Fene model

Although the FENE model has been shown earlier to be inappropriate for polymer systems on other grounds, it nevertheless illustrates an instance in which J(y) increases with shear rate naturally and without the need for postulating structural changes in the system. 

Starting from the FENE model (Warner Bird and collaborators) one obtains the following statements ... 

LJl) and van der Waals (LJ2) potentials were used for nonpolar bead-bead and bead-wall interactions, respectively. For polar interactions, exponential potential functions (EXP 1,2) were added to both bead-bead and bead-wall cases. For the bonding potential between adjacent beads in the chain, a finitely extensible nonlinear elastic (FENE) model was used. For example, PFPE Zdol... 

In addition to the excluded volume interaction, one also needs to add an attractive interaction binding the beads along the chain and for polymer brushes a wall-monomer interaction. The specific form of these interactions is not critical, except that the interaction between bonded beads should be such that the maximum extent of the bond is small enough such that bond crossing is inhibited. For the simulation results presented here, the finite extendible nonlinear elastic (FENE) model first introduced by Bird et al. [114] with suitably modified parameters [46,115] to avoid bond crossing was used. 

The adsorption of FENE-model chains on a structureless siuTace was studied by Milchev and Binder (307). The energetics was such that the polymer segments preferred to interact with themselves rather than with the siuTace, resulting in pol5mier droplet formation. By careful analysis of the fluctuations in the position of the free surface of the droplets, the authors were able to evaluate the polymer-vacuum interfacial tension. It would be of considerable interest to extend this method to more accurate force field descriptions of a variety of polymers. 

The FENE model with short chains of ten beads interacting with a truncated Lennard-Jones potential has been used for an MD investigation (381,382) of mode coupling theory just above the predicted critical temperatiue, which in turn is above the Tg. It is too early to say whether this theory will prove to be of practical use for polymer science, but simulations of this kind are probing deeply into the natiue of the glass transition, which can only help to illiuninate the physics of this important transition. 

A generic FENE model has also been used to study crazing in glassy polymers (383), with the resulting pictures looking very much hke what one sees in micrographs. Earlier work with a 2-D lattice model of a cross-linked system exhibited much the same phenomena (384). 

Polyelectrolytes. Multiple chain polyelectrolytes (qv) have been simulated by Limbach and Holm (419). They nsed a FENE model having beads that interact with both a truncated LJ potential and an electrostatic potential. Counte-... 

Very frequently, complex fluid manifest non-linear constitutive relations between the stress tensor and the velocity gradient. These relations can be a consequence of the fact that the relaxation time dependens on a scalar function of the deformation tensor like in the FENE models (Hinch, 1994 Rallison Hinch, 1988) or of the stress tensor, like in the case of the... 

These equations were applied to describe different situations. For isotropic friction, the dynamics of dumbell-like molecule solutions under simple shear conditions was analyzed to obtain a correction to the usual FENE models that arises through the kinetic contribution to the stress tensor. A second application for non-constant friction coefficient jS lead to the... 

Provided that F is not so large that the inertia term becomes irrelevant, the equations of motion can be integrated with any standard algorithm. Third- and fifth-order predictor-corrector and velodty-Verlet algorithms have been tested. For Lennard-Jones interaction, eq. (9.3), between all monomers and the FENE model between connected monomers, the equations of motion for each monomer can be stably intergrated with a time step At between 0.006-0.012r, where r = is the unit of... 

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 ... 

The inclusion of chain connectivity prevents polymer strands from crossing one another in the course of a computer simulation. In bead-spring polymer models, this typically means that one has to limit the maximal (or typical) extension of a spring connecting the beads that represent the monomers along the chain. This process is most often performed using the so-called finitely extensible, nonlinear elastic (FENE) type potentials44 of Eq. [17]... 

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 1.41. Potential energies for the bead-spring model LJ1—Lennard-Jones potential LJ2—van der Waals potential EXP1, EXP2—short-range polar potential FENE—finitely extensible nonlinear elastic potential.

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.)...

For this model, the second normal stress difference is zero at all shear rates. For the freely jointed chain, to which the FENE or FENE-P spring is an approximation, the polymer contribution to the shear viscosity at high shear rates is proportional to rather than... 

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... 

A simple generic bead spring model of chains can be used to study universal polymer properties that do not depend on specific chemical details. Bonds between neighbouring Lennard-Jones particles in a chain can be represented by the finite extension non-linear elastic (FENE) potential. 

A significant effort is under way to predict polymer DR in turbulent viscoelastic channel flow by direct numerical simulation using FENE, FENE-P, Gieskus, and other models (see, for example, Refs. ). While progress has been made by these and other authors, a review of these results is outside the scope of this entry. 

Finite-extensibility non-linear elastic (FENE) extension of dumbell model Multimode Zimm model... 

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. 

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