Viscosity temperature-invariant dependence

FIG. 28 Temperature-invariant dependence (Vinogradov function) of melt viscosity of HDPE vs y 7o) various melt temperatures from 140°C to 240 C. (From Ref. 13.)... 

V, is the molar volume of polymer or solvent, as appropriate, and the concentration is in mass per unit volume. It can be seen from Equation (2.42) that the interaction term changes with the square of the polymer concentration but more importantly for our discussion is the implications of the value of x- When x = 0.5 we are left with the van t Hoff expression which describes the osmotic pressure of an ideal polymer solution. A sol vent/temperature condition that yields this result is known as the 0-condition. For example, the 0-temperature for poly(styrene) in cyclohexane is 311.5 K. At this temperature, the poly(styrene) molecule is at its closest to a random coil configuration because its conformation is unperturbed by specific solvent effects. If x is greater than 0.5 we have a poor solvent for our polymer and the coil will collapse. At x values less than 0.5 we have the polymer in a good solvent and the conformation will be expanded in order to pack as many solvent molecules around each chain segment as possible. A 0-condition is often used when determining the molecular weight of a polymer by measurement of the concentration dependence of viscosity, for example, but solution polymers are invariably used in better than 0-conditions. 

Figure 6 shows the dependency ijl T) for eight different liquids with greatly different temperature coefficients of the viscosity, whose viscosities cover six decades within the range of T= 20-80°C. Figure 7 depicts the standard representation of this behavior. Surprisingly this proves that all these liquids behave similarly in the /r(I) respect. In addition, it proves that this standard representation is invariant to reference temperature. Water is a special juice it behaves like the other liquids only in the vicinity of the standardization range yo(2 —To) 0. 

Fig. 8 a Standard representation of the temperature dependence of viscosity in the form of the relationship x/[t0 =/[y0 (T- T0)]. The solid curve a represents the reference-invariant approximation by the y-fu notion (see section 8.2), whereas the dotted line corresponds to the engineering representation, eq. (8.9) from [27],... 

Fig. 11 Standard representation of the temperature dependence of viscosity of aqueous sugar solutions of different concentrations as well as their reference-invariant approximation using ip = -0.500 (solid curve). For key parameters a(x)...

The reference-invariant representations of the temperature dependence of viscosity in Fig. 8 a and b were obtained by ip = -0.179 and -0.106 respectively. 

Fig. 11 shows the reference-invariant representation of the temperature dependence of viscosity of aqueous sugar solutions of different concentration (x - mass portion of cane sugar). To obtain this correlation, p and (T - T0) had to be transformed by the transformation or key parameters a = 1/yo [K] and b [Pa s] as w = p/b and u = (T-T0)/a, whereby a and b are functions ofx, see auxiliary diagram in Fig. 11. The reference temperature is T0 = 20 °C.

Fluids for which a proportionality between r and exists are known as Newtonian fluids. For such fluids the dynamic viscosity // is a material constant, which is only temperature dependent. Their temperature dependence can be well described hy an Arrhenius relationship. For other liquids, in particular reference-invariant representations, see [431]. 

In this section we present the equations of motion and heat transfer for incompressible non-Newtonian fluids governed by the rheological equation of state (7.1.1) when the apparent viscosity p = p(h, T) arbitrarily depends on the second invariant I2 of the shear rate tensor and on the temperature T. This section contains some material from the books [47, 320, 443], For the continuity equation in cylindrical and spherical coordinates, see Supplement 5.3. 

Stability. As long as the temperature remains below Tg, the composition of the system is virtually fixed. This implies physical stability crystallization, for instance, will not occur. As mentioned, some chemical reactions may still proceed, albeit very slowly because of the high viscosity and the low temperature. The parameters Tg and i//s are, however, not invariable they are not thermodynamic quantities. Their values will depend to some extent on the history of the system, such as the initial solute concentration and the cooling rate. The curve in Figure 16.6 denoted rff (for fast freezing) shows what the relation may become if the system is cooled very fast. The Tg curve is now reached at a lower ice content, so the apparent Tg and i// values are smaller. However, the system now is physically not fully stable water can freeze very slowly until the true i// s is reached. 

Probably the most systematic and complete study on the influence of temperature on water transfer has been performed on mammalian red cells [10,20,28]. The dependence on temperature of both the tracer diffusional permeability coefficient (cotho) 3 nd the hydraulic conductivity (Lp) of water in human and dog red-cell membranes have been studied. The apparent activation energies calculated from these results for both processes are given in Table 2. The values for the apparent activation energies for water self-diffusion and for water transport in a lipid bilayer are also included in the table. For dog red cells, the value of 4.9 kcal/mol is not significantly different from that of 4.6-4.8 kcal/mol for the apparent activation energy of the water diffusion coefficient ( >,) in free solution. Furthermore, it can be shown that the product L — THOV )rt, where is the partial molar volume of water and the viscosity of water remains virtually independent of temperature for dog, hut not for the human red-cell membrane [20]. The similarity of the transmembrane diffusion with bulk water diffusion and the invariance of the... 

Most polymer processes are dominated by the shear strain rate. Consequently, the viscosity used to characterize the fluid is based on shear deformation measurement devices. The rheological models that are used for these types of flows are usually termed Generalized Newtonian Fluids (GNF). In a GNF model, the stress in a fluid is dependent on the second invariant of the stain rate tensor, which is approximated by the shear rate in most shear dominated flows. The temperature dependence of GNF fluids is generally included in the coefficients of the viscosity model. Various models are currently being used to represent the temperature and strain rate dependence of the viscosity. 

In this chapter we have developed the general constitutive equation for a viscous liquid. We found that by using the rate of deformation or strain rate tensor 2D, we can write Newton s viscosity law properly in three dimensions. By making the coefficient of 2D dependent on invariants of 2D, we can derive models like the power law. Cross, and Carreau. We also showed how to introduce a three-dimensional yield stress to describe plastic materials with models like those Bingham and Casson. We saw two ways to describe the temperature dependence of viscosity and the importance of shear heating. 

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