FENE model

{{about|finitely extensible nonlinear elastic|other uses|Fene (disambiguation){{!}}Fene}}

In polymer physics, the finite extensible nonlinear elastic (FENE) model, also called the FENE dumbbell model, represents the dynamics of a long-chained polymer. It simplifies the chain of monomers by connecting a sequence of beads with nonlinear springs.

Its direct extension the FENE-P model, is more commonly used in computational fluid dynamics to simulate turbulent flow. The P stands for the last name of physicist Anton Peterlin, who developed an important approximation of the model in 1966.{{Cite journal |last=Peterlin |first=A. |date=April 1966 |title=Hydrodynamics of macromolecules in a velocity field with longitudinal gradient |url=https://onlinelibrary.wiley.com/doi/10.1002/pol.1966.110040411 |journal=Journal of Polymer Science Part B: Polymer Letters |language=en |volume=4 |issue=4 |pages=287–291 |doi=10.1002/pol.1966.110040411 |issn=0449-2986|url-access=subscription }} The FENE-P model was introduced by Robert Byron Bird et al. in the 1980s.{{Cite journal |last1=Herrchen |first1=Markus |last2=Öttinger |first2=Hans Christian |date=1997 |title=A detailed comparison of various FENE dumbbell models |url=https://linkinghub.elsevier.com/retrieve/pii/S037702579601498X |journal=Journal of Non-Newtonian Fluid Mechanics |language=en |volume=68 |issue=1 |pages=17–42 |doi=10.1016/S0377-0257(96)01498-X|url-access=subscription }}

In 1991 the FENE-MP model (PM for modified Peterlin) was introduced and in 1988 the FENE-CR was introduced by M.D. Chilcott and J.M. Rallison.{{Cite journal |last1=Chilcott |first1=M. D. |last2=Rallison |first2=J. M. |date=1988-01-01 |title=Creeping flow of dilute polymer solutions past cylinders and spheres |url=https://www.sciencedirect.com/science/article/abs/pii/0377025788850626 |journal=Journal of Non-Newtonian Fluid Mechanics |volume=29 |pages=381–432 |doi=10.1016/0377-0257(88)85062-6 |issn=0377-0257|url-access=subscription }}

Formulation

The spring force in the FENE model is given Warner's spring force,{{Cite journal |last=Warner |first=Harold R. |date=1972 |title=Kinetic Theory and Rheology of Dilute Suspensions of Finitely Extendible Dumbbells |url=https://pubs.acs.org/doi/abs/10.1021/i160043a017 |journal=Industrial & Engineering Chemistry Fundamentals |language=en |volume=11 |issue=3 |pages=379–387 |doi=10.1021/i160043a017 |issn=0196-4313|url-access=subscription }} as

: $\textbf{F}_i=k\frac{\textbf{R}_i}{1-(R_i/L_{\rm max})^2}$ ,

where $R_i = |\textbf{R}_i|$ , k is the spring constant and Lmax the upper limit for the length extension.{{Cite book |last=Kröger |first=Martin |url=https://books.google.com/books?id=WAVzyygV5NQC&dq=fene+model&pg=PA15 |title=Models for Polymeric and Anisotropic Liquids |date=2005-09-15 |publisher=Springer Science & Business Media |isbn=978-3-540-26210-7 |language=en}} Total stretching force on i-th bead can be written as $\textbf{F}_i - \textbf{F}_{i-1}$ .

The Werner's spring force approximate the inverse Langevin function found in other models.

FENE-P model

The FENE-P model takes the FENE model and assumes the Peterlin statistical average for the restoring force as

: $\textbf{F}_i=k\frac{\textbf{R}_i}{1-\lang R^2_i/L_{\rm max}^2\rang}$ ,

where the $\lang\cdots\rang$ indicates the statistical average.

= Advantages and disanvatages =

FENE-P is one of few polymer models that can be used in computational fluid dynamics simulations since it removes the need of statistical averaging at each grid point at any instant in time. It is demonstrated to be able to capture some of the most important polymeric flow behaviors such as polymer turbulence drag reduction and shear thinning. It is the most commonly used polymer model that can be used in a turbulence simulation since direct numerical simulation of turbulence is already extremely expensive.

Due to its simplifications FENE-P is not able to show the hysteresis effects that polymers have, while the FENE model can.

References

[https://iopscience.iop.org/article/10.1088/1367-2630/9/10/360/meta Dynamics of dissolved polymer chains in isotropic turbulence]

External links

[http://sourceforge.net/projects/qpolymer/ QPolymer:] an open source (for Mac OS X) FENE model Brownian dynamics simulation software
[http://link.aps.org/abstract/PRL/v98/e024503 Stretching of Polymers in Isotropic Turbulence: A Statistical Closure]

Category:Polymers