The Elastic Dumbbell Model

In Fig. 3-5a, the polymer coil is modeled as a series of beads equally spaced along the polymer backbone and connected to each other by springs. The beads account for the viscous forces and the springs the elastic forces in the molecule the portion of the chain represented by a single spring is called a submolecule. The bead-spring model is 

F = IksTfi R, is that of a linear, or Hookean, spring, given in Eq. (3-10). It is an appropriate expression when the molecule is stretched to no more than about a third of its maximum extension. 

The third term is the Brownian force, F = kfiTS In /9R. It represents the average force exerted by bead 2 as a result of random bombardments by the surrounding (mostly solvent) molecules. 

The right sides of the last three equations were obtained by integrating the left sides by parts. 

Since the polymer contribution to the stress tensor is r = IvksTfi (RR), where v is the number of dumbbells per unit volume, we have, after collecting terms and multiplying 

---

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. 

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. 

Rheological Constitutive Equation of the 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. 

Note that it is possible to define other ctmvected derivatives (Bird etai, 1987 Larson, /988j. The upper-convected derivative arises most naturally from molecular theory. We will see this in Chapter II with the elastic dumbbell model. Note also V... 

The notation used for the elastic dumbbell model is shown in Figure 9. There are n dumbbells per unit volume, dissolved in a solvent with viscosity rj. The imposed velocity distribution for the solution is given by v = Vq + [k i ], in which Vq is independent of position, k = (Vv) is a position-independent traceless tensor, and r is the position vector such a velocity distribution is referred to as homogeneous since the velocity gradients are constant throughout the fluid. 

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. 

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 ... 

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. 

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. ... 

Similar to the elastic dumbbell case, we can obtain the various viscoelastic properties from the constitutive equation of the Rouse model. The main difference between the two models is that the elastic dumbbell... 

The elastic and the FENE dumbbell models lack interactions between different parts of the chain, and between different chains (such as cross-linking or entanglements in a concentrated polymer system). There are two more sophisticated theories network and reptation theories. The Phan-Thien-Tanner (PTT) model (Phan-Thien and Tanner 1977) was derived based on network theory (Lodge 1964). This approach is concerned with a balance between the creation and destruction of strands in the network of long chain polymer molecules. The result for the one-relaxation-tome form is... 

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. 

This is the constitutive equation or rheological equation of state for the elastic dumbbell suspensions. It is identical to the upper-convected Maxwell model, eq. 4.3.7. The molecular dynamics have led to a proper (frame-indifferent) time derivative and to a definition... 

Coppola et al. [142] calculated the dimensionless induction time, defined as the ratio of the quiescent nucleation rate over the total nucleation rate, as a function of the strain rate in continuous shear flow. They used AG according to different rheological models the Doi-Edwards model with the independent alignment assumption, DE-IAA [143], the linear elastic dumbbell model [154], and the finitely extensible nonlinear elastic dumbbell model with Peterlin s closure approximation, FENE-P [155]. The Doi-Edwards results showed the best agreement with experimental dimensionless induction times, defined as the time at which the viscosity suddenly starts to increase rapidly, normalized by the time at which this happens in quiescent crystallization [156-158]. 

---