Viscosity equation

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

For dilute dispersions of hard spheres, Einstein s viscosity equation predicts... 

Of the adjustable parameters in the Eyring viscosity equation, kj is the most important. In Sec. 2.4 we discussed the desirability of having some sort of natural rate compared to which rates of shear could be described as large or small. This natural standard is provided by kj. The parameter kj entered our theory as the factor which described the frequency with which molecules passed from one equilibrium position to another in a flowing liquid. At this point we will find it more convenient to talk in terms of the period of this vibration rather than its frequency. We shall use r to symbolize this period and define it as the reciprocal of kj. In addition, we shall refer to this characteristic period as the relaxation time for the polymer. As its name implies, r measures the time over which the system relieves the applied stress by the relative slippage of the molecules past one another. In summary. 

This result was published by Debye in 1946. Since we shall also encounter a light-scattering equation associated with his name, we shall refer to Eq. (2.56) and its variations as the Debye viscosity equation. 

The segmental friction factor introduced in the derivation of the Debye viscosity equation is an important quantity. It will continue to play a role in the discussion of entanglement effects in the theory of viscoelasticity in the next chapter, and again in Chap. 9 in connection with solution viscosity. Now that we have an idea of the magnitude of this parameter, let us examine the range of values it takes on. 

The viscosity given by Eq. (3.98) not only follows from a different model than the Debye viscosity equation, but it also describes a totally different experimental situation. Viscoelastic studies are done on solid samples for which flow is not measurable. A viscous deformation is present, however, and this result shows that it is equivalent to what would be measured directly, if such a measurement were possible. 

Inspection of Fig. 3.9 suggests that for polyisobutylene at 25°C, Ti is about lO hr. Use Eq. (3.101) to estimate the viscosity of this polymer, remembering that M = 1.56 X 10. As a check on the value obtained, use the Debye viscosity equation, as modified here, to evaluate M., the threshold for entanglements, if it is known that f = 4.47 X 10 kg sec at this temperature. Both the Debye theory and the Rouse theory assume the absence of entanglements. As a semi-empirical correction, multiply f by (M/M. ) to account for entanglements. Since the Debye equation predicts a first-power dependence of r) on M, inclusion of this factor brings the total dependence of 77 on M to the 3.4 power as observed. 

Suppose you wanted to estimate the viscosity of a polystyrene sample at 125°C using the Debye viscosity equation, but the only available value... 

Before turning to this, however, it is useful to introduce some additional vocabulary that is often employed in discussing solution viscosity. Equation (9.10) is a special case of the general function... 

An alternative point of view assumes that each repeat unit of the polymer chain offers hydrodynamic resistance to the flow such that f-the friction factor per repeat unit-is applicable to each of the n units. This situation is called the free-draining coil. The free-draining coil is the model upon which the Debye viscosity equation is based in Chap. 2. Accordingly, we use Eq. (2.53) to give the contribution of a single polymer chain to the rate of energy dissipation ... 

The relative viscosity of a dilute dispersion of rigid spherical particles is given by = 1 + ft0, where a is equal to [Tj], the limiting viscosity number (intrinsic viscosity) in terms of volume concentration, and ( ) is the volume fraction. Einstein has shown that, provided that the particle concentration is low enough and certain other conditions are met, [77] = 2.5, and the viscosity equation is then = 1 + 2.50. This expression is usually called the Einstein equation. 

For airblast-type atomizers, it has been speculated (33) that the Sauter mean diameter is governed by two factors, one controlled by air velocity and density, the other by Hquid viscosity. Equation 13 has been proposed for the estimation of equation 13, and B are constants whose values depend... 

The resistance to liquid flow aroimd particles may be presented by an equation similar to the viscosity equation but with considering the void fraction. Recall that the shear stress is expressed by the ratio of the drag force, R, to the active surface, K27td. The total sphere surface is Ttd and Kj is the coefficient accoimting for that part of the surface responsible for resistance. Considering the influence of void fraction as a function 2( ). we obtain ... 

The correction of mean free path, hi by the nanoscale effect function results in a smaller mean free path, or a smaller Knudsen number in other word. As a matter of fact, a similar effect is able to be achieved even if we use the conventional definition of mean free path, / = irSn, and the Chapmann-Enskog viscosity equation, /r = (5/16)... 

Many polymer properties can be expressed as power laws of the molar mass. Some examples for such scaling laws that have already been discussed are the scaling law of the diffusion coefficient (Equation (57)) and the Mark-Houwink-Sakurada equation for the intrinsic viscosity (Equation (36)). Under certain circumstances scaling laws can be employed advantageously for the determination of molar mass distributions, as shown by the following two examples. 

Eimco High-Capacity thickener, 22 66 Einsteinium (Es), 1 463-491, 464t electronic configuration, l 474t ion type and color, l 477t metal properties of, l 482t Einstein relation, 22 238. See also Einstein s viscosity equation filled networks and, 22 571, 572 Einstein s coefficient, 14 662 Einstein s equation, 7 280 21 716 23 99 Einstein s law, 19 108 Einstein s viscosity equation, 22 54. 

The velocity profile must have a form like that shown in Figure 1.17. The velocity is zero at the pipe wall and increases to a maximum at the centre. From Example 1.8, it is known that the shear stress vanishes on the centre-line r = 0, so from Newton s law of viscosity (equation 1.45) the velocity gradient must be zero at the centre. 

When a fluid flows past a solid surface, the velocity of the fluid in contact with the wall is zero, as must be the case if the fluid is to be treated as a continuum. If the velocity at the solid boundary were not zero, the velocity gradient there would be infinite and by Newton s law of viscosity, equation 1.44, the shear stress would have to be infinite. If a turbulent stream of fluid flows past an isolated surface, such as an aircraft wing in a large wind tunnel, the velocity of the fluid is zero at the surface but rises with increasing distance from the surface and eventually approaches the velocity of the bulk of the stream. It is found that almost all the change in velocity occurs in a very thin layer of fluid adjacent to the solid surface ... 

Towards the middle of 1929, Mark was clearly close to establishing a viscosity equation. He and H. Fikentscher published a somewhat complex relationship of viscosity, and molecular volume (33). It was based on the Einstein relationship of viscosity and solute concentration. 

In the following four years Mark successively reported on the viscosity and molecular weight of cellulose (40), Staudinger s Law (41), high polymer solutions (42), and the effect of viscosity on polymerization rates (43). Confident of his findings, he proposed (at the same time as R. Houwink) the general viscosity equation now known as the Mark-Houwink Equation (44, 45). 

Mark and Houwink were the first to formulate the equation in the power form and to demonstrate its validity by means of empirical values. In reality, the Mark-Houwink Equation is simply the Einstein viscosity equation, which assumed spheres, transferred to particles with size dependent particle density. 

Hence to calculate an increase in viscosity, Equation (3.51) must be integrated up to the value of interest so... 

This result, that the low frequency limit of the in phase component of the viscosity equates to the viscosity of the dashpot, means that for a single Maxwell model it is possible to replace rj by rj(0). Thus far we have concentrated on the description of experimental responses to the application of a strain. Similar constructions can be developed for the application of a stress. For example the application of an oscillating stress to a sample gives rise to an oscillating strain. We can define a complex compliance J which is the ratio of the strain to the stress. We will explore the relationship between different experiments and the resulting models in Section 4.6. 

Einstein viscosity equation phys chem An equation which gives the viscosity of a sol in terms of the volume of dissolved particles divided by the total volume. Tn,stTn vis kas-od-e i.kwa-zhon ... 

This is a theoretical equation that was derived from free volume theory. If extruding materials at lower than normal temperatures, the higher sensitivity of the viscosity to temperature is an issue that needs to be considered. The engineering-based viscosity equation developed by Adams and Campbell [18] has been shown to hold for all nominal processing temperatures, from within a few degrees of Tg [26, 27] to conventional extruder melt temperatures. The Adams-Campbell model limiting shear temperature dependence is ... 

Note that the relationship contains an absolute value of the shear rate since it can be either positive or negative. The manufacturer has supplied parameters for the proposed 0.8 dg/min (190 °C, 2.16 kg) Ml resin and a lower viscosity resin with an Ml of 2 dg/min. The viscosity equation parameters are provided in Table 7.1. 

Staudinger showed that the intrinsic viscosity or LVN of a solution ([tj]) is related to the molecular weight of the polymer. The present form of this relationship was developed by Mark-Houwink (and is known as the Mark Houwink equation), in which the proportionality constant K is characteristic of the polymer and solvent, and the exponential a is a function of the shape of the polymer in a solution. For theta solvents, the value of a is 0.5. This value, which is actually a measure of the interaction of the solvent and polymer, increases as the coil expands, and the value is between 1.8 and 2.0 for rigid polymer chains extended to their full contour length and zero for spheres. When a is 1.0, the Mark Houwink equation (3.26) becomes the Staudinger viscosity equation. 

Another relationship often used in determining [17] is called the inherent viscosity equation and is given in the following equation ... 

The stiffness of the main chain of a polymer is of great importance for the solution viscosity the stiffer the chain is, the higher is the viscosity for polymers with the same molecular weight (see Sect. 2.3.3.3.1 for the dependency of K and a in the viscosity equation on the shape of macromolecules in solution). 

---