# Active Brownian Filamentous Polymers under Shear Flow

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model: Active Brownian Filament/Polymer

#### 2.1. Equation of Motion

#### 2.2. Eigenfunction Expansion

#### 2.3. Mode–Amplitude Correlation Functions

#### 2.4. Inextensibility and Stretching Coefficient $\lambda $

- (i)
- Passive semiflexible polymer in shear flow, i.e., $Pe=0$ (for details, cf. Ref. [88])
- -
- For $pL\gg 1$ and $\mu \gg 1$$$\begin{array}{c}\hfill {\mu}^{3}-{\mu}^{5/2}-\frac{{\pi}^{4}W{i}^{2}}{540pL}=0,\phantom{\rule{8.53581pt}{0ex}}\stackrel{Wi\gg 1}{\u27f9}\mu =W{i}^{2/3}{\left(\right)}^{\frac{{\pi}^{4}}{540pL}}1/3\end{array}$$
- -
- For $pL<1$ and $\mu \gg 1$$$\begin{array}{c}\hfill \mu =\frac{W{i}^{2/3}}{pL}{\left(\right)}^{\frac{4}{15}}1/3\end{array}$$

- (ii)
- Active flexible polymer at weak shear flow, i.e., $Wi\ll 1$, $pL\gg 1$, and ${\tau}_{n}={\tau}_{R}/{\mu}_{0}{n}^{2}$ [52,53]. For later use, we denote the Lagrangian multiplier at $Wi=0$ and $Pe>0$ by ${\mu}_{0}$
- -
- For $1\ll Pe<\infty $ and ${\mu}_{0}\to \infty $$$\begin{array}{c}\hfill {\mu}_{0}=\frac{P{e}^{4/3}}{pL}\frac{L}{6l\Delta}\end{array}$$
- -
- For $Pe\to \infty $, i.e., ${\mu}_{0}\to \infty $$$\begin{array}{c}\hfill {\mu}_{0}=\frac{Pe}{pL\Delta}\sqrt{\frac{{L}^{3}}{54{l}^{3}}}\end{array}$$

- (iii)
- Active flexible polymer in shear flow, ${\tau}_{n}={\tau}_{R}/\mu {n}^{2}$,
- -
- For $1<Pe,Wi<\infty $, ${L}^{3}/3\pi {l}^{3}pL\Delta \mu \gg 1$, and $pL\gg 1$ (with Equation (25))$$\begin{array}{c}\hfill \mu =W{i}^{2/3}{\mu}_{0}^{2/3}{\left(\right)}^{\frac{P{e}^{2}}{36\Delta pL}}1/3=W{i}^{2/3}\frac{P{e}^{14/9}}{pL\Delta}{\left(\right)}^{\frac{L}{36l}}2/3\end{array}$$
- -
- For $Pe\to \infty $, i.e., ${\mu}_{0}=Pe\sqrt{{L}^{3}/54{l}^{3}}/pL\Delta \to \infty $$$\begin{array}{c}\hfill \mu ={\mu}_{0}\frac{1}{\sqrt{2}}\sqrt{1+\sqrt{1+\frac{8W{i}^{2}}{{\pi}^{2}}}}\phantom{\rule{4pt}{0ex}}\stackrel{Wi\gg 1}{\u27f6}\phantom{\rule{4pt}{0ex}}W{i}^{1/2}\frac{Pe}{pL\Delta}\sqrt{\frac{\sqrt{2}{L}^{3}}{54\pi {l}^{3}}}\end{array}$$

Hence, in the limit $Wi\to \infty $, $\mu $ exhibits a crossover from a $\mu \sim W{i}^{2/3}$ dependence for $1\ll Pe<\infty $ to a dependence $\mu \sim W{i}^{1/2}$ for $Pe\to \infty $. The latter characteristics are different from the passive case and are a consequence of the coupling between activity and shear flow.

## 3. Dynamics and Conformations

#### 3.1. Relaxation Times

#### 3.2. Radius of Gyration

#### 3.3. Alignment

## 4. Rheology: Viscosity

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ramaswamy, S. The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys.
**2010**, 1, 323–345. [Google Scholar] [CrossRef] - Romanczuk, P.; Bär, M.; Ebeling, W.; Lindner, B.; Schimansky-Geier, L. Active Brownian Particles. Eur. Phys. J. Spec. Top.
**2012**, 202, 1–162. [Google Scholar] [CrossRef] - Marchetti, M.C.; Joanny, J.F.; Ramaswamy, S.; Liverpool, T.B.; Prost, J.; Rao, M.; Simha, R.A. Hydrodynamics of soft active matter. Rev. Mod. Phys.
**2013**, 85, 1143. [Google Scholar] [CrossRef] - Elgeti, J.; Winkler, R.G.; Gompper, G. Physics of microswimmers—Single particle motion and collective behavior: A review. Rep. Prog. Phys.
**2015**, 78, 056601. [Google Scholar] [CrossRef] [PubMed] - Bechinger, C.; Di Leonardo, R.; Löwen, H.; Reichhardt, C.; Volpe, G.; Volpe, G. Active particles in complex and crowded environments. Rev. Mod. Phys.
**2016**, 88, 045006. [Google Scholar] [CrossRef] - Marchetti, M.C.; Fily, Y.; Henkes, S.; Patch, A.; Yllanes, D. Minimal model of active colloids highlights the role of mechanical interactions in controlling the emergent behavior of active matter. Curr. Opin. Colloid Interface Sci.
**2016**, 21, 34–43. [Google Scholar] [CrossRef][Green Version] - Winkler, R.G.; Elgeti, J.; Gompper, G. Active Polymers—Emergent Conformational and Dynamical Properties: A Brief Review. J. Phys. Soc. Jpn.
**2017**, 86, 101014. [Google Scholar] [CrossRef] - Lauga, E.; Powers, T.R. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys.
**2009**, 72, 096601. [Google Scholar] [CrossRef][Green Version] - Vicsek, T.; Zafeiris, A. Collective motion. Phys. Rep.
**2012**, 517, 71–140. [Google Scholar] [CrossRef] - Wysocki, A.; Winkler, R.G.; Gompper, G. Cooperative motion of active Brownian spheres in three-dimensional dense suspensions. EPL
**2014**, 105, 48004. [Google Scholar] [CrossRef][Green Version] - Zöttl, A.; Stark, H. Emergent behavior in active colloids. J. Phys. Condens. Matter
**2016**, 28, 253001. [Google Scholar] [CrossRef][Green Version] - Duman, O.; Isele-Holder, R.E.; Elgeti, J.; Gompper, G. Collective dynamics of self-propelled semiflexible filaments. Soft Matter
**2018**, 14, 4483–4494. [Google Scholar] [CrossRef] [PubMed][Green Version] - Martin-Gomez, A.; Levis, D.; Diaz-Guilera, A.; Pagonabarraga, I. Collective motion of active Brownian particles with polar alignment. Soft Matter
**2018**, 14, 2610–2618. [Google Scholar] [CrossRef] [PubMed][Green Version] - Muddana, H.S.; Sengupta, S.; Mallouk, T.E.; Sen, A.; Butler, P.J. Substrate Catalysis Enhances Single-Enzyme Diffusion. J. Am. Chem. Soc.
**2010**, 132, 2110–2111. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dey, K.K.; Das, S.; Poyton, M.F.; Sengupta, S.; Butler, P.J.; Cremer, P.S.; Sen, A. Chemotactic Separation of Enzymes. ACS Nano
**2014**, 8, 11941–11949. [Google Scholar] [CrossRef] [PubMed] - Nédélec, F.J.; Surrey, T.; Maggs, A.C.; Leibler, S. Self-organization of microtubules and motors. Nature
**1997**, 389, 305. [Google Scholar] [CrossRef] [PubMed] - Howard, J. Mechanics of Motor Proteins and the Cytoskeleton; Sinauer Associates: Sunderland, MA, USA, 2001. [Google Scholar]
- Kruse, K.; Joanny, J.F.; Jülicher, F.; Prost, J.; Sekimoto, K. Asters, Vortices, and Rotating Spirals in Active Gels of Polar Filaments. Phys. Rev. Lett.
**2004**, 92, 078101. [Google Scholar] [CrossRef] [PubMed] - Bausch, A.R.; Kroy, K. A bottom-up approach to cell mechanics. Nat. Phys.
**2006**, 2, 231. [Google Scholar] [CrossRef] - Jülicher, F.; Kruse, K.; Prost, J.; Joanny, J.F. Active behavior of the cytoskeleton. Phys. Rep.
**2007**, 449, 3–28. [Google Scholar] [CrossRef] - Schaller, V.; Weber, C.; Semmrich, C.; Frey, E.; Bausch, A.R. Polar patterns of driven filaments. Nature
**2010**, 467, 73. [Google Scholar] [CrossRef] [PubMed] - Prost, J.; Jülicher, F.; Joanny, J.F. Active gel physics. Nat. Phys.
**2015**, 11, 111. [Google Scholar] [CrossRef] - Berg, H.C. E. Coli in Motion; Biological and Medical Physics Series; Springer: New York, NY, USA, 2004. [Google Scholar]
- Guasto, J.S.; Rusconi, R.; Stocker, R. Fluid Mechanics of Planktonic Microorganisms. Ann. Rev. Fluid Mech.
**2011**, 44, 373–400. [Google Scholar] [CrossRef] - Howse, J.R.; Jones, R.A.L.; Ryan, A.J.; Gough, T.; Vafabakhsh, R.; Golestanian, R. Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk. Phys. Rev. Lett.
**2007**, 99, 048102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Volpe, G.; Buttinoni, I.; Vogt, D.; Kümmerer, H.J.; Bechinger, C. Microswimmers in patterned environments. Soft Matter
**2011**, 7, 8810–8815. [Google Scholar] [CrossRef] - Thutupalli, S.; Seemann, R.; Herminghaus, S. Swarming behavior of simple model squirmers. New J. Phys.
**2011**, 13, 073021. [Google Scholar] [CrossRef][Green Version] - Hagen, B.; Kümmel, F.; Wittkowski, R.; Takagi, D.; Löwen, H.; Bechinger, C. Gravitaxis of asymmetric self-propelled colloidal particles. Nat. Commun.
**2014**, 5, 4829. [Google Scholar] [CrossRef] [PubMed][Green Version] - Maass, C.C.; Krüger, C.; Herminghaus, S.; Bahr, C. Swimming Droplets. Annu. Rev. Condens. Matter Phys.
**2016**, 7, 171–193. [Google Scholar] [CrossRef] - Copeland, M.F.; Weibel, D.B. Bacterial swarming: A model system for studying dynamic self-assembly. Soft Matter
**2009**, 5, 1174–1187. [Google Scholar] [CrossRef] [PubMed] - Selander, E.; Jakobsen, H.H.; Lombard, F.; Kiørboe, T. Grazer cues induce stealth behavior in marine dinoflagellates. Proc. Natl. Acad. Sci. USA
**2011**, 108, 4030–4034. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sohn, M.H.; Seo, K.W.; Choi, Y.S.; Lee, S.J.; Kang, Y.S.; Kang, Y.S. Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry. Mar. Biol.
**2011**, 158, 561–570. [Google Scholar] [CrossRef] - Yan, J.; Han, M.; Zhang, J.; Xu, C.; Luijten, E.; Granick, S. Reconfiguring active particles by electrostatic imbalance. Nat. Mater.
**2016**, 15, 1095. [Google Scholar] [CrossRef] [PubMed] - Löwen, H. Active colloidal molecules. EPL
**2018**, 121, 58001. [Google Scholar] [CrossRef] - Liverpool, T.B.; Maggs, A.C.; Ajdari, A. Viscoelasticity of Solutions of Motile Polymers. Phys. Rev. Lett.
**2001**, 86, 4171. [Google Scholar] [CrossRef] [PubMed] - Sarkar, D.; Thakur, S.; Tao, Y.G.; Kapral, R. Ring closure dynamics for a chemically active polymer. Soft Matter
**2014**, 10, 9577–9584. [Google Scholar] [CrossRef] [PubMed] - Chelakkot, R.; Gopinath, A.; Mahadevan, L.; Hagan, M.F. Flagellar dynamics of a connected chain of active, polar, Brownian particles. J. R. Soc. Interface
**2014**, 11, 20130884. [Google Scholar] [CrossRef] [PubMed] - Loi, D.; Mossa, S.; Cugliandolo, L.F. Non-conservative forces and effective temperatures in active polymers. Soft Matter
**2011**, 7, 10193–10209. [Google Scholar] [CrossRef] - Harder, J.; Valeriani, C.; Cacciuto, A. Activity-induced collapse and reexpansion of rigid polymers. Phys. Rev. E
**2014**, 90, 062312. [Google Scholar] [CrossRef] [PubMed] - Ghosh, A.; Gov, N.S. Dynamics of Active Semiflexible Polymers. Biophys. J.
**2014**, 107, 1065–1073. [Google Scholar] [CrossRef] [PubMed] - Shin, J.; Cherstvy, A.G.; Kim, W.K.; Metzler, R. Facilitation of polymer looping and giant polymer diffusivity in crowded solutions of active particles. New J. Phys.
**2015**, 17, 113008. [Google Scholar] [CrossRef][Green Version] - Isele-Holder, R.E.; Elgeti, J.; Gompper, G. Self-propelled worm-like filaments: Spontaneous spiral formation, structure, and dynamics. Soft Matter
**2015**, 11, 7181–7190. [Google Scholar] [CrossRef] [PubMed] - Isele-Holder, R.E.; Jager, J.; Saggiorato, G.; Elgeti, J.; Gompper, G. Dynamics of self-propelled filaments pushing a load. Soft Matter
**2016**, 12, 8495–8505. [Google Scholar] [CrossRef] [PubMed] - Laskar, A.; Adhikari, R. Brownian microhydrodynamics of active filaments. Soft Matter
**2015**, 11, 9073–9085. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jiang, H.; Hou, Z. Motion transition of active filaments: Rotation without hydrodynamic interactions. Soft Matter
**2014**, 10, 1012–1017. [Google Scholar] [CrossRef] [PubMed] - Babel, S.; Löwen, H.; Menzel, A.M. Dynamics of a linear magnetic “microswimmer molecule”. EPL
**2016**, 113, 58003. [Google Scholar] [CrossRef][Green Version] - Kaiser, A.; Löwen, H. Unusual swelling of a polymer in a bacterial bath. J. Chem. Phys.
**2014**, 141, 044903. [Google Scholar] [CrossRef] [PubMed][Green Version] - Valeriani, C.; Li, M.; Novosel, J.; Arlt, J.; Marenduzzo, D. Colloids in a bacterial bath: Simulations and experiments. Soft Matter
**2011**, 7, 5228–5238. [Google Scholar] [CrossRef] - Suma, A.; Gonnella, G.; Marenduzzo, D.; Orlandini, E. Motility-induced phase separation in an active dumbbell fluid. EPL
**2014**, 108, 56004. [Google Scholar] [CrossRef] - Cugliandolo, L.F.; Gonnella, G.; Suma, A. Rotational and translational diffusion in an interacting active dumbbell system. Phys. Rev. E
**2015**, 91, 062124. [Google Scholar] [CrossRef] [PubMed] - Winkler, R.G. Dynamics of flexible active Brownian dumbbells in the absence and the presence of shear flow. Soft Matter
**2016**, 12, 3737–3749. [Google Scholar] [CrossRef] [PubMed] - Eisenstecken, T.; Gompper, G.; Winkler, R.G. Conformational Properties of Active Semiflexible Polymers. Polymers
**2016**, 8, 304. [Google Scholar] [CrossRef] - Eisenstecken, T.; Gompper, G.; Winkler, R.G. Internal dynamics of semiflexible polymers with active noise. J. Chem. Phys.
**2017**, 146, 154903. [Google Scholar] [CrossRef] [PubMed] - Siebert, J.T.; Letz, J.; Speck, T.; Virnau, P. Phase behavior of active Brownian disks, spheres, and dimers. Soft Matter
**2017**, 13, 1020–1026. [Google Scholar] [CrossRef] [PubMed] - Küchler, N.; Löwen, H.; Menzel, A.M. Getting drowned in a swirl: Deformable bead-spring model microswimmers in external flow fields. Phys. Rev. E
**2016**, 93, 022610. [Google Scholar] [CrossRef] [PubMed] - Kokot, G.; Das, S.; Winkler, R.G.; Gompper, G.; Aranson, I.S.; Snezhko, A. Active turbulence in a gas of self-assembled spinners. Proc. Natl. Acad. Sci. USA
**2017**, 114, 12870–12875. [Google Scholar] [CrossRef] [PubMed][Green Version] - Eisenstecken, T.; Ghavami, A.; Mair, A.; Gompper, G.; Winkler, R.G. Conformational and dynamical properties of semiflexible polymers in the presence of active noise. AIP Conf. Proc.
**2017**, 1871, 050001. [Google Scholar][Green Version] - Laskar, A.; Singh, R.; Ghose, S.; Jayaraman, G.; Kumar, P.B.S.; Adhikari, R. Hydrodynamic instabilities provide a generic route to spontaneous biomimetic oscillations in chemomechanically active filaments. Sci. Rep.
**2013**, 3, 1964. [Google Scholar] [CrossRef] [PubMed] - Vandebroek, H.; Vanderzande, C. Dynamics of a polymer in an active and viscoelastic bath. Phys. Rev. E
**2015**, 92, 060601. [Google Scholar] [CrossRef] [PubMed] - Sanchez, T.; Chen, D.T.N.; DeCamp, S.J.; Heymann, M.; Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature
**2012**, 491, 431. [Google Scholar] [CrossRef] [PubMed] - Schaller, V.; Weber, C.; Frey, E.; Bausch, A.R. Polar pattern formation: Hydrodynamic coupling of driven filaments. Soft Matter
**2011**, 7, 3213–3218. [Google Scholar] [CrossRef][Green Version] - Abkenar, M.; Marx, K.; Auth, T.; Gompper, G. Collective behavior of penetrable self-propelled rods in two dimensions. Phys. Rev. E
**2013**, 88, 062314. [Google Scholar] [CrossRef] [PubMed] - Denk, J.; Huber, L.; Reithmann, E.; Frey, E. Active Curved Polymers Form Vortex Patterns on Membranes. Phys. Rev. Lett.
**2016**, 116, 178301. [Google Scholar] [CrossRef] [PubMed] - Peruani, F. Active Brownian rods. Eur. Phys. J. Spec. Top.
**2016**, 225, 2301–2317. [Google Scholar] [CrossRef][Green Version] - Needleman, D.; Dogic, Z. Active matter at the interface between materials science and cell biology. Nat. Rev. Mater.
**2017**, 2, 201748. [Google Scholar] [CrossRef] - Prathyusha, K.R.; Henkes, S.; Sknepnek, R. Dynamically generated patterns in dense suspensions of active filaments. Phys. Rev. E
**2018**, 97, 022606. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brangwynne, C.P.; Koenderink, G.H.; MacKintosh, F.C.; Weitz, D.A. Nonequilibrium Microtubule Fluctuations in a Model Cytoskeleton. Phys. Rev. Lett.
**2008**, 100, 118104. [Google Scholar] [CrossRef] [PubMed][Green Version] - Weber, C.A.; Suzuki, R.; Schaller, V.; Aranson, I.S.; Bausch, A.R.; Frey, E. Random bursts determine dynamics of active filaments. Proc. Natl. Acad. Sci. USA
**2015**, 112, 10703–10707. [Google Scholar] [CrossRef] [PubMed][Green Version] - Weber, S.C.; Spakowitz, A.J.; Theriot, J.A. Nonthermal ATP-dependent fluctuations contribute to the in vivo motion of chromosomal loci. Proc. Natl. Acad. Sci. USA
**2012**, 109, 7338–7343. [Google Scholar] [CrossRef] [PubMed][Green Version] - Javer, A.; Long, Z.; Nugent, E.; Grisi, M.; Siriwatwetchakul, K.; Dorfman, K.D.; Cicuta, P.; Cosentino Lagomarsino, M. Short-time movement of E. coli chromosomal loci depends on coordinate and subcellular localization. Nat. Commun.
**2013**, 4, 3003. [Google Scholar] [CrossRef] [PubMed] - Zidovska, A.; Weitz, D.A.; Mitchison, T.J. Micron-scale coherence in interphase chromatin dynamics. Proc. Natl. Acad. Sci. USA
**2013**, 110, 15555. [Google Scholar] [CrossRef] [PubMed] - Winkler, R.G.; Wysocki, A.; Gompper, G. Virial pressure in systems of spherical active Brownian particles. Soft Matter
**2015**, 11, 6680–6691. [Google Scholar] [CrossRef] [PubMed][Green Version] - Das, S.; Gompper, G.; Winkler, R.G. Confined active Brownian particles: Theoretical description of propulsion-induced accumulation. New J. Phys.
**2018**, 20, 015001. [Google Scholar] [CrossRef] - Barry, M.T.; Rusconi, R.; Guasto, J.S.; Stocker, R. Shear-induced orientational dynamics and spatial heterogeneity in suspensions of motile phytoplankton. J. R. Soc. Interface
**2015**, 12, 20150791. [Google Scholar] [CrossRef] [PubMed][Green Version] - Winkler, R.G.; Reineker, P.; Harnau, L. Models and equilibrium properties of stiff molecular chains. J. Chem. Phys.
**1994**, 101, 8119–8129. [Google Scholar] [CrossRef] - Harnau, L.; Winkler, R.G.; Reineker, P. Dynamic properties of molecular chains with variable stiffness. J. Chem. Phys.
**1995**, 102, 7750–7757. [Google Scholar] [CrossRef] - Harnau, L.; Winkler, R.G.; Reineker, P. Dynamic Structure Factor of Semiflexible Macromolecules in Dilute Solution. J. Chem. Phys.
**1996**, 104, 6355–6368. [Google Scholar] [CrossRef] - Bawendi, M.G.; Freed, K.F. A Wiener integral model for stiff polymer chains. J. Chem. Phys.
**1985**, 83, 2491–2496. [Google Scholar] [CrossRef] - Battacharjee, S.M.; Muthukumar, M. Statistical mechanics of solutions of semiflexible chains: A path integral formulation. J. Chem. Phys.
**1987**, 86, 411–418. [Google Scholar] [CrossRef] - Ha, B.Y.; Thirumalai, D. A mean-field model for semiflexible chains. J. Chem. Phys.
**1995**, 103, 9408–9412. [Google Scholar] [CrossRef] - Winkler, R.G.; Reineker, P. Finite Size Distribution and Partition Functions of Gaussian Chains: Maximum Entropy Approach. Macromolecules
**1992**, 25, 6891–6896. [Google Scholar] [CrossRef] - Winkler, R.G. Deformation of Semiflexible Chains. J. Chem. Phys.
**2003**, 118, 2919–2928. [Google Scholar] [CrossRef][Green Version] - Fodor, É.; Nardini, C.; Cates, M.E.; Tailleur, J.; Visco, P.; van Wijland, F. How Far from Equilibrium Is Active Matter? Phys. Rev. Lett.
**2016**, 117, 038103. [Google Scholar] [CrossRef] [PubMed] - Fily, Y.; Marchetti, M.C. Athermal Phase Separation of Self-Propelled Particles with No Alignment. Phys. Rev. Lett.
**2012**, 108, 235702. [Google Scholar] [CrossRef] [PubMed] - Bialké, J.; Speck, T.; Löwen, H. Crystallization in a Dense Suspension of Self-Propelled Particles. Phys. Rev. Lett.
**2012**, 108, 168301. [Google Scholar] [CrossRef] [PubMed] - Kaiser, A.; Babel, S.; ten Hagen, B.; von Ferber, C.; Löwen, H. How does a flexible chain of active particles swell? J. Chem. Phys.
**2015**, 142, 124905. [Google Scholar] [CrossRef] [PubMed][Green Version] - Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, UK, 1986. [Google Scholar]
- Winkler, R.G. Conformational and rheological properties of semiflexible polymers in shear flow. J. Chem. Phys.
**2010**, 133, 164905. [Google Scholar] [CrossRef] [PubMed] - Huang, C.C.; Winkler, R.G.; Sutmann, G.; Gompper, G. Semidilute polymer solutions at equilibrium and under shear flow. Macromolecules
**2010**, 43, 10107. [Google Scholar] [CrossRef] - Bird, R.B.; Armstrong, R.C.; Hassager, O. Dynamics of Polymer Liquids; John Wiley & Sons: New York, NY, USA, 1987; Volume 1. [Google Scholar]
- Schroeder, C.M.; Teixeira, R.E.; Shaqfeh, E.S.G.; Chu, S. Dynamics of DNA in the flow-gradient plane of steady shear flow: Observations and simulations. Macromolecules
**2005**, 38, 1967–1978. [Google Scholar] [CrossRef] - Lyulin, A.V.; Adolf, D.B.; Davies, G.R. Brownian dynamics simulations of linear polymers under shear flow. J. Chem. Phys.
**1999**, 111, 758–771. [Google Scholar] [CrossRef] - Jendrejack, R.M.; de Pablo, J.J.; Graham, M.D. Stochastic simulations of DNA in flow: Dynamics and the effects of hydrodynamic interactions. J. Chem. Phys.
**2002**, 116, 7752–7759. [Google Scholar] [CrossRef][Green Version] - Liu, S.; Ashok, B.; Muthukumar, M. Brownian dynamics simulations of bead-rod-chain in simple shear flow and elongational flow. Polymer
**2004**, 45, 1383–1389. [Google Scholar] [CrossRef] - Aust, C.; Kröger, M.; Hess, S. Structure and dynamics of dilute polymer solutions under shear flow via nonequilibrium molecular dynamics. Macromolecules
**1999**, 32, 5660–5672. [Google Scholar] [CrossRef] - Eslami, H.; Müller-Plathe, F. Viscosity of Nanoconfined Polyamide-6,6 Oligomers: Atomistic Reverse Nonequilibrium Molecular Dynamics Simulation. J. Phys. Chem. B
**2010**, 114, 387–395. [Google Scholar] [CrossRef] [PubMed] - Singh, S.P.; Chatterji, A.; Gompper, G.; Winkler, R.G. Dynamical and rheological properties of ultrasoft colloids under shear flow. Macromolecules
**2013**, 46, 8026–8036. [Google Scholar] [CrossRef]

**Figure 1.**Illustration of the continuous semiflexible active polymer (ABPO) in shear flow. The arrows and colors indicate the orientation of the active velocity $\mathit{v}(s,t)$.

**Figure 2.**Stretching coefficient $\mu =2\lambda /3p$ normalized by the value ${\mu}_{0}$ of the active, non-sheared system as function of the Weissenberg number $Wi$ for the Péclet numbers $Pe=0,\phantom{\rule{4pt}{0ex}}0.6,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}30,\phantom{\rule{4pt}{0ex}}{10}^{2},3\times {10}^{2},$ and ∞ (bright to dark color); (

**left**) $pL=L/2{l}_{p}=0.1$ (stiff) and (

**right**) $pL={10}^{2}$ (flexible polymer). The number of active sites is $L/l={10}^{2}$ and the diffusion coefficient ratio $\Delta =0.3$.

**Figure 3.**Stretching coefficient $\mu =2\lambda /3p$ as a function of the Péclet number $Pe$ for the stiffness $pL={10}^{-2},\phantom{\rule{4pt}{0ex}}{10}^{-1},\phantom{\rule{4pt}{0ex}}{10}^{0},\phantom{\rule{4pt}{0ex}}{10}^{1},$ and ${10}^{2}$ (bottom to top). The Weissenberg numbers are (

**left**) $Wi=0$ [52] and (

**right**) $Wi={10}^{2}$. The number of active sites is $L/l={10}^{2}$ and $\Delta =0.3$.

**Figure 4.**Longest polymer relaxation time ${\tau}_{1}$ normalized by the longest relaxation time ${\tau}_{0}$ of the active, non-sheared system ($Wi=0$) as function of the Weissenberg number $Wi$ for the Péclet numbers $Pe=0,\phantom{\rule{4pt}{0ex}}0.6,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}30,\phantom{\rule{4pt}{0ex}}{10}^{2},3\times {10}^{2},$ and ∞ (bright to dark color); (

**left**) $pL=0.1$ and (

**right**) $pL={10}^{2}$. The number of active sites is $L/l={10}^{2}$ and $\Delta =0.3$.

**Figure 5.**Mode-number dependence of the relaxation times ${\tau}_{n}$ normalized by the longest relaxation time ${\tau}_{1}$ for the Péclet numbers $Pe=0,\phantom{\rule{4pt}{0ex}}{10}^{1},\phantom{\rule{4pt}{0ex}}{10}^{2}$, and ${10}^{3}$ (different colors and symbols; from left to right), and the Weissenberg numbers $Wi=0,\phantom{\rule{4pt}{0ex}}{10}^{1},\phantom{\rule{4pt}{0ex}}{10}^{2}$, and ${10}^{3}$ (different tone, bright to dark, for every color). The persistence length is $pL=1$ and $L/l={10}^{2}$.

**Figure 6.**Radius of gyration-tensor component ${G}_{xx}$ along the flow direction normalized by the value ${G}_{xx}^{0}$ at zero shear as function of the Weissenberg number $Wi$ for the Péclet numbers $Pe=0,\phantom{\rule{4pt}{0ex}}0.6,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}30,\phantom{\rule{4pt}{0ex}}{10}^{2},3\times {10}^{2},$ and ∞ (bright to dark color); (

**left**) $pL=0.1$ and (

**right**) $pL={10}^{2}$. The number of active sites is $L/l={10}^{2}$ and $\Delta =0.3$.

**Figure 7.**Radius of gyration-tensor component ${G}_{yy}$ along the gradient direction normalized by the value ${G}_{yy}^{0}$ at zero shear as function of the Weissenberg number $Wi$ for the Péclet numbers $Pe=0,\phantom{\rule{4pt}{0ex}}0.6,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}30,\phantom{\rule{4pt}{0ex}}{10}^{2},3\times {10}^{2},$ and ∞ (bright to dark color); (

**left**) $pL=0.1$ and (

**right**) $pL={10}^{2}$. The number of active sites is $L/l={10}^{2}$ and $\Delta =0.3$.

**Figure 8.**Shear-induced polymer alignment, characterized by the angle ${\chi}_{G}$ between the eigenvector of the gyration tensor with the largest eigenvalue and the flow direction, as function of the Weissenberg number $Wi$. The Péclet numbers are $Pe=0,\phantom{\rule{4pt}{0ex}}0.6,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}30,\phantom{\rule{4pt}{0ex}}{10}^{2},3\times {10}^{2},$ and ∞ (bright to dark color); (

**left**) $pL=0.1$ and (

**right**) $pL={10}^{2}$. The number of active sites is $L/l={10}^{2}$ and $\Delta =0.3$.

**Figure 9.**Zero-shear viscosity ${\eta}_{p}^{0}$ normalized by the zero-shear viscosity ${\eta}_{p}^{00}$ of a passive polymer as function of the Péclet number $Pe$ for the polymer stiffness $pL=100,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}1,\phantom{\rule{4pt}{0ex}}0.1$, and $0.01$ (top to bottom at $Pe={10}^{4}$, dark to bright color). The number of active sites is $L/l={10}^{2}$ and $\Delta =0.3$.

**Figure 10.**Shear viscosity ${\eta}_{p}$ normalized by the viscosity ${\eta}_{p}^{0}$ of a non-sheared, active polymer as function of the Weissenberg number $Wi$ for the Péclet numbers $Pe=0,\phantom{\rule{4pt}{0ex}}0.6,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}30,\phantom{\rule{4pt}{0ex}}{10}^{2},3\times {10}^{2},$ and ∞ (bright to dark color); (

**left**) $pL=0.1$ and (

**right**) $pL={10}^{2}$. The number of active sites is $L/l={10}^{2}$ and $\Delta =0.3$.

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## Share and Cite

**MDPI and ACS Style**

Martín-Gómez, A.; Gompper, G.; Winkler, R.G.
Active Brownian Filamentous Polymers under Shear Flow. *Polymers* **2018**, *10*, 837.
https://doi.org/10.3390/polym10080837

**AMA Style**

Martín-Gómez A, Gompper G, Winkler RG.
Active Brownian Filamentous Polymers under Shear Flow. *Polymers*. 2018; 10(8):837.
https://doi.org/10.3390/polym10080837

**Chicago/Turabian Style**

Martín-Gómez, Aitor, Gerhard Gompper, and Roland G. Winkler.
2018. "Active Brownian Filamentous Polymers under Shear Flow" *Polymers* 10, no. 8: 837.
https://doi.org/10.3390/polym10080837