Dumbbell model

Hantel,/. dumbbell, hantelfdrmig, a. dumbbell-shaped. Hantelmodell, n. dumbbell model, hanthaben, v.t. hantieren, v.t. hantieren, v.t. work, handle, manipulate,... 

In an attempt to describe the behavior at large chain deformations, de Gennes [7] incorporated into the dumbbell model the FENE spring law along with a variable bead friction coefficient which increases linearly with the interbead distance ... 

Both the many-bead and dumbbell models with internal friction predict limiting viscosities at high frequencies, tfa, and high shear rates t]m. The theories predict that tfm and tjx are related, such that... 

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. 

However, normally, the groups of both types are present in synthetic and natural monomer units of water-soluble polymers (Scheme 2), which suggests that the units are amphiphilic rather than hydrophobic or hydrophilic. Vasilevskaya et al. [22,23] reported a dumbbell model of the monomer unit in a chain in which a new representation of monomer units was proposed. In this representation, the amphiphilic character of the monomer units was... 

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 ... 

G. G. Fuller and L. G. Leal, The effects of conformation-dependent friction and internal viscosity on the dynamics of the nonlinear dumbbell model for a dilute polymer solution, J. Non-Newt. Fluid Mech. 8, 271 (1981). 

In the simplest case, at N = 1, the considered subchain model of a macromolecule reduces to the dumbbell model consisting of two Brownian particles connected with an elastic force. It can be called relaxator as well. The re-laxator is the simplest model of a macromolecule. Moreover, the dynamics of a macromolecule in normal co-ordinates is equivalent to the dynamics of a set of independent relaxators with various coefficients of elasticity and internal viscosity. In this way, one can consider a dilute solution of polymer as a suspension of independent relaxators which can be considered here to be identical for simplicity. The latter model is especially convenient for the qualitative analysis of the effects in polymer solutions under motion. 

It is instructive to start with the simplest possible model of a rotating diatomic molecule, the so-called dumbbell model, as illustrated in figure 6.19. The two atoms, of masses ni and m2, are regarded as point-like, and are fastened a distance R apart... 

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

The simplest model of a rotating diatomic molecule is a rigid rotor or dumbbell model in which the two atoms of mass and m2 are considered to be joined by a rigid, weightless rod. The allowed energy levels for a rigid rotor may be shown by quantum mechanics to be... 

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...

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. 

The derivation of the constitutive equation for the dumbbell model can also be extended to allow multiple beads and springs. From the force-balance equations for such a model, one obtains (Bird et al. 1987b Larson 1988)... 

If a dilute polymer solution is subjected to a imidirectional or steady flow with a velocity gradient large enough to stretch out the polymer molecule, nonlinear viscoelastic effects are observed. The simple Hookean dumbbell model, described in Section 3.4.4, can predict... 

EXTENSIONAL FLOW. In steady extensional flows, such as uniaxial extension, the single-relaxation-time Hookean dumbbell model and the multiple-relaxation-time Rouse and Zimm models predict that the steady-state extensional viscosity becomes infinite at a finite strain rate, s. With the dumbbell model, this occurs when the frictional drag force that stretches the dumbbell exceeds the contraction-producing force of the spring—that is, when the extension rate equals the critical value Sc. ... 

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.)...

When a nonlinear spring law is used in the dumbbell model, the Smoluchowski equation (3-28) is changed to... 

Figure 3.19 The polymer contribution to the steady-state uniaxial extensional viscosity r divided by the polymer contribution to the zero-shear viscosity rjp = r/o — fjj for the dumbbell model with a nonlinear FENE spring and various values of B = ipL. (From Bird et al. Dynamics of Polymeric Liquids, Vol. 2, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

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... 

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