Dumbbells, FENE

The equilibrium distribution of bead separation distances in the FENE dumbbell is ... 

It seems possible to rule out finite extensibility as an explanation of shear rate dependence in the viscosity, based simply on the equilibrium properties of polymer chains and the parallel between t] and t] in their departures from >/0. Experimentally, the mean square end-to-end vector obeys Gaussian statistics in 0-solvents spring constant K in FENE dumbbells is chosen to match this condition, then K - 3kT/(r2. The parameter b is therefore given by... 

Accordingly, given the necessity from equilibrium coil dimensions that bt> 1, the shear rate and frequency departures predicted by FENE dumbbells are displaced from each other. Moreover, the displacement increases with chain length. This is a clearly inconsistent with experimental behavior at all levels of concentration, including infinite dilution. Thus, finite extensibility must fail as a general model for the onset of nonlinear viscoelastic behavior in flexible polymer systems. It could, of course, become important in some situations, such as in elongational and shear flows at very high rates of deformation. 

Figure 3.2 Trouton ratio, Tr, of uniaxial extensional viscosity to zero-shear viscosity jq after start-up of steady uniaxial extension at a rate of 1 sec i for a Boger fluid consisting of a 0.185 wt% solution of flexible polyisobutylene (Mu, = 2.11 x 10 ) in a solvent composed mostly of viscous polybutene with some added kerosene (solid line). The dashed line is a fit of a multimode FENE dumbbell model, where each mode is represented by a FENE dumbbell model, with a spring law given by Eq. (3-56), without preaveraging, as described in Section 3.6.2.2.I. The relaxation times were obtained by fitting the linear viscoelastic data, G (co) and G"(cu). The slowest mode, with ri = 5 sec, dominates the behavior at large strains the best fit is obtained by choosing for it an extensibility parameter of = 40,000. The value of S — = 3(0.82) n/C(x, predicted from the...

Figure 3.17 Birefringence as a function of the eigenvalue of the velocity gradient tensor, G, for planar flows generated in a four-roll mill, for dilute solutions of polystyrenes of three different molecular weights in polychlorinated biphenyl solvent. Here G is the strain rate and a the flow type parameter. For planar extension, a — 1 and G = is the extension rate for simple shear, a = 0 and G = y is the shear rate. The different symbols correspond to a values of 1.0 (0)> 0.8 (A), 0.5 (-1-), and 0.25 (diamonds). The curves are theoretical predictions from the FENE dumbbell model, including conformation-dependent drag (discussed in Section 3.6.2.2.2). (From Fuller and Leal 1980, reprinted with permission from Steinkopff Publishers.)...

Comparisons of the predictions of the FENE dumbbell model with measurements of the extensional viscosity of dilute solutions have been fairly encouraging. Figure 3-2 compares the Trouton ratio predicted by a multimode FENE dumbbell model with experimental data for a Roger fluid Good agreement is obtained if one uses a value of the... 

This process has been examined theoretically by a number of authors (29-31), who derived constitutive equations based upon finitely extendable nonlinear elastic (FENE) dumbbell models (29), bead-rod models (30), and bead-spring models (31). There is general agreement that a large increase in elongational viscosity should be expected. 

As we have seen, there exist a number of treatments of the increase in extensional viscosity of the solution corresponding to the coil-stretch transition. In particular, the Warner FENE dumbbell model (29) and the Kramers bead-rod model (30) predict increases in normalized extensional viscosity of the order of N, the number of statistical segment units in the flexible chain. The normalized extensional viscosity (T)e ) compares the increase in extensional viscosity due to the polymer to three times (the Trouton ratio) the corresponding increase in the simple shear viscosity and is given by... 

Other solutions for the singlet configuration-space distribution function are those for the steady-state, homogeneous potential flow of elastic dumbbells with any kind of spring (DPL, Eq. (13.2-14)), and the first few terms in a perturbation solution for steady-state, homogeneous flow of FENE dumbbells (DPL, Eq. (13.2-15)). 

However, in his theory he did not take into account all three contributions to the heat-flux vector. Results analogous to Eqs. (16,28-16,33) have also been obtained for FENE dumbbells [31a]. 

Serveral authors [1 - 3] describe the non-Newtonian flow behaviour of dilute polymer solutions in porous media as an increase of extensional viscosity based on finitely extensible, nonlinear-elastic dumbbells (FENE-dumbbell — theory [4]). Conformity between theory and experiment is partially so good that the relationships found can be used for characterization of polymer solutions. A few significant examples are demonstrated in the following with the aid of selected polymer-solvent-temperature systems. 

If modified coefficients, Tjg and Deg, are used, as suggested in Refs. [1, 2] on the basis of the FENE dumbbell model. Fig. lb is obtained. The onset behaviour is described by the onset Deborah number, De o e,0 " with g 0 critical elongation rate of the porous media flow and T = relaxation time of the polymer solution, whilst the maximum value of the attainable increase of the extensional viscosity in normalized form only depends on the... 

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... 

Finitely extensible nonlinear elastic (FENE) dumbbells... 

Hookean dumbbells are infinitely extensible, and real polymer molecules are not this suggests that finitely extensible springs ought to be used, and consequently the FENE dumbbell model has been much studied. The spring force is taken to be... 

Figure 10 PMMA solution data reported by D. D. Joseph, G. S. Beavers, A. Cers, C. Dewald, A. Hoger and P. T. Than, J. Rheoly 1984, 28, 325, along with FENE dumbbell curve fits by L. E. Wedgewood (see L. E. Wedgewood and R. B. Bird, in Integration of Fundamental Polymer Science and Technology , ed. L. A. Kleintjens and P. J. Lemstra, Elsevier, Amsterdam, 1986, p. 337). The FENE-dumbbell parameters are also shown, where a = and yo = s+wfc7U, h/(h + 5)...

In addition to the Curtiss-Bird theory for Kramers bead-rod chains, a simpler theory for FENE dumbbells has also been worked out and compared with experimental data. The simplicity of this model enables one to see the structure of the theory without the mathematical complications. 

To simulate the vortex formation in the single mcirochannel, we used the FENE dumbbell in the CONNFFESSIT approach. Here three controlling parameters are chosen as Wi = 6.61, b = 500, and P = 6.67 n 10 The value of p is calculated from the viscosity data in Fig. 2(a). We also compared the simulation results between the viscoelastic and Newtonian fluid flow. 

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