Finitely extensible nonlinear elastic

Derived from molecular arguments, Eq. (14) is correct for any extension ratio of the freely-jointed chain. In spite of its generality, the use of Eq. (14) is limited due to mathematical complexity. To account for the finite extensibility of the chain, the approximate finitely extensible nonlinear elastic (FENE) law proposed by Warner has gained popularity due to its ease of computation [33] ... 

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]... 

H. R. Warner, Jr., Kinetic Theory and Rheology of Dilute Suspensions of Finitely Extendible Dumbbells, Ind. Eng. Chem. Fundam., 11,379-387 (1972) also, R. L. Christiansen and R. B. Bird, Dilute Solution Rheology Experimental Results and Finitely Extensible Nonlinear Elastic Dumbbell Theory, J. Non-Newt. Fluid Mech., 3, 161-177 (1977/1978). 

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... 

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.

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. 

The Gaussian bond (1.4) can easily be stretched to high extension, and allows unphysical mutual passing of bonds. To prevent this unreaUstic mechanical property, the model potential, called the finitely extensible nonlinear elastic potential (FENE), and described by... 

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... 

In case of tetrahedral non polar liquid i.e. methane) each molecule in CG level represented by one bead and atomistically the molecule consisted of four atoms. The non-bonded interactions between the atoms were treated by Weeks-Chandler-Andersen potential (potential form is given in equation 38) and bonded interaction of all the atoms of a molecules by finitely extensible nonlinear elastic bonds as given in equation 39. 

However, the first DNS based on a microscopically originated constitutive equation for the polymer dynamics (the finite extensibility nonlinear elastic with the Peterlin approximation (FENE-P) model [46]) was conducted by Sureshkumar et al. [47]. In this work, for a fixed friction Reynolds number and other rheological parameters, drag reduction was observed as the Weissenberg number increased beyond a critical onset value. Moreover, accompanying drag reduction, characteristic changes were observed in the velocity and vorticity mean and rms values, the Reynolds stress, and... 

Lee et al. [21] conducted molecular dynamics simulations of the flow of a com-positionally symmetric diblock copolymer into the galleries between two siUcate sheets whose surfaces were modified by grafted surfactant chains. In these simulations they assumed that block copolymers and surfactants were represented by chains of soft spheres connected by an finitely extensible nonlinear elastic potential, non-Hookean dumbbells [22], which had been employed earlier in the simulations of the dynamics of polymer blends and block copolymers by Grest et al. [23] and Murat et al. [24]. To describe the interactions among the four components, namely the surfaces, the surfactant, and two blocks, Lee et al. [21] employed a Lennard-Jones potential having the energy parameters which are associated with the type of interactions often employed for lattice systems such as in the Flory-Huggins theory. 

Coppola et al. [142] calculated the dimensionless induction time, defined as the ratio of the quiescent nucleation rate over the total nucleation rate, as a function of the strain rate in continuous shear flow. They used AG according to different rheological models the Doi-Edwards model with the independent alignment assumption, DE-IAA [143], the linear elastic dumbbell model [154], and the finitely extensible nonlinear elastic dumbbell model with Peterlin s closure approximation, FENE-P [155]. The Doi-Edwards results showed the best agreement with experimental dimensionless induction times, defined as the time at which the viscosity suddenly starts to increase rapidly, normalized by the time at which this happens in quiescent crystallization [156-158]. 

Finitely extensible nonlinear elastic (FENE) dumbbells... 

The bond potential within the confinement region r,, + — ro < R (the total symmetric fluctuation width is 2R and centered about ro) is typically modeled by the finitely extensible nonlinear elastic (FENE) potential [50], which we introduce here in the form [51]... 

Finitely extensible nonlinear elastic Lennard-Jones polymers... 

The mesoscale model consists of a set of crosslink nodes (i.e., junctions) connected via single finite-extensible nonlinear elastic (FENE) bonds (that can be potentially cross-linked and/or scissioned), which represent the chain segments between crosslinks. In addition, there is a repulsive Lennard-Jones interaction between all crosslink positions to simulate volume exclusion effects. The Eennard-Jones and FENE interaction parameters were adjusted and the degree of polymerization (p) for a given length of a FENE bond calibrated until the MWD computed from our network matched the experimental MWD of the virgin material [112]. 

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