value 17 - Comparison of Eqs. (2.28) and (2.29) shows that t7/t7im values decrease from the Newtonian limit by the fraction sinh" (j37)/j37. 

From plots of these data, estimate the Newtonian viscosity of each of the solutions and the approximate rate of shear at which non-Newtonian behavior sets in. Are these two quantities better correlated with the molecular weight of the polymer or the molecular weight of the arms ... 

Suppose we consider a spring and dashpot connected in series as shown in Fig. 3. 7a such an arrangement is called a Maxwell element. The spring displays a Hookean elastic response and is characterized by a modulus G. The dashpot displays Newtonian behavior with a viscosity 77. These parameters (superscript ) characterize the model whether they have any relationship to the... 

Rheology. Both PB and PMP melts exhibit strong non-Newtonian behavior thek apparent melt viscosity decreases with an increase in shear stress (27,28). Melt viscosities of both resins depend on temperature (24,27). The activation energy for PB viscous flow is 46 kj /mol (11 kcal/mol) (39), and for PMP, 77 kJ/mol (18.4 kcal/mol) (28). Equipment used for PP processing is usually suitable for PB and PMP processing as well however, adjustments in the processing conditions must be made to account for the differences in melt temperatures and rheology. 

Gla.ss Ca.pilla.ry Viscometers. The glass capillary viscometer is widely used to measure the viscosity of Newtonian fluids. The driving force is usually the hydrostatic head of the test Hquid. Kinematic viscosity is measured directly, and most of the viscometers are limited to low viscosity fluids, ca 0.4—16,000 mm /s. However, external pressure can be appHed to many glass viscometers to increase the range of measurement and enable the study of non-Newtonian behavior. Glass capillary viscometers are low shear stress instmments 1—15 Pa or 10—150 dyn/cm if operated by gravity only. The rate of shear can be as high as 20,000 based on a 200—800 s efflux time. 

The increase in fuel viscosity with temperature decrease is shown for several fuels in Figure 9. The departure from linearity as temperatures approach the pour point illustrates the non-Newtonian behavior created by wax matrices. The freezing point appears before the curves depart from linearity. It is apparent that the low temperature properties of fuel are closely related to its distillation range as well as to hydrocarbon composition. Wide-cut fuels have lower viscosities and freezing points than kerosenes, whereas heavier fuels used in ground turbines exhibit much higher viscosities and freezing points. 

Chocolate does not behave as a tme Hquid owing to the presence of cocoa particles and the viscosity control of chocolate is quite compHcated. This non-Newtonian behavior has been described (28). When the square root of the rate of shear is plotted against the square root of shear stress for chocolate, a straight line is produced. With this Casson relationship method (29) two values are obtained, Casson viscosity and Casson yield value, which describe the flow of chocolate. The chocolate industry was slow in adopting the Casson relationship but this method now prevails over the simpler MacMichael viscometer. Instmments such as the Carri-Med Rheometer and the Brookfield and Haake Viscometers are now replacing the MacMichael. 

Elasticity is another manifestation of non-Newtonian behavior. Elastic Hquids resist stress and deform reversibly provided that the strain is not too large. The elastic modulus is the ratio of the stress to the strain. Elasticity can be characterized usiag transient measurements such as recoil when a spinning bob stops rotating, or by steady-state measurements such as normal stress ia rotating plates. 

All fluids for which the viscosity varies with shear rate are non-Newtonian fluids. For uou-Newtouiau fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distiuc tiou from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids. 

Nazem [31] has reported that mesophase pitch exhibits shear-thinning behavior at low shear rates and, essentially, Newtonian behavior at higher shear rates. Since isotropic pitch is Newtonian over a wide range of shear rates, one might postulate that the observed pseudoplasticity of mesophase is due to the alignment of liquid crystalline domains with increasing shear rate. Also, it has been reported that mesophase pitch can exhibit thixotropic behavior [32,33]. It is not clear, however, if this could be attributed to chemical changes within the pitch or, perhaps, to experimental factors. 

The viscosity of a fluid arises from the internal friction of the fluid, and it manifests itself externally as the resistance of the fluid to flow. With respect to viscosity there are two broad classes of fluids Newtonian and non-Newtonian. Newtonian fluids have a constant viscosity regardless of strain rate. Low-molecular-weight pure liquids are examples of Newtonian fluids. Non-Newtonian fluids do not have a constant viscosity and will either thicken or thin when strain is applied. Polymers, colloidal suspensions, and emulsions are examples of non-Newtonian fluids [1]. To date, researchers have treated ionic liquids as Newtonian fluids, and no data indicating that there are non-Newtonian ionic liquids have so far been published. However, no research effort has yet been specifically directed towards investigation of potential non-Newtonian behavior in these systems. 

All three methods discussed above appear to provide equally high quality ionic liquid viscosity data. However, the rotational viscometer could potentially provide additional information concerning the Newtonian behavior of the ionic liquids. The capillary method has been by far the most commonly used to generate the ionic liquid viscosity data found in the literature. This is probably due to its low cost and relative ease of use. 

According to the structure of this equation the quantity cp indicates the influence of the filler on yield stress, and t r on Newtonian (more exactly, quasi-Newtonian due to yield stress) viscosity. Both these dependences Y(cp) andr r(cp) were discussed above. Non-Newtonian behavior of the dispersion medium in (10) is reflected through characteristic time of relaxation X, i.e. in the absence of a filler the flow curve of a melt is described by the formula ... 

Viscosity is usually understood to mean Newtonian viscosity in which case the ratio of shearing stress to the shearing strain is constant. In non-Newtonian behavior, which is the usual case for plastics, the ratio varies with the shearing stress (Fig. 8-5). Such ratios are often called the apparent viscosities at the corresponding shearing stresses. Viscosity is measured in terms of flow in Pa s, with water as the base standard (value of 1.0). The higher the number, the less flow. 

The non-Newtonian behavior of a plastic melt makes its flow through a die somewhat complicated. One characteristic of plastic is... 

In the molten state, a Newtonian behavior was observed, a consequence of lack of entanglements. The melt behavior is also dependent on die structure of die endgroups. 

Toothpaste flow is an extreme example of non-Newtonian flow. Problem 8.2 gives a more typical example. Molten polymers have velocity profiles that are flattened compared with the parabolic distribution. Calculations that assume a parabolic profile will be conservative in the sense that they will predict a lower conversion than would be predicted for the actual profile. The changes in velocity profile due to variations in temperature and composition are normally much more important than the fairly subtle effects due to non-Newtonian behavior. 

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