Rigid dumbbell model

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

The convection-diffusion equation for y (u, f) will be of the same form as the rigid dumbbell model of section 7.1.6.2 except that the diffusivity must be replaced by Dr(u, i) to give... 

This result is referred to as the Giesekus expression [62,86] and can be used to develop the form of the stress tensor for the rigid dumbbell model. Equation (7.63) for the rate of change of the second-moment tensor for this model is used to give the following result ... 

Figure 6.19. The rigid dumbbell model of a diatomic molecule.

Figure 6.17 Normalized intrinsic viscosity [r ]/[)7]o for a dilute solution of poly(y-benzyl-L-glutamate) (PBLG) = 208,000) in m-cresol. The line is a calculation for the rigid-dumbbell model, with the relaxation time t = lj6Dro adjusted to the value 10- sec to obtain a fit. The stress tensor for a suspension of rigid dumbbells is given by Eq. (6-36) with Cstr replaced by k T/Dro-(From Bird et al. 1987 data from Yang 1958, Dynamics of Polymeric Liquids, VoL 2, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

Fig. 3 a. Plot of the ratio (t) — ris)/(ri0 — /s) versus the shear rate k (sec-1). The solid triangles are the data of Wada (73 a) for tobacco mosaic virus in water at a concentration of 1.14 grams per liter. The solid curve is a plot of Eq. (6.7) for the rigid dumbbell model with the time constant A chosen to be 1 sec., which represents... 

That is, the rigid dumbbell model predicts that the elongational viscosity when plotted versus elongation rate should have a nonzero slope at k = 0. 

The quantity in Eq. (22.8) is rj (to) of the small-amplitude oscillatory experiment and is obtained, as expected, in the limit of vanishingly small k. It should be noted that when Act) = kQ and (AOT)=Ak, the expressions for tj of Eq. (22.8) and xxJ(—QY) of Eq. (21.6) are equivalent except for the final term in the series. Bird and Harris (4) found these two series to be equivalent to any order of terms from a calculation made using an integral continuum model. From the above result for the rigid dumbbell model we conclude that Bird and Harris result is a fortuitous one and that, in general, these two series are not equal. The only data to date on the transverse superposition experiment are those of Simmons (69 a), which show that tf —tjs as a function of cw decreases with increasing k, and that the curves of rf" as a function of to go through a maximum. 

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. ... 

Mendoza et al. (2003) studied experimentally the influence of processing conditions on the spatial distribution of the molecular orientation in injection molded isotactic polypropylene (iPP) plates. They found that the anisotropy of injection molded semi-crystalline polymers is governed by the orientation of the crystalline phase, and the distribution of the orientation strongly depends on the shear rate. Doufas et al. (2000), followed by Zheng and Kennedy (2001), have applied a rigid dumbbell model to simulate crystalline orientation in injection molded semicrystalline polymers. The model reads, in the form used by Zheng and Kennedy (2001, 2004) ... 

Doufas et al. (1999, 2000) proposed a two-phase model based on a modified Giesekus model for the amorphous melt phase and a rigid dumbbell model for the semi-crystalline phases. In the modified Giesekus model the relaxation time is a function of relative crystallinity a ... 

The rigid dumbbell model and the multibead rod models are shown in Figure 11. For these models we use a distribution function f(i, t% which gives the probability that a molecule is in the orientation u at time t, where u is the unit vector designating the orientation. 

If the Rotne-Prager-Yamakawa hydrodynamic interaction is included in the rigid dumbbell model, it is found that the equation for the distribution function is identical to equation (83) except that the time constant A must be replaced by... 

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