Abstract
We performed 3D visualisation of a Newtonian droplet embedded in an immiscible Newtonian liquid after a strain jump with a home-built counter-rotating shear device. The use of different linear polyurethanes for the droplet liquid allowed us to cover almost three decades of viscosity ratios (K) and to obtain a distinct interface with PDMS matrices with the same interfacial tension for all droplet/matrix pairs. During the droplet retraction, the major axis (L) showed universal time dependence. The apparent Hencky strain of L decayed linearly at large deformations and exponentially at small deformations. After large strain steps, the droplet axis along the vorticity direction (W) deflated and then inflated and the time dependence could be well described by a log normal function. The full width at half maximum was proportional to the droplet relaxation time for all K. The amplitude and the position of the minimum of W were proportional to the affine deformation. The results revealed interesting scaling law behaviour of the droplet retraction after large strain jumps.
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References
Almusallam AS, Larson RG, Solomon MJ (2000) A constitutive model for the prediction of ellipsoidal droplet shapes and stresses in immiscible blends. J Rheol 45:1055–1083
Assighaou S (2006) Comportement sous grandes déformations de gouttelettes de polymères dans une matrice newtonienne. Université du Maine
Assighaou S, Benyahia L (2008) Universal retraction process of a droplet shape after a large strain jump. Phys Rev E: Stat Phys, Plasmas, Fluids 77:036305
Assighaou S, Benyahia L, Du GP-L (2007) Dispositif d’observation de gouttes sous déformation: développement et validation. Rhéologie 11:45–55
Bentley BJ, Leal LG (1986) A computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear shear flows. J Fluid Mech 167:219–240
Chan PCH, Leal LG (1979) Motion of a deformable drop in a 2nd-order fluid. J Fluid Mech 92:131–170
Delaby I, Ernst B, Germain Y, Muller R (1994) Droplet deformation in polymer blends during uniaxial elongational flow: influence of viscosity ratio for large capillary numbers. J Rheol 38:1705–1720
Favelukis M, Lavrenteva OM, Nir A (2005) Deformation and breakup of a non-Newtonian slender drop in an extensional flow. J Non-Newton Fluid Mech 125:49–59
Grace HP (1982) Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems. Chem Eng Commun 14:225–227
Guido S, Villone M (1998) Three-dimensional shape of a drop under simple shear flow. J Rheol 42:395–415
Guido S, Villone M (1999) Measurement of interfacial tension by drop retraction analysis. J Colloid Interface Sci 209:247–250
Hayashi R, Takahashi M, Kajihara T, Yamane H (2001a) Application of large double-step shear to analyse deformation and shape recovery of a polymer droplet in an immiscible polymer matrix. J Rheol 45:627–639
Hayashi R, Takahashi M, Yamane H, Jinnai H, Watanabe H (2001b) Dynamic interfacial properties of polymer blends under large step strains: shape recovery of a single droplet. Polymer 42:757–764
Jansseune T, Vinckier I, Moldenaers P, Mewis J (2001) Transient stresses in immiscible model polymer blends during start-up flows. J Non-Newton Fluid Mech 99:167–181
Leal LG (1980) Particle motions in a viscous fluid. Annu Rev Fluid Mech 12:435–476
Lerdwijijarud W, Larson RG, Sirivat A, Solomon MJ (2003) Influence of weak elasticity of dispersed phase on droplet behavior in sheared polybutadiene/poly(dimethyl siloxane) blends. J Rheol 47:37–58
Luciani A, Champagne MF, Utracki LA (1997) Interfacial tension coefficient from the retraction of ellipsoidal drops. J Polym Sci, Part B: Polym Phys 35:1393–1403
Mighri F, Ajji A, Carreau PJ (1997) Influence of elastic properties on drop deformation in elongational flow F. J Rheol 41:1183–1201
Mighri F, Carreau PJ, Ajji A (1998) Influence of elastic properties on drop deformation and breakup in shear flow. J Rheol 42:1477–1490
Oldroyd JG (1953) The elastic and viscous properties of emulsions and suspensions. Proc R Soc Lond, Ser A 218:122–132
Palierne JF (1990) Linear rheology and viscoelastic emulsions with interfacial tension. Rheol Acta 29:204–214
Prochazka F, Nicolai T, Durand D (2000) Molar mass distribution of linear and branched polyurethane studied by size exclusion chromatography. Macromolecules 33:1703–1709
Rallison JM (1984) The deformation of small viscous drops and bubbles in shear flow. Annu Rev Fluid Mech 16:45–66
Renardy Y, Renardy M, Assighaou S, Benyahia L (2009) Numerical simulation of drop retraction after a strain jump. Phys Rev E: Stat Phys, Plasmas, Fluids 79:4
Sibillo V, Simeone M, Guido S, Greco F, Maffettone PL (2006) Start-up and retraction dynamics of a Newtonian drop in a viscoelastic matrix under simple shear flow. J Non-Newton Fluid Mech 134:27–32
Sigillo I, Santo Ld, Guido S, Grizzuti N (1997) Comparative measurements of interfacial tension in a model polymer blend. Polym Eng Sci 37:1540–1549
Stone HA (1994) Dynamics of drop deformation and breakup in viscous fluids. Annu Rev Fluid Mech 26:95–102
Taylor GI (1932) The viscosity of a fluid containing small drops of another fluid. Proc R Soc Lond, Ser A 138:41–48
Taylor GI (1934) The formation of emulsions in definable fields of flow. Proc R Soc Lond, Ser A 146:201–523
Tjahjadi M, Stone HA, Ottino JM (1992) Satellite and subsatellite formation in capillary breakup. J Fluid Mech 243:297–317
Tjahjadi M, Ottino JM, Stone HA (1994) Estimating interfacial tension via relaxation of drop shapes and filament breakup. AlChE J 40:385
Tretheway DC (2001) Deformation and relaxation of Newtonian drops in planar extensional flows of a Boger fluid. J Non-Newton Fluid Mech 99:81–108
Verdier C (2000) Coalescence of polymer droplets: experiments on collision. C R Acad Sci, Sér IV Phys Astrophys 1:119–126
Verdier C, Brizard M (2002) Understanding droplet coalescence and its use to estimate interfacial tension. Rheol Acta 41:514–523
Xing P, Bousmina M, Rodrigue D, Kamal MR (2000) Critical experimental comparison between five techniques for the determination of interfacial tension in polymer blends: model system of polystyrene/polyamide-6. Macromolecules 33:8020–8034
Yamane H, Takahashi M, Hayashi R, Okamoto K (1998) Observation of deformation and recovery of poly(isobutylene) droplet in a poly(isobutylene)/poly(dimethyl siloxane) blend after application of step shear strain. J Rheol 42:567–580
Acknowledgements
The authors would like to thank T. Nicolai and J-F. Tassin for the fruitful discussions. Special acknowledgments were to Y. and M. Renardy for performing numerical simulations around this work. This research was supported by Le Conseil Régional des Pays de la Loire.
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Assighaou, S., Benyahia, L. Scaling law behaviour of the retraction of a Newtonian droplet after a strain jump in a Newtonian matrix. Rheol Acta 49, 677–686 (2010). https://doi.org/10.1007/s00397-009-0429-x
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DOI: https://doi.org/10.1007/s00397-009-0429-x