# Capillary Thinning of Viscoelastic Threads of Unentangled Polymer Solutions

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## Abstract

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## 1. Introduction

## 2. Capillary Thinning for the Oldroyd-B Model

## 3. Finite Extensibility Effects

#### 3.1. FENE-P Model

#### 3.2. Equations of Motion in the Entrance Zone

#### 3.3. Flow in the Thread and the Thinning Law

## 4. Discussion

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Entov, V.M. Elastic effects in flows of dilute polymer solutions. In Progress and Trends in Rheology II; Giesekus, H., Hibberd, M.F., Eds.; Springer: Berlin, Germany, 1988; pp. 260–261. [Google Scholar]
- Sattler, R.; Wagner, C.; Eggers, J. Blistering Pattern and Formation of Nanofibers in Capillary Thinning of Polymer Solutions. Phys. Rev. Lett.
**2008**, 100, 164502. [Google Scholar] [CrossRef] - Bazilevsky, A.V.; Rozhkov, A.N. Dynamics of the Capillary Breakup of a Bridge in an Elastic Fluid. Fluid Dyn.
**2015**, 50, 800–811. [Google Scholar] [CrossRef] - Malkin, A.Y.; Semakov, A.V.; Skvortsov, I.Y.; Zatonskikh, P.; Kulichikhin, V.G.; Subbotin, A.V.; Semenov, A.N. Spinnability of dilute polymer solutions. Macromolecules
**2017**, 50, 8231–8244. [Google Scholar] [CrossRef] - McKinley, G.H.; Tripathi, A. How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer. J. Rheol.
**2000**, 44, 653–670. [Google Scholar] [CrossRef][Green Version] - Anna, S.L.; McKinley, G.H. Elasto-capillary thinning and breakup of model elastic liquids. J. Rheol.
**2001**, 45, 115–138. [Google Scholar] [CrossRef][Green Version] - Clasen, C.; Eggers, J.; Fontelos, M.A.; Li, J.; McKinley, G.H. The beads-on-string structure of viscoelastic threads. J. Fluid Mech.
**2006**, 556, 283–308. [Google Scholar] [CrossRef][Green Version] - Entov, V.M.; Hinch, E.J. Effect of a Spectrum Relaxation Times on the Capillary Thinning of a Filament Elastic Liquids. J. Non-Newton. Fluid Mech.
**1997**, 72, 31–53. [Google Scholar] [CrossRef] - Renardy, M. A numerical study of the asymptotic evolution and breakup of Newtonian and viscoelastic jets. J. Non-Newton. Fluid Mech.
**1995**, 59, 267–282. [Google Scholar] [CrossRef] - Eggers, J.; Fontelos, M.A. Singularities: Formation, Structure, and Propagation; Cambridge University Press: Cambridge, UK, 2015. [Google Scholar]
- Zhou, J.; Doi, M. Dynamics of viscoelastic filaments based on Onsager principle. Phys. Rev. Fluids
**2018**, 3, 084004. [Google Scholar] [CrossRef][Green Version] - Eggers, J.; Herrada, M.A.; Snoeijer, J.H. Self-similar breakup of polymeric threads as described by the Oldroyd-B model. J. Fluid Mech.
**2020**, 887, A19. [Google Scholar] [CrossRef] - Snoeijer, J.H.; Pandey, A.; Herrada, M.A.; Eggers, J. The relationship between viscoelasticity and elasticity. Proc. R. Soc. A
**2020**, 476, 20200419. [Google Scholar] [CrossRef] [PubMed] - Bird, R.B.; Armstrong, R.C.; Hassager, O. Dynamics of Polymeric Fluids; Wiley: New York, NY, USA, 1987. [Google Scholar]
- Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford University Press: New York, NY, USA, 1986. [Google Scholar]
- Khokhlov, A.R. Concept of quasimonomers and its application to some problems of polymer statistics. Polymer
**1978**, 19, 1387–1396. [Google Scholar] [CrossRef] - Schaefer, D.; Joanny, J.-F.; Pincus, P. Dynamics of semiflexible polymers in solution. Macromolecules
**1980**, 13, 1280–1289. [Google Scholar] [CrossRef] - Stelter, M.; Brenn, G.; Yarin, A.L.; Singh, R.P.; Durst, F. Validation and application of a novel elongational device for polymer solutions. J. Rheol.
**2000**, 44, 595–616. [Google Scholar] [CrossRef] - Warner, H.R., Jr. Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind. Eng. Chem. Fund.
**1972**, 11, 379–387. [Google Scholar] [CrossRef] - Bird, R.B.; Dotson, P.J.; Johnson, N.L. Polymer solution rheology based on a finitely extensible bead-spring chain model. J. Non-Newton. Fluid Mech.
**1980**, 7, 213–235. [Google Scholar] [CrossRef] - Purnode, B.; Crochet, M.J. Polymer solution characterization with the FENE-P model. J. Non-Newton. Fluid Mech.
**1998**, 77, 1–20. [Google Scholar] [CrossRef] - Entov, V.M.; Yarin, A.L. Influence of elastic stresses on the capillary breakup of jets of dilute polymer solutions. Fluid Dyn.
**1984**, 19, 21–29. [Google Scholar] [CrossRef] - Turkoz, E.; Lopez-Herrera, J.M.; Eggers, J.; Arnold, C.B.; Deike, L. Axisymmetric simulation of viscoelastic filament thinning with the Oldroyd-B model. J. Fluid Mech.
**2018**, 851, R2. [Google Scholar] [CrossRef][Green Version] - Deblais, A.; Herrada, M.A.; Eggers, J.; Bonn, D. Self-similarity in the breakup of very dilute viscoelastic solutions. J. Fluid Mech.
**2020**, 904, R2. [Google Scholar] [CrossRef] - Kibbelaar, H.V.M.; Deblais, A.; Burla, F.; Koenderink, G.H.; Velikov, K.P.; Bonn, D. Capillary thinning of elastic and viscoelastic threads: From elastocapillarity to phase separation. Phys. Rev. Fluids
**2020**, 5, 092001. [Google Scholar] [CrossRef] - Dunlap, P.N.; Leal, L.G. Dilute polystyrene solutions in extensional flows: Birefringence and flow modification. J. Non-Newton. Fluid Mech.
**1987**, 23, 5–48. [Google Scholar] [CrossRef] - Subbotin, A.V.; Semenov, A.N. Capillary-induced Phase Separation in Ultrathin Jets of Rigid-chain Polymer Solutions. JETP Lett.
**2020**, 111, 55–61. [Google Scholar] [CrossRef] - Subbotin, A.V.; Semenov, A.N. Dynamics of Dilute Polymer Solutions at the Final Stages of Capillary Thinning. Macromolecules
**2022**, 55, 2096–2108. [Google Scholar] [CrossRef] - Semenov, A.N.; Subbotin, A.V. Phase Separation Kinetics in Unentangled Polymer Solutions Under High-Rate Extension. J. Polym. Sci. Part B Phys. Ed.
**2017**, 55, 623–637. [Google Scholar] [CrossRef] - Eggers, J.; Villermaux, E. Physics of liquid jets. Rep. Prog. Phys.
**2008**, 71, 036601. [Google Scholar] [CrossRef][Green Version] - Deblais, A.; Velikov, K.P.; Bonn, D. Pearling Instabilities of a Viscoelastic Thread. Phys. Rev. Lett.
**2018**, 120, 194501. [Google Scholar] [CrossRef] [PubMed][Green Version] - Eggers, J. Instability of a polymeric thread. Phys. Fluids
**2014**, 26, 033106. [Google Scholar] [CrossRef][Green Version] - Semenov, A.N.; Subbotin, A.V. Phase Separation in Dilute Polymer Solutions at High-Rate Extension. J. Polym. Sci. Part B Phys. Ed.
**2016**, 54, 1066–1073. [Google Scholar] - Semenov, A.N.; Subbotin, A.V. Multiple droplets formation in ultrathin bridges of rigid rod dispersions. J. Rheol.
**2020**, 64, 13–27. [Google Scholar] - De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, USA, 1979. [Google Scholar]
- Dinic, J.; Sharma, V. Macromolecular relaxation, strain, and extensibility determine elastocapillary thinning and extensional viscosity of polymer solutions. Proc. Natl. Acad. Sci. USA
**2019**, 116, 8766–8774. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhou, J.; Doi, M. Dynamics of a viscoelastic liquid filament connected to two mobile droplets. Phys. Fluids
**2020**, 32, 043101. [Google Scholar] - Rubinstein, M.; Colby, R. Polymer Physics; Oxford University Press: New York, NY, USA, 2003. [Google Scholar]
- Tirtaatmadja, V.; McKinley, G.H.; Cooper-White, J.J. Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration. Phys. Fluids
**2006**, 18, 043101. [Google Scholar] [CrossRef] - Helfand, E.; Fredrickson, G.H. Large fluctuations in polymer solutions under shear. Phys. Rev. Lett.
**1989**, 62, 2468–2471. [Google Scholar] [CrossRef] [PubMed] - Doi, M.; Onuki, A. Dynamic coupling between stress and composition in polymer solutions and blends. J. Phys. II
**1992**, 2, 1631–1656. [Google Scholar] [CrossRef] - Milner, S.T. Dynamical theory of concentration fluctuations in polymer solutions under shear. Phys. Rev. E
**1993**, 48, 3674–3691. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**A thinning filament of radius ${a}_{0}$ and length ${L}_{f}$ connecting two semi-spherical droplets (a cross-section along the main axis is shown here); ${z}_{m}$ corresponds to the middle of the thread. ${L}_{t}\ll {L}_{f}$ is the size of the filament/droplet transition regions (shown in cyan).

**Figure 2.**The thread/droplet transition region; $r=a\left(z\right)$ defines the free surface in cylindrical coordinates $(z,r,\theta )$; ${a}_{0}$ and ${v}_{0}$ are the thread radius and the flow velocity at the end of the uniform thread region. Note that a longitudinal cross-section including the cylindrical symmetry axis $(z$) is shown here. Thus, the upper curve corresponds to the cylindrical angle $\theta =0$, the lower curve to $\theta ={180}^{\circ}$, and the r-axis coincides with the Cartesian ${x}_{1}$-axis. The inset clarifies the cylindrical coordinates used here.

**Figure 4.**Coordinates u and ${u}_{\perp}$ attached to point O of a generic streamline shown as thick curve. Note that some actual streamlines are depicted in Figure 3.

**Figure 7.**Time-dependence of the thread radius: ${a}_{0}$ (upper solid curve) and $ln{a}_{0}$ (lower curve) vs. $t/\tau $. The dashed line indicates the asymptotic exponential law, Equation (12).

**Figure 8.**The dependence $a\left(z\right)/{l}_{e}$ vs. $z/{l}_{e}$ in the transition zone: our data (crosses), the similarity solution for the bead-string structure obtained using a neo-Hookean elastic model [12] (solid line); ${l}_{e}={2}^{1/3}{a}_{0}$ is the elasto-capillary length.

**Figure 9.**The time dependence of the thread thickness $d=2{a}_{0}$. Red solid curves: theoretical results (cf. Section 3.3). Thick black curves: experimental data for semidilute aqueous solutions of Praestol-2540 with different concentrations from 125 to 1000 ppm (cf. Figure 5 of ref. [18]). Red circles indicate the onset of significant deviations between theoretical and experimental curves. The theoretical breakup points are indicated with red arrows for each red curve. For the two highest concentrations, we also indicate the reduced time $t/\tau $ (with $t=0$ corresponding to the theoretical breakup). Thin red dashed lines indicate the exponential asymptotic behavior at short times (according to Equation (12)).

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**MDPI and ACS Style**

Semenov, A.; Nyrkova, I.
Capillary Thinning of Viscoelastic Threads of Unentangled Polymer Solutions. *Polymers* **2022**, *14*, 4420.
https://doi.org/10.3390/polym14204420

**AMA Style**

Semenov A, Nyrkova I.
Capillary Thinning of Viscoelastic Threads of Unentangled Polymer Solutions. *Polymers*. 2022; 14(20):4420.
https://doi.org/10.3390/polym14204420

**Chicago/Turabian Style**

Semenov, Alexander, and Irina Nyrkova.
2022. "Capillary Thinning of Viscoelastic Threads of Unentangled Polymer Solutions" *Polymers* 14, no. 20: 4420.
https://doi.org/10.3390/polym14204420