Example Concentration-Dependent Viscosity

The viscosity of a mixture depends upon its concentration, although often that dependence is small. To resolve the previous problem and allow the viscosity to depend upon concentration you use the formula 

Then the viscosity of the material coming into the top of the T-sensor is 2 whereas the viscosity coming into the bottom is 1. This model was solved by Marlina Lukman when she was a senior chemical engineering student at the University of Washington. To modify the model, perform the following steps. 

Step 1 Choose the Navier-Stokes equation under the Multiphysics/Model Navigator option. Then choose Physics/Subdomain Settings. Select the domain and type in the formula for viscosity 1 -I- c. If you have several concentration variables, be sure to use the variable names assigned by FEMLAB (or changed by you) in this formula. 

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A comparison of values of yield stress for filled polymers of the same nature but of different molecular weights is of fundamental interest. An example of experimental results very clearly answering the question about the role of molecular weight is given in Fig. 9, where the concentration dependences of yield stress are presented for two filled poly(isobutilene)s with the viscosity differing by more than 103 times. As is seen, a difference between molecular weights and, as a result, a vast difference in the viscosity of a polymer, do not affect the values of yield stress. 

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. 

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. 

Several studies have considered the influence of filler type, size, concentration and geometry on shear yielding in highly loaded polymer melts. For example, the dynamic viscosity of polyethylene containing glass spheres, barium sulfate and calcium carbonate of various particle sizes was reported by Kambe and Takano [46]. Viscosity at very low frequencies was found to be sensitive to the network structure formed by the particles, and increased with filler concentration and decreasing particle size. However, the effects observed were dependent on the nature of the filler and its interaction with the polymer melt. 

The effects of interaction on viscoelastic properties at low concentrations depend on the Simha parameter. For example, Ferry has pointed out the importance of c[jj] for the transition from Zimm-like to Rouse-like behavior in the dynamic properties and in the observed values of J (15). The shear rate dependence of viscosity undergoes a corresponding transition as a function of... 

Polymer entanglement has been an important concept governing the physical properties of polymers. For example, UHMW-PE single-crystalline materials can be super-drawn up to a draw ratio of 300.30 This high drawability is assumed to be due to the existence of entanglement.31 The first evaluation of entanglement was offered by Porter and Johnson for the MW dependence of melt viscosity and the polymer concentration dependence for the solution viscosity.32 33... 

A mathematical expression relating forces and deformation motions in a material is known as a constitutive equation. However, the establishment of constitutive equations can be a rather difficult task in most cases. For example, the dependence of both the viscosity and the memory effects of polymer melts and concentrated solutions on the shear rate renders it difficult to establish constitute equations, even in the cases of simple geometries. A rigorous treatment of the flow of these materials requires the use of fluid mechanics theories related to the nonlinear behavior of complex materials. However, in this chapter we aim only to emphasize important qualitative aspects of the flow of polymer melts and solutions that, conventionally interpreted, may explain the nonlinear behavior of polymers for some types of flows. Numerous books are available in which the reader will find rigorous approaches, and the corresponding references, to the subject matter discussed here (1-16). 

In binary systems, the above-accepted ideal behavior of the concentration dependence of viscosity can be used e.g. in the estimation of the dissociation degree of the thermal dissociation of the additive compound AB, which are formed in the binary system A-B. As an example, the binary system KF-K2NbF7 was chosen, in which the additive compound KsNbFg is formed. 

Besides being enormously expanded and spheroidal, the mucins also appear very flexible evidence for this derives from the ease with which these molecules are deformed with shear, as shown in streaming-birefringence experiments, from the concentration dependence of viscosity, and the temperature-dependence of sedimentation coeflScients. Mikkelsen and co-workers confirmed these observations by using the dependence on field strength of electric birefringence relaxation-phenomena. Evidence for local flexibility in the carbohydrate side-chains has come from, for example, n.m.r.-spectral studies. ... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Physical parameters in constitutive laws are function of pressure and temperature. For example concentration of vapour under planar surface (in psychrometric law), surface tension (in retention curve), dynamic viscosity (in Darcy s law), are strongly dependent on temperature. 

Subtype IIIb-1 isotherms with a curve is a special case of subtype Illb isotherms. The curve is caused by viscosity influence. Isotherm Illb turns to subtype IIIb-1 at higher temperatures. A concentration dependence of conductivity for stybium (III) chloride-methanol is an example of the systems. 

There are numerous semiempirical and empirical relations proposed for describing the concentration dependence of suspension viscosity. For example, the first category of relations has the algebraic form [44] ... 

According to Eq. (2.8) or (2.10) the concentration dependence of has two parameters, [rj and 0m- Both are measures of a specific physical quantity (respectively, shape and packing) and may be independently determined, for example, [rj] from viscosity of diluted suspensions and 0m from dry packing of solid particles. For anisometric particles, the magnitude of these parameters may also be theoretically predicted see the rheological summary in a quite recent monograph [64]. Once [rj] and 0m are known, Eq. (2.8) will correctly describe the rjr versus 0 dependence for complex industrial systems, for example, PVC (poly(vinyl chloride)) emulsions and plastisols, mica-reinforced polyolefins, and sealant formulations [44,65]. However, in some suspensions and blends, r] and 0m may vary with composition [66]. 

The definition of the intrinsic viscosity states that the measurements for the determination of the intrinsic viscosity should be performed at a shear rate y— 0. At higher shear rates, the viscosity might become shear rate dependent (so called non-Newtonian flow behavior). Examples for the occurrence of a shear rate dependent viscosity at low shear rates and for relatively low polymer concentrations are given in Fig. 5.8. 

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