Elastic dumbbell model

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. 

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. 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

We will discuss Eq. (3.20) further in Chapters 6 and 7, where it will be used to obtain the rheological constitutive equations of the elastic dumbbell model and the Rouse chain model. 

Molecular Theory of Polymer Viscoelasticity — Elastic Dumbbell Model... 

Equation (6.70) indicates that the viscosity is independent of the shear rate Aq. However, it is well known that the polymeric liquid exhibits non-Newtonian behavior, namely, that the viscosity value decreases with increasing shear rate after the rate reaches a certain value. This discrepancy is a weak point of the elastic dumbbell model and arises from an inherent weakness in the Gaussian distribution assumed for the connector vector. We can see the cause of this deficiency from the following analysis of how the dumbbell configuration changes with shear rate. 

An important concept in continuum mechanics is the objectivity, or admissibility, of the constitutive equation. There are the covariant and contravariant ways of achieving objectivity. The molecular theories the elastic dumbbell model of this chapter, the Rouse model to be studied in the next chapter, and the Zimm model which includes the preaveraged hydrodynamic interaction, all give the result equivalent to the contravariant way. In this appendix, we limit our discussion of continuum mechanics to what is needed for the molecular theories studied in Chapters 6 and 7. More detailed discussions of the subject, particularly about the convected coordinates, can be found in Refs. 5 and 6. 

The elastic dumbbell model studied iu Chapter 6 is both structurally and djmamicaUy too simple for a poljmier. However, the derivation of its constitutive equation illustrates the main theoretical steps involved. In this chapter we shall apply these theoretical results to a Gaussian chain (or Rouse chain) containing many bead-spring segments (Rouse segments). First we obtain the Smoluchowski equation for the bond vectors. After transforming to the normal coordinates, the Smoluchowski equation for each normal mode is equivalent in form to the equation for the elastic dumbbell. 

In Chapter 6, we derived the stress tensor for the elastic dumbbell model. By following the derivation steps given there, one obtains the following result for the Rouse chain model with N beads per chain, which is equivalent to Eq. (6.50) for the elastic dumbbell model ... 

In describing flowing polymeric liquids it is probably not feasible to use detailed models that describe the locations of all the atoms in the polymer molecules. Consequently, it is necessary to use some kind of mechanical models that portray the overall molecular architecture. Bead-spring models have been widely used with considerable success for relating macroscopic properties to the main features of the molecular architecture. Even the simplest of these models - the elastic dumbbell models - are capable of describing polymer orientation and polymer stretching. More complicated chain, nng, and star models reflect better the molecular structure and allow for the portrayal of the most important internal molecular motions as well. 

For example, a linearly elastic dumbbell model representation (Figure 13.19) of polymer molecules predicts Eq. (14) to apply, where X = /4H is a time constant and is the bead friction factor. 

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