Elastic dumbbell

Next we summarize the parameters used to characterize the dumbbell suspension  

H = the spring constant of the Hookean spring, which we assume joins the two beads in other words, we are assuming that the tension in the connector is given by a linear spring law = HR, where force is proportional to separation 

So what do we have so far We have a model macromolecule that (1) has viscous drag interactions with the surrounding solvent, (2) has variable R, which describes how the macromolecule stretches and aligns when subjected to flow, (3) accounts for elasticity with a spring connector, and (4) includes the possibility for concentration and molecular weight effects through 

To develop a theory for dilute solutions, we can consider that the dumbbells all move independently. We further assume that the effect of the suspended dumbbells on the rheological properties of the solution can be obtained by finding the statistical contribution of one dumbbell. The dumbbell on which we focus our attention will 

If the dumbbell is in a solvent with v = 0 everywhere (a fluid at rest), the dumbbell nonetheless will rotate continually because of the Brownian forces acting on it. However, all angular orientations are equally likely. On the other hand, if the solution is being sheared, then the dumbbells will tend to be aligned and all orientations are not equally likely. This idea leads to the notion of a distribution function, which we will call I (R,t). This function has the meaning  

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Fig. 6. Bead-spring and elastic dumbbell representation of a polymer chain in solution...

To study processes which affect the end-to-end vector r, it is sometimes informative to consider only the two beads localized at each chain end and connected by a single spring. This model, known as the elastic dumbbell, was originally proposed by Kuhn over half a century ago [40] and constitutes the simplest model of chain dynamics in flow. 

Internal viscosity (Section 4) provides another possible source of shear-rate dependence. For sufficiently rapid disturbances, a spring-bead model with internal viscosity acts like a rigid body for sufficiently slow disturbances it is flexible and indefinitely extensible. The analytical difficulties for coupled, non-linear spring-bead systems are equally severe in linear spring-bead systems with internal viscosity. Even the elastic dumbbell with internal viscosity has only been solved exactly in the limit of small e (559), where e is the ratio of internal friction coefficient to molecular (external) friction coefficient Co n. For this case, the viscosity decreases with shear rate. 

The time constant t0 is the relaxation time of the elastic dumbbell, which would be (rj0 — t] M/cRT if the dumbbells were independent. Williams makes the dimensional argument described earlier (Part 6) in favor of a different form, the result being... 

It is interesting to examine the bead-spring models to see what flow-induced configurational changes would be required in order to develop N2 values of the proper magnitude and sign. In the Rouse model, the components of the stress tensor are related directly to averages of the internal coordinates of the beads. For the simplest case of the elastic dumbbell ... 

Other modifications to the elastic dumbbell have been considered, such as the concept of internal viscosity, where an additional spring force proportional to the rate of... 

The two contributions to the rate of rotation, li, of the rod are convection and Brownian diffusion. Unlike the elastic dumbbell, where the springs were allowed to deform by the flow, the fixed separation of the beads in the rigid dumbbell must be maintained. For that reason, the vector u can rotate, but it cannot stretch. This constraint is satisfied by ensur-... 

H. R. Warner, Jr., Kinetic Theory and Rheology of Dilute Suspensions of Finitely Extendible Dumbbells, Ind. Eng. Chem. Fundam., 11,379-387 (1972) also, R. L. Christiansen and R. B. Bird, Dilute Solution Rheology Experimental Results and Finitely Extensible Nonlinear Elastic Dumbbell Theory, J. Non-Newt. Fluid Mech., 3, 161-177 (1977/1978). 

It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

Equations (3-32)-(3-34) are equivalent to the so-called Oldroyd-B equation. The Oldroyd-B equation is a simple, but qualitatively useful, constitutive equation for dilute solutions of macromolecules (see Section 3.6.2). Refinements to the simple elastic dumbbell model, such as the effects of the nonlinearity of the force-extension relationship at high extensions, are discussed in Section 3.6.2.2.I. 

Worked Examples 3.1 and 3.2 (at the end of this chapter) show how calculations of stress in simple flows are carried out using the temporary network model and the elastic dumbbell model. 

From Eq. (3-32) for a dilute solution of Hookean elastic dumbbells with relaxation time T and modulus G, calculate polymer contributions to the extensional viscosity as a function of time after start-up of steady extension at extension rate e. 

The starting point of a molecular constitutive theory is a simple mechanical model for the molecule that captures its most salient traits. Thus, flexible polymer molecules have been represented by elastic dumbbells and bead-spring chains, and rigid polymers by rigid dumbbells and rigid rods. For its simplicity, the evolution of the model molecule is easily described by a convection-diffusion equation. Then a Fokker-Planck equation is written for the probability distribution function of an ensemble of these molecules. Finally, the macroscopic stress tensor is derived in terms of the distribution function. This kinetic theory framework was pioneered by Kirkwood (see, for example, Ref. ). 

In Table 3 we summarize the expressions for the material functions for the various experiments discussed in Parts II, III, and IV. The rigid dumbbells seem to give results which are in qualitative agreement with experiment, whereas the elastic dumbbells do not. Specific points of difference are ... 

In steady-state shear flow the rigid dumbbells give a shear-rate dependent t] and 6, whereas these non-Newtonian effects are absent for the elastic dumbbells. 

In small-amplitude oscillatory motion, the rigid dumbbells give a non-zero high-frequency asymptote for t] whereas the elastic dumbbells do not. 

In stress relaxation after cessation of steady shear flow, the elastic dumbbells give no dependence of the relaxation process on the steady-state shear rate, but the rigid dumbbells do. In addition the elastic dumbbells show the shear and normal stresses relaxing with exactly the same... 

Table 3. Comparison of rigid and elastic dumbbell results... 

In stress growth at inception of steady shearing flow, the rigid dumbbells give a stress expression which is dependent on the steady-state shear rate however, elastic dumbbells do not. Also the rigid dumbbell model predicts stress overshoot, a phenomenon which the elastic dumbbell model cannot describe. 

In constrained recoil after steady shear flow, the elastic dumbbells give a value of yw/K which is independent of k, whereas the rigid dumbbell model contains a dependence on k. ... 

In stress growth at inception of steady elongational flow, both the rigid dumbbells and elastic dumbbells give expressions for... 

For rigid or elastic dumbbells the diffusion equation is then ... 

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