# Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### The Governing Equations

## 3. Results

#### 3.1. Mean-Field Force under Elongation

#### 3.2. Mean-Field Force under Shear

#### 3.3. FENE Solution

#### 3.4. Comparison with Molecular Dynamics Simulations

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CDRT | Concentration dependent relaxation time |

FENE | Finitely extensible nonlinear elastic |

FENE-P | Finitely extensible nonlinear elastic with the Peterlin approximation |

FP | Fokker–Planck |

## References

- Warner, H.R. Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind. Eng. Chem. Fundam.
**1972**, 11, 379–387. [Google Scholar] [CrossRef] - Adelman, S.A.; Freed, K.F. Microscopic theory of polymer internal viscosity: Mode coupling approximation for the rouse model. J. Chem. Phys.
**1977**, 64, 1380–1393. [Google Scholar] [CrossRef] - Allegra, G. Internal viscosity in polymer chains: A critical analysis. J. Chem. Phys.
**1986**, 84, 5881–5890. [Google Scholar] [CrossRef] - Farago, J.; Meyer, H.; Baschnagel, J.; Semenov, A.N. Hydrodynamic and viscoelastic effects in polymer diffusion. J. Condens. Matter Phys.
**2012**, 82, 284105–340000. [Google Scholar] [CrossRef] - Ma, H.B.; Graham, M.D. Theory of shear-induced migration in dilute polymer solutions near solid boundaries. Phys. Fluids
**2005**, 17, 083103. [Google Scholar] [CrossRef] [Green Version] - Ottinger, H.C. A model of dilute polymer solutions with hydrodynamic interaction and finite extensibility. I. Basic equations and series expansions. J. Non-Newton. Fluid Mech.
**1987**, 26, 207–246. [Google Scholar] [CrossRef] - Bird, R.B.; Wiest, J.M. Anisotropic effects in dumbbell kinetic-theory. J. Rheol.
**1985**, 29, 519–532. [Google Scholar] [CrossRef] - Delgado-Buscolioni, R. Dynamics of a single tethered polymer under shear flow. AIP Conf. Proc.
**2007**, 913, 114–120. [Google Scholar] - Haber, C.; Ruiz, S.A.; Wirtz, D. Shape anisotropy of a single random-walk polymer. Proc. Natl. Acad. Sci. USA
**2000**, 97, 10792–10795. [Google Scholar] [CrossRef] [Green Version] - Hur, J.S.; Shaqfeh, E.S.G.; Larson, R.G. Brownian dynamics simulations of single dna molecules in shear flow. J. Rheol.
**2000**, 44, 713–742. [Google Scholar] [CrossRef] [Green Version] - Perkins, T.T.; Smith, D.E.; Chu, S. Single polymer dynamics in an elongational flow. Science
**1997**, 276, 2016–2021. [Google Scholar] [CrossRef] [Green Version] - Schneggenburger, C.; Kröger, M.; Hess, S. An extended FENE dumbbell theory for concentration dependent shear-induced anisotropy in dilute polymer solutions. J. Non-Newton. Fluid Mech.
**1996**, 62, 235–251. [Google Scholar] [CrossRef] - Kröger, M.; De Angelis, E. An extended FENE dumbbell model theory for concentration dependent shear-induced anisotropy in dilute polymer solutions: Addenda. J. Non-Newton. Fluid Mech.
**2005**, 125, 87–90. [Google Scholar] [CrossRef] - Keunings, R. On the Peterlin approximation for finitely extensible dumbbells. J. Non-Newton. Fluid Mech.
**1997**, 68, 85–100. [Google Scholar] [CrossRef] - Van Heel, A.P.G.; Hulsen, M.A.; van den Brule, B.H.A.A. On the selection of parameters in the fene-p model. J. Non-Newton. Fluid Mech.
**1998**, 75, 253–271. [Google Scholar] [CrossRef] - Link, A.; Springer, J. Light-scattering from dilute polymer-solutions in shear-flow. Macromolecules
**1993**, 26, 464–471. [Google Scholar] [CrossRef] - Babcock, H.P.; Smith, D.E.; Hur, J.S.; Shaqfeh, E.S.G.; Chu, S. Relating the microscopic and macroscopic response of a polymeric fluid in a shearing flow. Phys. Rev. Lett.
**2000**, 85, 2018–2021. [Google Scholar] [CrossRef] [PubMed] - Hur, J.S.; Shaqfeh, E.S.G.; Babcock, H.P.; Smith, D.E.; Chu, S. Dynamics of dilute and semidilute DNA solutions in the start-up of shear flow. J. Rheol.
**2001**, 45, 421–450. [Google Scholar] [CrossRef] - Clasen, C.; Plog, J.P.; Kulicke, W.M.; Owens, M.; Macosko, C.; Scriven, L.E.; Verani, M.; McKinley, G.H. How dilute are dilute solutions in extensional flows? J. Rheol.
**2006**, 50, 849–881. [Google Scholar] [CrossRef] [Green Version] - Ng, R.C.Y.; Leal, L.G. Concentration effects on birefringence and flow modification of semidilute polymer-solutions in extensional flows. J. Rheol.
**1993**, 37, 443–468. [Google Scholar] [CrossRef] - Feng, J.; Leal, L.G. Numerical simulations of the flow of dilute polymer solutions in a four-roll mill. J. Non-Newton. Fluid Mech.
**1997**, 72, 187–218. [Google Scholar] [CrossRef] - Kai-Wen, H.; Sasmal, C.; Prakash, J.R.; Schroeder, C.M. Direct observation of DNA dynamics in semidilute solutions in extensional flow. J. Rheol.
**2017**, 61, 151–167. [Google Scholar] - Stoltz, C.; de Pablo, J.J.; Graham, M.D. Concentration dependence of shear and extensional rheology of polymer solutions: Brownian dynamics simulations. J. Rheol.
**2006**, 50, 137–167. [Google Scholar] [CrossRef] - Ng, R.C.Y.; Leal, L.G. A study of the interacting FENE dumbbell model for semidilute polymer-solutions in extensional flows. J. Rheol.
**1993**, 32, 25–35. [Google Scholar] [CrossRef] - Bird, R.B.; Curtiss, C.F.; Armstrong, R.C.; Hassager, O. Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, 2nd ed.; Wiley-Interscience: New York, NY, USA, 1987; pp. 1–672. [Google Scholar]
- Underhill, P.T.; Doyle, P.S. Accuracy of bead-spring chains in strong flows. J. Non-Newton. Fluid Mech.
**2007**, 145, 109–123. [Google Scholar] [CrossRef] - Cohen, A. A Padé approximant to the inverse Langevin function. Rheol. Acta
**1991**, 30, 270–273. [Google Scholar] [CrossRef] - Lielens, G.; Halin, P.; Jaumain, I.; Keunings, R.; Legat, V. New closure approximations for the kinetic theory of finitely extensible dumbbells. J. Non-Newton. Fluid Mech.
**1998**, 76, 249–279. [Google Scholar] [CrossRef] - Lielens, G.; Keunings, R.; Legat, V. The FENE-L and FENE-LS closure approximations to the kinetic theory of finitely extensible dumbbells. J. Non-Newton. Fluid Mech.
**1999**, 87, 179–196. [Google Scholar] [CrossRef] - Bird, R.B.; Curtiss, C.F.; Armstrong, R.C.; Hassager, O. Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory, 2nd ed.; Wiley-Interscience: New York, NY, USA, 1987; pp. 1–464. [Google Scholar]
- Kröger, M. Models for Polymeric and Anisotropic Liquids, 1st ed.; Lecture Notes in Physics 675; Springer: Berlin/Heidelberg, Germany, 2005; pp. 1–215. [Google Scholar]
- Kuzuu, N. Constitutive equation for nematic liquid-crystals under weak velocity-gradient derived from a molecular kinetic-equation. J. Phys. Soc. Jpn.
**1983**, 52, 3486–3494. [Google Scholar] [CrossRef] - Peterlin, A. Hydrodynamics of macromolecules in a velocity field with longitudinal gradient. J. Polym. Sci. B Polym. Phys.
**1966**, 4, 287–291. [Google Scholar] [CrossRef] - Ottinger, H.C. Stochastic Processes in Polymeric Fluids: Tools and Examples for Developing Simulation Algorithms, 1st ed.; Springer: Berlin, Germany; New York, NY, USA, 1996; pp. 1–384. [Google Scholar]
- Chaffin, S. Non-Newtonian Fluids in Complex Geometries. Ph.D. Thesis, University of Sheffield, Sheffield, UK, 2016. [Google Scholar]
- Hess, S. Rheological properties via nonequilibrium molecular dynamics: From simple towards polymeric liquids. J. Non-Newton. Fluid Mech.
**1987**, 23, 305–319. [Google Scholar] [CrossRef] - Herrchen, M. A detailed comparison of various fene dumbbell models. J. Non-Newton. Fluid Mech.
**1997**, 68, 17–42. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The radially averaged probability distribution $\overline{\psi}$ for elongational flow for $\dot{\epsilon}=1$ with $f=0,0.5$ and $0.75$, denoted by the solid, dashed, and dot-dashed lines, respectively, in the upper quadrant $0<\theta <\frac{\pi}{4}$. (

**b**) The radially averaged distribution under shear flow, with $\dot{\gamma}=1$.

**Figure 2.**Numerical solution for the radius of gyration $\left(\right)$ plotted against the non-dimensional elongation rate $\dot{\epsilon}$ for the FENE-P (dashed lines) and FENE (solid lines) for (

**a**) $f=0$, (

**b**) $f=0.5$, and (

**c**) $f=0.75$. Circular markers indicate the exact analytic solution, where it exists.

**Figure 3.**Numerical solution for the radius of gyration $\left(\right)$ plotted against the non-dimensional shear rate $\dot{\gamma}$ for the FENE-P (dashed lines) and FENE (solid lines) models for (

**a**) $f=0$, (

**b**) $f=0.5$, and (

**c**) $f=0.75$.

**Figure 4.**The probability density function $\psi $ for the full nonlinear FENE solution with $\dot{\epsilon}=1$ and $b=10$ for (

**a**) $f=0$ and (

**b**) $f=0.5$.

**Figure 5.**The probability density function $\psi $ for the full nonlinear FENE solution with $\dot{\gamma}=1$ for $f=0$ and $b=10$ with (

**a**) $f=0$ and (

**b**) $f=0.5$.

**Figure 6.**The mean horizontal extension scaled by the infinitely dilute case against (

**a**) $\dot{\epsilon}$ and (

**b**) ${\dot{\epsilon}}_{c}$ from simulations that include the mean-field term. The results are given for $\frac{c}{{c}^{*}}=0,1,2$ denoted by the solid, dashed and dotted lines, respectively.

**Figure 7.**The mean horizontal extension scaled by the infinitely dilute case plotted against $\dot{\epsilon}$ where the mean-field term has been set to zero. The results are given for $\frac{c}{{c}^{*}}=0,1,2$, denoted by the solid, dashed, and dotted lines, respectively.

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Chaffin, S.; Rees, J.
Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation. *J* **2021**, *4*, 341-355.
https://doi.org/10.3390/j4030026

**AMA Style**

Chaffin S, Rees J.
Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation. *J*. 2021; 4(3):341-355.
https://doi.org/10.3390/j4030026

**Chicago/Turabian Style**

Chaffin, Stephen, and Julia Rees.
2021. "Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation" *J* 4, no. 3: 341-355.
https://doi.org/10.3390/j4030026