FENE springs

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 ... 

Neighboring beads are connected via a FENE spring as follows... 

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.)...

Still other postulates for the anisotropy tensors can be made. If it is assumed that the anisotropy tensors are of the form AS-VBQQ/Q, then one is led to the encapsulated dumbbell model . If the FENE spring connector is used, then an approximate constitutive equation can be obtained, and some comparisons of this model have been made with experimental data on concentrated systems. 

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]... 

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... 

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.

A reasonable approximation for the force between two adjacent particles is given by the so-called FENE (finitely extendable non-linear elastic) spring force law (Bird et al. 1987a)... 

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...

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... 

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. 

Massah, H. Hanratty, T.J. Added stresses because of the presence of FENE-P bead-spring chains in a random velocity field. J. Fluid Mech. 1997, 337, 67-101. 

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. 

Here, 1/bb represent the bonds between consecutive backbone sites, which are connected by nonlinear elastic springs (FENE). The functional form of the potential 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)). 

For the case of viscous anisotropic polymer model, almost all turbulence statistics and power spectra calculated agree in qualitative sense with experimental results. Dimitropolous and co-workers (88) did DNS for fully turbulent channel flow of a polymer solution using the finitely extensible nonlinear elastic head spring dumbbell model with Peterlin approximation (FENE-P) and the Giesekus... 

Further models for polymer dynamics include the incorporation of stiffness parameters for both local and collective modes, and the approach of Bird and co-workers using the finitely extensible non-linear elastic (FENE) dumb-bell. The latter has been used to reproduce the non-Newtonian viscosity observed with polymer solutions even at the 6-temperature at high shear rates (frequencies), but not given by the simple (infinitely extensible) bead spring. 

As discussed in previous chapters, the choice of the bond / o rf( r/ ) and short-range —r l) potential varies from simulation to simulation. Off-lattice models, for example, have used the harmonic-spring potential, the FENE (finitely extendable, nonlinear elastic) potential, the rigid bond with fixed valence angles, and the freely-jointed chain model to represent the bonding interaction between adjacent monomers. For the short-range... 

The polymers were modeled as bead-spring chains. The attractive part of the bond potential is described by the FENE potential... 

For a FENE-type model with nonlinear elastic force of the dumbell, the restituting elastic force of a molecule can be represented in the form F = —where fo is th characteristic spring restitution coefficient per unit mass and Fq = L / L — trA) with L the maximum length of the polymer. The evolution equation for the elastic contribution is given in terms of the deformation or conformational tensor A = (RR), which is now related to the elastic part of the stress tensor by the formula The equation is... 

The most popular and efficient off-lattice models are of bead-spring type (Fig. 1.3[c]) and are not only used for MC but also for MD and Brownian dynamics (BD) simulations. " " It often is advantageous not to use a simple harmonic potential for the bond lengths as in eq. (1.2) but rather allow only a finite extensibility of the chains. In the MD work one works with the so-called FENE potential ... 

The simulation techniques used for polyelectrolytes in solution are extensions of the standard methods used for neutral polymers. The polymer chain is modeled as a set of connected beads. The beads are charged depending on the charge fraction, but otherwise the details of the monomer structure are neglected. Various means of connecting the bonded monomers are used. In lattice Monte Carlo the bonds are of course fixed. Two sets of simulations have used the rotational isomeric state model. Other simulations have used Hookean springs or the finite-extendable-nonlinear-elastic (FENE) potential. No important dependence on the nature of the bonds is expected at this level of modeling the polymer chain. 

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