Temperature dependence of the viscosity

Not only the viscosity but also the temperature dependence of the viscosity can be related to the structure of mineral oil fractions. 

The viscosity-temperature relationship of mineral oil fractions, and in fact of many other liquids, can be represented by a formula proposed by Cornelissen and Waterman24 

The constants A, B and x in the formula can be calculated from three viscosity values vv v2 and vz, determined at three different temperatures Tv T2 and Tz, respectively. The three following equations are obtained  

A more detailed table than Table II has been published25. x can also be calculated25a by means of the equation 

Once the value of x is known the value of A can be determined by means of the equation that can be derived from equations (a) and (c) by subtraction 

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The flow behavior of the polymer blends is quite complex, influenced by the equilibrium thermodynamic, dynamics of phase separation, morphology, and flow geometry [2]. The flow properties of a two phase blend of incompatible polymers are determined by the properties of the component, that is the continuous phase while adding a low-viscosity component to a high-viscosity component melt. As long as the latter forms a continuous phase, the viscosity of the blend remains high. As soon as the phase inversion [2] occurs, the viscosity of the blend falls sharply, even with a relatively low content of low-viscosity component. Therefore, the S-shaped concentration dependence of the viscosity of blend of incompatible polymers is an indication of phase inversion. The temperature dependence of the viscosity of blends is determined by the viscous flow of the dispersion medium, which is affected by the presence of a second component. 

FIGURE 21.9 Temperature dependence of the viscosity constant K (solid) and K (dashed). Both constants are divided by those in solvent for scaling. 

As in the case of the diffusion properties, the viscous properties of the molten salts and slags, which play an important role in the movement of bulk phases, are also very structure-sensitive, and will be referred to in specific examples. For example, the viscosity of liquid silicates are in the range 1-100 poise. The viscosities of molten metals are very similar from one metal to another, but the numerical value is usually in the range 1-10 centipoise. This range should be compared with the familiar case of water at room temperature, which has a viscosity of one centipoise. An empirical relationship which has been proposed for the temperature dependence of the viscosity of liquids as an Arrhenius expression is... 

Figure 2 Sketch of typical temperature dependencies of the viscosity r of glass-forming systems. The viscosimetric Tg of a material is defined by the viscosity reaching 1013 Poise. Strong glass formers show an Arrhenius temperature dependence, whereas fragile glass formers follow reasonably well a Vogel-Fulcher (VF) law predicting a diverging viscosity at some temperature T0.

Fig. 12.1 Illustration of the temperature sensitivity of 15N relaxation parameters, Rlf R2t and NOE, as indicated. Shown are the relative deviations in these relaxation parameters from their values at 25 °C as a function of temperature in the range of + 3 °C. The expected variations in / ] and R2 due to temperature deviations of as little as +1 °C are already greater than the typical level of experimental precision ( % ) of these measurements (indicated by the dashed horizontal lines). For simplicity, only temperature variation of the overall tumbling time of the molecule (due to temperature dependence of the viscosity of water) is taken into account the effect of temperature variations on local dynamics is not considered here.

Fig. 4.24 Temperature dependence of the characteristic times obtained from the fits of Spair(Q,t) to stretched exponentials with =0.41 at Qmax=l-48 A (filled circle) and 2.71 A (empty circle). Dashed-dotted line corresponds to the Vogel-Fulcher-like temperature dependence of the viscosity and the solid line to the Arrhenius-like temperature dependence of the dielectric -relaxation. (Reprinted with permission from [189]. Copyright 1996 The American Physical Society)...

An even more useful property of supercritical fluids involves the near temperature-independence of the solvent viscosity and, consequently, of the line-widths of quadrupolar nuclei. In conventional solvents the line-widths of e. g. Co decrease with increasing temperature, due to the strong temperature-dependence of the viscosity of the liquid. These line-width variations often obscure chemical exchange processes. In supercritical fluids, chemical exchange processes are easily identified and measured [249]. As an example. Figure 1.45 shows Co line-widths of Co2(CO)g in SCCO2 for different temperatures. Above 160 °C, the line-broadening due to the dissociation of Co2(CO)g to Co(CO)4 can be easily discerned [249]. 

Martinez and Angell [4] have recently reviewed the parallelism between the temperature dependence of the viscosity and that of S, and they have shown that this correlation is strong for the small molecule glass formers considered in their study. Since these findings are relevant to the entropy theory of glass formation. 

The variation of the transport rate of [3H]PVP with temperature (temperature range studied was from 5 to 35 °C) utilizing the standard system in the Sundelof cell could, to a major extent, be explained by the temperature dependence of the viscosity of the solvent50). 

Although most physical properties (e.g., viscosity, density, heat conductivity and capacity, and surface tension) must be regarded as variable, it is of particular value that viscosity can be varied by many orders of magnitude under certain process conditions (5,11). In the following, dimensional analysis will be applied exemplarily to describe the temperature dependency of the viscosity and the viscosity of non-Newtonian fluids (pseudoplastic and viscoelastic, respectively) as influenced by the shear stress. 

Figure 6 Temperature dependency of the viscosity, /x(7), for eight different liquids. (Baysilon is a silicone oil of the BAYER AG, Germany.)...

The temperature dependence of the viscosity is seen to have the largest effect on the temperature dependence of D. ... 

Evaluate the viscosity of ethane using these Lennard-Jones parameters over the temperature range 300 to 700 K. Fit the temperature dependence of the viscosity using Eq. 12.114, and report the values of the polynomial fitting coefficients that you obtained. 

Finally, the data published by Gee (30) permit one to evaluate the sharpness of a transition involving floor temperature. Gee studied the temperature dependence of the viscosity of liquid sulfur and observed its sudden, steep increase at a critical temperature followed by its decrease at still higher temperatures. He developed the first, relatively complete theory of equilibrium polymerization of liquid sulfur (30) from which he estimated the chain length of the polymeric sulfur at various temperatures. His results have been recently confirmed by experimental measurements of magnetic susceptibility of the liquid sulphur (50) and its electron spin resonance (57). 

Telegina et al. 72> showed that the activation energy for the viscous flow of a polyester oligomer filled with glass microspheres is 46.9 kJ/mol, while that of an epoxy oligomer is 78.3 kJ/mol. They also established the important fact that the addition of microspheres to an oligomer composition does not change the temperature viscosity coefficient. This means that the viscosity of a mixture with microspheres can be controlled, if the temperature dependence of the viscosity of the binder is known. 

To understand the mechanism of polyblending, experiments have been carried out with polymeric solution. W. Borchard and G. Rehage mixed two partially miscible polymer solutions, measured the temperature dependence of the viscosity, and determined the critical point of precipitation. When two incompatible polymers, dissolved in a common solvent, are intimately mixed, a polymeric oil-in-oil emulsion is formed. Droplet size of the dispersed phase and its surface chemistry, along with viscosity of the continuous phase, determine the stability of the emulsion. Droplet deformation arising from agitation has been measured on a dispersion of a polyurethane solution with a polyacrylonitrile solution by H. L. Doppert and W. S. Overdiep, who calculated the relationship between viscosity and composition. 

In practical applications, the increase of viscous friction with speed is often lower than expected from Eq. (11.9). The explanation is that friction leads to an increased temperature of the lubricant which reduces the viscosity. For most lubricants the temperature dependence of the viscosity is given by... 

The quantity / is usually called the characteristic length of the interaction potential, and is also employed in the more realistic wave-mechanical treatment. Many authors employ the symbol, a, which is equal to /-1. The form of the molecular interaction potential can be determined in terms of a suitable model, from experimental measurements of the temperature dependence of the viscosity of a gas, whence the characteristic length can be estimated. 

The results of the calculations shown in Fig. 2.32 represent a complete quantitative solution of the problem, because they show the decrease in the induction period in non-isothermal curing when there is a temperature increase due to heat dissipation in the flow of the reactive mass. The case where = 0 is of particular interest. It is related to the experimental observation that shear stress is almost constant in the range t < t. In this situation the temperature dependence of the viscosity of the reactive mass can be neglected because of low values of the apparent activation energy of viscous flow E, and Eq. (2.73) leads to a linear time dependence of temperature ... 

If the species X and B are to react on every encounter, it is obvious that no significant activation energy can be required. This does not imply that the experimental activation energy (EA) will be zero, even for the diffusion-controlled step, because of the temperature dependence of the viscosity of the medium. The variation of the viscosity of liquids with temperature normally follows equation (5) where b and B are constants. Even where this equation is not well obeyed (e.g. water), it is still convenient to define a mean value of B for a particular range of temperature. From (4) and (5), the variation of ken with temperature is as shown in (6). When this expression for ken is substituted in... 

A further test of this interpretation is provided by the activation energies of the reactions that occur at the limiting rate. The observed activation energies for the nitration of mesitylene and naphthalene in 67.1% sulphuric acid are 75.3 and 64.8 kJ mol"1 respectively (Coombes et al., 1968). On the interpretation in terms of reaction on encounter the activation energy should be the sum of the AH° term for the formation of the nitronium ion (43 kJ mol"1 in 68.3% sulphuric acid) (Hartshorn et al., 1972), and the term derived from the temperature dependence of the viscosity of the solvent (24 kJ mol"1 see (8) and Table 2). Thus, the expected activation energy is ca. 67 kJ mol-1, a value in reasonable agreement with the experimental results. 

Most dimensional analyses deal with problems with linear material properties. However, in polymer processing, the viscosity is temperature as well as rate of deformation dependent. In addition, other properties are temperature and pressure dependent. In Example 4.4, one such non-linearity was introduced, namely the temperature dependance of the viscosity. In a similar way the rate of deformation dependence of the viscosity may also be introduced. Choosing the power-law model for the viscosity... 

Here, we can see that an increase in temperature will reduce the viscosity by an amount controlled by the material constant a —the temperature dependence of the viscosity— and the actual temperature rise, AT. Hence, the effect is controlled by the product aAT. Taking eqn. (6.242) and dropping the 3/8 term we can say that... 

To illustrate the effect of thermal gradients and temperature dependent viscosity, we can plot a dimensionless velocity, ux/U0 as a function of dimensionless position, y/h, for various values of thermal imbalance between the surfaces, i. Note that Q, the product between the temperature dependence of the viscosity and the temperature imbalance is also a dimensionless quantity. This gives a fully dimensionless graph that can be used to assess many case scenarios. Figure 6.59 presents dimensionless velocity distributions across the plates for various dimensionless temperature imbalances, ft. 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

Incorporating both the effect of convection in the Him and the temperature dependence of the viscosity into the model improves the agreement between predictions and experimental measurements. It should be noted, however, that experimental conditions were such that viscous dissipation was insignificant and the temperature drop across the film was relatively small. Consequently, non-Newtonian effects, and effects due to the temperature dependence of viscosity, were less significant than were convection effects. This may not be the case in many practical situations, in particular with polymers, whose viscosity is more temperature sensitive than that of HDPE. 

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