A linear stability analysis is carried out for viscoelastic filaments (formed by an unentangled polymer solution) during capillary thinning in the regime of unfolded polymer coils taking into account the relative motion of the solvent and the polymer. The conditions for the onset of filament instability with respect to axisymmetric modulation of its surface are found. The analysis is valid for relatively fast processes occurring at times shorter than the characteristic thinning time. It is shown that the growth rate of such pearling instability is determined by the osmotic modulus of the solution and the degree of orientation of macromolecules. In the case of nonassociative polymers, the instability develops (with the growth rate exceeding the rate of filament thinning) when the longitudinal length of stretched polymer chains exceeds the diameter of the filament. The theory is also applicable to soft gels and associative polymer solutions with very long relaxation times. The predictions of the theory are in agreement with experimental data.
REFERENCES
1.
Goldin
, M.
, J.
Yerushalmi
, R.
Pfeffer
, and R.
Shinnar
, “Breakup of laminar capillary jet of a viscoelastic fluid
,” J. Fluid Mech.
38
, 689
–711
(1969
). 2.
Bazilevskii
, A. V.
, S. I.
Voronkov
, V. M.
Entov
, and A. N.
Rozhkov
, “Orientation effects in the breakup of jets and threads of dilute polymer solutions
,” Sov. Phys. Dokl
26
, 333
–336
(1981
).3.
Eggers
, J.
, and E.
Villermaux
, “Physics of liquid jets
,” Rep. Prog. Phys.
71
, 036601
(2008
). 4.
Li
, J.
, and M. A.
Fontelos
, “Drop dynamics on the beads-on-string structure for viscoelastic jets: A numerical study
,” Phys. Fluids
15
, 922
–937
(2003
). 5.
McKinley
, G. H.
, Visco-elasto-capillary thinning and break-up of complex fluids
, in Rheologycal Review
(The British Society of Rheology
, Aberystwyth
, 2005
), pp. 1
–48
.6.
McKinley
, G. H.
, and T.
Sridhar
, “Filament-stretching rheometry of complex fluids
,” Annu. Rev. Fluid Mech.
34
, 375
–415
(2002
). 7.
Clasen
, C.
, J.
Eggers
, M. A.
Fontelos
, J.
Li
, and G. H.
McKinley
, “The beads-on-string structure of viscoelastic threads
,” J. Fluid Mech.
556
, 283
–308
(2006
). 8.
Bhat
, P. P.
, S.
Appathurai
, M. T.
Harris
, M.
Pasquali
, G. H.
McKinley
, and O. A.
Basaran
, “Formation of beads-on-a-string structures during break-up of viscoelastic filaments
,” Nat. Phys.
6
, 625
–631
(2010
). 9.
Sattler
, R.
, C.
Wagner
, and J.
Eggers
, “Blistering pattern and formation of nanofibers in capillary thinning of polymer solutions
,” Phys. Rev. Lett.
100
, 164502
(2008
). 10.
Malkin
, A. Y.
, A. V.
Semakov
, I.
Yu. Skvortsov
, P.
Zatonskikh
, V. G.
Kulichikhin
, A. V.
Subbotin
, and A. N.
Semenov
, “Spinnability of dilute polymer solutions
,” Macromolecules
50
, 8231
–8244
(2017
). 11.
Bazilevsky
, A. V.
, V. M.
Entov
, and A. N.
Rozhkov
, Liquid filament microrheometer and some of its applications
, in Third European Rheology Conference and Golden Jubilee Meeting of the British Society of Rheology
, edited by D. R.
Oliver
(Springer Netherlands
, Dordrecht
, 1990
), pp. 41
–43
.12.
McKinley
, G. H.
, and A.
Tripathi
, “How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer
,” J. Rheol.
44
, 653
–670
(2000
). 13.
Anna
, S. L.
, and G. H.
McKinley
, “Elasto-capillary thinning and breakup of model elastic liquids
,” J. Rheol.
45
, 115
–138
(2001
). 14.
Stelter
, M.
, G.
Brenn
, A. L.
Yarin
, R. P.
Singh
, and F.
Durst
, “Validation and application of a novel elongational device for polymer solutions
,” J. Rheol.
44
, 595
–616
(2000
). 15.
Stelter
, M.
, G.
Brenn
, A. L.
Yarin
, R. P.
Singh
, and F.
Durst
, “Investigation of the elongational behavior of polymer solutions by means of an elongational rheometer
,” J. Rheol.
46
, 507
–527
(2002
). 16.
Bazilevskii
, A.
, V.
Entov
, and A.
Rozhkov
, “Breakup of an Oldroyd liquid bridge as a method for testing the rheological properties of polymer solutions
,” Polym. Sci. Ser. AC/C
43
, 716
–726
(2001
).17.
Bazilevskii
, A. B.
, and A. N.
Rozhkov
, “Dynamics of capillary breakup of elastic jets
,” Fluid Dyn.
49
, 827
–843
(2014
). 18.
Zell
, A.
, S.
Gier
, S.
Rafaï
, and C.
Wagner
, “Is there a relation between the relaxation time measured in CaBER experiments and the first normal stress coefficient?
,” J. Non-Newtonian Fluid Mech.
165
, 1265
–1274
(2010
). 19.
Dinic
, J.
, Y.
Zhang
, L. N.
Jimenez
, and V.
Sharma
, “Extensional relaxation time of dilute, aqueous, polymer solutions
,” ACS Macro Lett.
4
, 804
–808
(2015
). 20.
Dinic
, J.
, L. N.
Jimenez
, and V.
Sharma
, “Pinch-off dynamics and dripping-onto-substrate (DoS) rheometry of complex fluids
,” Lab Chip
17
, 460
–473
(2017
). 21.
Dinic
, J.
, and V.
Sharma
, “Macromolecular relaxation, strain, and extensibility determine elastocapillary thinning and extensional viscosity of polymer solutions
,” Proc. Natl. Acad. Sci. U.S.A.
116
, 8766
–8774
(2019
). 22.
Keshavarz
, B.
, V.
Sharma
, E. C.
Houze
, M. R.
Koerner
, J. R.
Moore
, P. M.
Cotts
, P.
Threlfall-Holmes
, and G. H.
McKinley
, “Studying the effects of elongational properties on atomization of weakly viscoelastic solutions using Rayleigh Ohnesorge jetting extensional rheometry (ROJER)
,” J. Non-Newtonian Fluid Mech.
222
, 171
–189
(2015
). 23.
Sharma
, V.
, S. J.
Haward
, J.
Serdy
, B.
Keshavarz
, A.
Soderlund
, P.
Threlfall-Holmes
, and G. H.
McKinley
, “The rheology of aqueous solutions of ethyl hydroxy-ethyl cellulose (EHEC) and its hydrophobically modified analogue (hmEHEC): extensional flow response in capillary break-up, jetting (ROJER) and in a cross-slot extensional rheometer
,” Soft Matter
11
, 3251
–3270
(2015
). 24.
Christanti
, Y.
, and L. M.
Walker
, “Surface tension driven jet break up of strain-hardening polymer solutions
,” J. Non-Newton. Fluid Mech.
100
, 9
–26
(2001
). 25.
Oliveira
, M. S. N.
, and G. H.
McKinley
, “Iterated stretching and multiple beads-on-a-string phenomena in dilute solutions of highly extensible flexible polymers
,” Phys. Fluids
17
, 071704
(2005
). 26.
Oliveira
, M. S. N.
, R.
Yeh
, and G. H.
McKinley
, “Iterated stretching, extensional rheology and formation of beads-on-a-string structures in polymer solution
,” J. Non-Newton. Fluid Mech.
137
, 137
–148
(2006
). 27.
Arnolds
, O.
, H.
Buggisch
, D.
Sachsenheimer
, and N.
Willenbacher
, “Capillary breakup extensional rheometry (CaBER) on semi-dilute and concentrated polyethyleneoxide (PEO) solutions
,” Rheol. Acta
49
, 1207
–1217
(2010
). 28.
Sattler
, R.
, S.
Gier
, J.
Eggers
, and C.
Wagner
, “The final stages of capillary break-up of polymer solutions
,” Phys. Fluids
24
, 023101
(2012
). 29.
Bazilevskii
, A. V.
, and A. N.
Rozhkov
, “Dynamics of the capillary breakup of a bridge in an elastic fluid
,” Fluid Dyn.
50
, 800
–811
(2015
). 30.
Deblais
, A.
, K. P.
Velikov
, and D.
Bonn
, “Pearling instabilities of a viscoelastic thread
,” Phys. Rev. Lett.
120
, 194501
(2018
). 31.
Kibbelaar
, H. V. M.
, A.
Deblais
, F.
Burla
, G. H.
Koenderink
, K. P.
Velikov
, and D.
Bonn
, “Capillary thinning of elastic and viscoelastic threads: From elastocapillarity to phase separation
,” Phys. Rev. Fluids
5
, 092001(R)
(2020
). 32.
Semakov
, A. V.
, V. G.
Kulichikhin
, A. K.
Tereshin
, S. V.
Antonov
, and A. Y.
Malkin
, “On the nature of phase separation of polymer solutions at high extension rates
,” J. Polym. Sci. Part B: Polym. Phys.
53
, 559
–565
(2015
). 33.
Kulichikhin
, V.
, I. Y.
Skvortsov
, A. V.
Subbotin
, S. V.
Kotomin
, and A.
Ya. Malkin
, “A novel technique for fiber formation: Mechanotropic spinning—Principle and realization
,” Polymers
10
, 1
–19 (2018
).34.
Tirtaatmadja
, V.
, G. H.
McKinley
, and J. J.
Cooper-White
, “Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration
,” Phys. Fluids
18
, 043101
(2006
). 35.
Clasen
, C.
, J. P.
Plog
, W.-M.
Kulicke
, M.
Owens
, C.
Macosko
, L. E.
Scriven
, M.
Verani
, and G. H.
McKinley
, “How dilute are dilute solutions in extensional flows?
,” J. Rheol.
50
, 849
–881
(2006
). 36.
Sur
, S.
, and J.
Rothstein
, “Drop breakup dynamics of dilute polymer solutions: Effect of molecular weight, concentration, and viscosity
,” J. Rheol.
62
, 1245
–1259
(2018
). 37.
Middleman
, S.
, “Stability of a viscoelastic jet
,” Chem. Eng. Sci.
20
, 1037
–1040
(1965
). 38.
Gordon
, S. L.
, and M.
Gottlieb
, “Surface-tension-driven breakup of viscoelastic liquid threads
,” J. Fluid Mech.
120
, 245
–266
(1982
). 39.
Yarin
, A. L.
, Free Liquid Jets and Films: Hydrodynamics and Rheology
(Wiley
, New York
, 1993
).40.
Entov
, V. M.
, and E. J.
Hinch
, “Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid
,” J. Non-Newton. Fluid Mech.
72
, 31
–53
(1997
). 41.
Mora
, S.
, T.
Phou
, J. M.
Fromental
, L. M.
Pismen
, and Y.
Pomeau
, “Capillarity driven instability of a soft solid
,” Phys. Rev. Lett.
105
, 214301
(2010
). 42.
Snoeijer
, J. H.
, A.
Pandey
, M. A.
Herrada
, and J.
Eggers
, “The relationship between viscoelasticity and elasticity
,” Proc. R. Soc. A Math. Phys. Eng. Sci.
476
, 20200419
(2020
). 43.
Pandey
, A.
, M.
Kansal
, M. A.
Herrada
, J.
Eggers
, and J. H.
Snoeijer
, “Elastic Rayleigh–plateau instability: Dynamical selection of nonlinear states
,” Soft Matter
17
, 5148
–5161
(2021
). 44.
Amarouchene
, Y.
, D.
Bonn
, J.
Meunier
, and H.
Kellay
, “Inhibition of the finite-time singularity during droplet fission of a polymeric fluid
,” Phys. Rev. Lett.
86
, 3558
–3561
(2001
).45.
Eggers
, J.
, M. A.
Herrada
, and J. H.
Snoeijer
, “Self-similar breakup of polymeric threads as described by the Oldroyd-B model
,” J. Fluid Mech.
887
, A19
–A31 (2020
). 46.
Deblais
, A.
, M. A.
Herrada
, J.
Eggers
, and D.
Bonn
, “Self-similarity in the breakup of very dilute viscoelastic solutions
,” J. Fluid Mech.
904
, R2
(2020
). 47.
Semenov
, A.
, and I.
Nyrkova
, “Capillary thinning of viscoelastic threads of unentangled polymer solutions
,” Polymers
14
, 1
–22 (2022
). 48.
Subbotin
, A. V.
, and A. N.
Semenov
, “Dynamics of dilute polymer solutions at the final stages of capillary thinning
,” Macromolecules
55
, 2096
–2108
(2022
). 49.
Chang
, H.-C.
, E. A.
Demekhin
, and E.
Kalaidin
, “Iterated stretching of viscoelastic jets
,” Phys. Fluids
11
, 1717
–1737
(1999
). 50.
Turkoz
, E.
, J. M.
Lopez-Herrera
, J.
Eggers
, C. B.
Arnold
, and L.
Deike
, “Axisymmetric simulation of viscoelastic filament thinning with the Oldroyd-B model
,” J. Fluid Mech.
851
, R2
–R13 (2018
). 51.
Eggers
, J.
, “Instability of a polymeric thread
,” Phys. Fluids
26
, 033106
(2014
). 52.
Subbotin
, A. V.
, and A. N.
Semenov
, “Phase separation in dilute polymer solutions at high-rate extension
,” J. Polym. Sci. Part B Polym. Phys.
54
, 1066
–1073
(2016
). 53.
Semenov
, A. N.
, and A. V.
Subbotin
, “Phase separation kinetics in unentangled polymer solutions under high-rate extension
,” J. Polym. Sci. Part B Polym. Phys.
55
, 623
–637
(2017
). 54.
Donets
, S.
, and J.-U.
Sommer
, “Molecular dynamics simulations of strain-induced phase transition of poly(ethylene oxide) in water
,” J. Phys. Chem. B
122
, 392
–397
(2018
). 55.
Donets
, S.
, O.
Guskova
, and J.-U.
Sommer
, “Flow-induced formation of thin PEO fibres in water and their stability after the strain release
,” J. Phys. Chem. B
124
, 9224
–9229
(2020
). 56.
Subbotin
, A. V.
, and A. N.
Semenov
, “Multiple droplets formation in ultrathin bridges of rigid rod dispersions
,” J. Rheol.
64
, 13
–27
(2020
). 57.
Subbotin
, A. V.
, and A. N.
Semenov
, “Dynamics of annular solvent droplets under capillary thinning of non-entangled polymer solutions
,” J. Rheol.
67
, 53
–65
(2023
). 58.
Bazilevskii
, A. V.
, V. M.
Entov
, M. M.
Lerner
, and A. N.
Rozhkov
, “Failure of polymer solution filaments
,” Polym. Sci. Ser. A
39
, 316
–324
(1997
).59.
Semenov
, A. N.
, and A. R.
Khokhlov
, “Statistical physics of liquid-crystalline polymers
,” Sov. Phys. Usp.
31
, 988
–1014
(1988
). 60.
Lifshitz
, I. M.
, A. Y.
Grosberg
, and A. R.
Khokhlov
, “Structure of a polymer globule formed by saturating bonds
,” Sov. Phys. JETP
44
, 855
–860
(1976
).61.
Rubinstein
, M.
, and A. N.
Semenov
, “Thermoreversible gelation in solutions of associating polymers. 1. Statics
,” Macromolecules
31
, 1373
–1385
(1998
).62.
Doi
, M.
, and S. F.
Edwards
, The Theory of Polymer Dynamics
(Oxford University
, New York
, 1986
).63.
Brochard
, F.
, and P.-G.
de Gennes
, “Dynamical scaling for polymers in theta solvents
,” Macromolecules
10
, 1157
–1161
(1977
). 64.
Helfand
, E.
, and G. H.
Fredrickson
, “Large fluctuations in polymer solutions under shear
,” Phys. Rev. Lett.
62
, 2468
–2471
(1989
). 65.
Milner
, S. T.
, “Dynamical theory of concentration fluctuations in polymer solutions under shear
,” Phys. Rev. E
48
, 3674
–3691
(1993
). 66.
Doi
, M.
, Dynamics and patterns in complex fluids: New aspects of the physics-chemistry interface
,” in Springer Proceedings in Physics
, edited by A.
Onuki
and K.
Kawasaki
(Springer-Verlag
, Berlin
, 1990
), Vol. 52.67.
Doi
, M.
, and A.
Onuki
, “Dynamic coupling between stress and composition in polymer solutions and blends
,” J. Phys. II France
2
, 1631
–1656
(1992
).68.
Landau
, L. D.
, and E. M.
Lifshitz
, Statistical Physics, Part 1
(Pergamon
, New York
, 1980
).69.
Bird
, R. B.
, R. C.
Armstrong
, and O.
Hassager
, Dynamics of Polymeric Fluids
(Wiley
, New York
, 1987
).70.
Peterlin
, A.
, “Hydrodynamics of macromolecules in a velocity field with longitudinal gradient
,” J. Polym. Sci. Part B Polym. Lett.
4
, 287
–291
(1966
). 71.
Entov
, V. M.
, Elastic effects in flows of dilute polymer solutions
, in Progress and Trends in Rheology II
, edited by H.
Giesekus
and M. F.
Hibberd
(Springer
, Berlin
, 1988
), pp. 260
–261
.72.
Bohdanecký
, M.
, V.
Petrus
, and B.
Sedláček
, “Estimation of the characteristic ratio of polyacrylamide in water and in a mixed theta-solvent
,” Die Makromol. Chem.
184
, 2061
–2073
(1983
). 73.
Laurent
, T. C.
, and J.
Gergely
, “Light scattering studies on hyaluronic acid
,” J. Biol. Chem.
212
, 325
–333
(1955
). 74.
Buhler
, E.
, and F.
Boué
, “Chain persistence length and structure in Hyaluronan solutions: Ionic strength dependence for a model semirigid polyelectrolyte
,” Macromolecules
37
, 1600
–1610
(2004
). 75.
De Gennes
, P.-G.
, Scaling Concepts in Polymer Physics
(Cornell University
, Ithaca, NY
, 1979
).76.
77.
Gatej
, I.
, M.
Popa
, and M.
Rinaudo
, “Role of the pH on Hyaluronan behavior in aqueous solution
,” Biomacromolecules
6
, 61
–67
(2005
). 78.
Rubinstein
, M.
, and A. N.
Semenov
, “Thermoreversible gelation in solutions of associating polymers: 2. Linear dynamics
,” Macromolecules
31
, 1386
–1397
(1998
). © 2023 The Society of Rheology.
2023
The Society of Rheology
You do not currently have access to this content.