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Gravity-induced collisions of spherical drops covered with compressible surfactant

  • ALEXANDER Z. ZINCHENKO (a1), MICHAEL A. ROTHER (a2) and ROBERT H. DAVIS (a1)

Abstract

Gravity-induced collisions of two spherical drops covered with an insoluble surfactant at low Reynolds numbers are considered. Unlike in previous collision studies, the present work accounts for nonlinear coupling between the surfactant distribution and drop hydrodynamics by solving the full unsteady convective–diffusion equation for the surfactant transport. Our method includes high-order three-dimensional multipole expansions for hydrodynamics and a Galerkin-type approach for the surfactant transport with implicit marching. The efficiency of the algorithm allows for calculating thousands of trajectories to very close contact and determining the collision efficiency (related to the critical initial horizontal offset) by trial and error. The solution is valid for arbitrary surface Péclet (Pes) and Marangoni (Ma) numbers and sets limitations on approximations used in prior work for collision-efficiency calculations. Two limiting cases are observed: at small Pes or large Ma, the variation in surfactant coverage is small, and the results for the incompressible surfactant model are recovered, while for large Pes and small Ma, the collision efficiency approaches the clean-interface value. For moderate drop-size ratios (radius ratio k ≤ 0.5), the results generally fall between these limits. At larger size ratios, however, the collision efficiency may even exceed the geometrical Smoluchowski limit for both drops and bubbles. Moreover, with even moderate redistribution of the surfactant, equal-sized drops can move relative to one another and collide. These novel effects do not exist for clean drops or drops covered with an incompressible surfactant, and they are due to the nonlinear coupling between surfactant dynamics and flow. This surfactant-enhanced coalescence takes place, for example, in a physical system of air bubbles in water if the surfactant surface concentration is dilute (Γ ≈ 1×10−9 mol m−2, much smaller than the typical maximum-packing value of 10−5−10−6 mol m−2).

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Agrawal, S. & Wasan, D. 1979 The effect of interfacial viscosities on the motion of drops and bubbles. Chem. Engng J. 18, 215223.
Alves, S. S., Orvalho, S. P. & Vasconcelos, J. M. T. 2005 Effect of bubble contamination on rise velocity and mass transfer Chem. Engng Sci. 60, 19.
Beitel, A. & Heidegger, W. J. 1971 Surfactant effects on mass transfer from drops subject to interfacial instability. Chem. Engng Sci. 26, 711717.
Bławzdziewicz, J., Cristini, V. & Loewenberg, M. 1999 a Near-contact motion of surfactant-covered spherical drops: ionic surfactant. J. Colloid Interface Sci. 211, 355366.
Bławzdziewicz, J., Wajnryb, E. & Loewenberg, M. 1999 b Hydrodynamic interactions and collision efficiencies of spherical drops covered with an incompressible surfactant film. J. Fluid Mech. 395, 2959.
Bławzdziewicz, J., Vlahovska, P. & Loewenberg, M. 2000 Rheology of a dilute emulsion of surfactant-covered spherical drops. Physica A 276, 5085.
Chen, J. & Stebe, K. J. 1996 Marangoni retardation of the terminal velocity of a settling droplet: the role of surfactant physico-chemistry. J. Colloid Interface Sci. 178, 144155.
Chesters, A. K. & Bazhlekov, I. B. 2000 Effect of insoluble surfactants on drainage and rupture of a film between drops interacting under a constant force. J. Colloid Interface Sci. 230, 229243.
Cichocki, B., Felderhof, B. U. & Schmitz, R. 1988 Hydrodynamic interactions between two spherical particles. Physico-Chem. Hydrodyn. 10, 383403.
Cristini, V., Bławzdziewicz, J. & Loewenberg, M. 1998 Near-contact motion of surfactant-covered spherical drops. J. Fluid Mech. 366, 259287.
Cristini, V. & Tan, Y. 2004 Theory and numerical simulation of droplet dynamics in complex flows – a review. Lab on a Chip 4, 257264.
Cuenot, B., Magnaudet, J. & Spennato, B. 1997 The effects of slightly soluble surfactants on the flow around a spherical bubble. J. Fluid Mech. 339, 2553.
Danov, K. D., Valkovska, D. S. & Ivanov, I. B. 1999 Effect of surfactants on the film drainage. J. Colloid Interface Sci. 211, 291303.
Davis, R. H., Schonberg, J. A. & Rallison, J. M. 1989 The lubrication force between two viscous drops. Phys. Fluids 1, 7781.
Edge, R. M. & Grant, C. D. 1972 The motion of drops in water contaminated with a surface active agent. Chem. Engng Sci. 27, 17091721.
Eggleton, C. D., Pawar, Y. & Stebe, K. J. 1999 Insoluble surfactants on a drop in an extensional flow: a generalization of the stagnated surface limit to deforming interfaces. J. Fluid Mech. 385, 7999.
Eggleton, C. D., Tsai, T. M. & Stebe, K. J. 2001 Tip streaming from a drop in the presence of surfactants. Phys. Rev. Lett. 87, 048302.
Elzinga, E. R. & Banchero, J. T. 1961 Some observations on the mechanics of drops in liquid–liquid systems. AIChE J. 7, 394399.
Fdhila, R. & Duineveld, P. C. 1996 The effect of surfactants on the rise of a spherical bubble at high Reynolds and Peclet numbers. Phys. Fluids 8, 310321.
Frumkin, A. & Levich, V. 1947 O vliyanii poverkhnosto-aktivnikh veshestv na dvizhenie na granitse zhidkikh sred. Zhur. Fizic. Khimii 21, 11831204.
Garner, F. H. & Skelland, H. P. 1955 Some factors affecting droplet behavior in liquid–liquid systems. Chem. Engng Sci. 4, 149158.
Griffith, R. 1962 The effect of surfactants on the terminal velocity of drops and bubbles. Chem. Engng Sci. 17, 10571070.
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Nijhoff.
Harper, J. F. 1988 The near stagnation region of a bubble rising steadily in a dilute surfactant solution. Q. J. Mech. Appl. Math. 41, 204213.
Harper, J. F. 2007 Bubble rise in a liquid with a surfactant gas in particular carbon dioxide J. Fluid Mech. 581, 157165.
He, Z., Maldarelli, C. & Dagan, Z. 1991 The size of stagnant caps of bulk soluble surfactant on the interfaces of translating liquid droplets. J. Colloid Interface Sci. 146, 442451.
Hetsroni, S. & Haber, S. 1978 Low Reynolds number motion of two drops submerged in an unbounded arbitrary velocity field. Intl J. Multiphase Flow 4, 117.
Holbrook, J. A. & LeVan, M. D. 1983 a Retardation of droplet motion by surfactant. Part 1. Theoretical development and asymptotic solutions. Chem. Engng Commun. 20, 191207.
Holbrook, J. A. & LeVan, M. D. 1983 b Retardation of droplet motion by surfactant. Part 2. Numerical solutions for exterior diffusion, surface diffusion, and adsorption kinetics. Chem. Engng Commun. 20, 273290.
Horton, T. J., Fritsch, T. R. & Kintner, R. C. 1965 Experimental determination of circulation velocities inside drops. Can. J. Chem. Engng 43, 143146.
Hu, Y. T., Pine, D. J. & Leal, L. G. 2000 Drop deformation, breakup, and coalescence with compatibilizer. Phys. Fluids A 12, 484489.
Hudson, S. D., Jamieson, A. M. & Burkhart, B. E. 2003. The effect of surfactant on the efficiency of shear-induced drop coalescence, J. Colloid Interface Sci. 265 (2), 409421.
Jones, R. B. & Schmitz, R. 1988 Mobility matrix for arbitrary spherical particles in solution. Physica A 149, 373394.
Karsa, D. R. (Ed.) 2000 Surface Active Behaviour of Performance Surfactants. Sheffield Academic Press.
Korn, G. A. & Korn, T. M. 1968 Mathematical Handbook for Scientists and Engineers. McGraw-Hill.
Kushner, J. IV, Rother, M. A. & Davis, R. H. 2001 Buoyancy-driven interactions of viscous drops with deforming interfaces. J. Fluid Mech. 446, 253269.
Levich, A. 1962 Physiochemical Hydrodynamics. Prentice Hall.
Li, D. 1996 Coalescence between small bubbles: effects of surface tension gradient and surface viscosities. J. Colloid Interface Sci. 181, 3444.
Li, X. & Mao, Z. 2001 The effect of surfactant on the motion of a buoyancy-driven drop at intermediate Reynolds numbers: a numerical approach. J. Colloid Interface Sci. 240, 307322.
Li, X. & Pozrikidis, C. 1997 The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow. J. Fluid Mech. 341, 165194.
Manga, M. & Stone, H. A. 1993 Buoyancy-driven interactions between two deformable viscous drops. J. Fluid Mech. 256, 647683.
Manga, M. & Stone, H. A. 1995 Collective hydrodynamics of deformable drops and bubbles in dilute low Reynolds number suspensions. J. Fluid Mech. 300, 231263.
Mo, G. & Sangani, A. S. 1994 A method for computing Stokes flow interactions among spherical objects. Phys. Fluids 6, 1637.
Mousa, H. & van de Ven, T. G. M. 1991 Stability of water-in-oil emulsions in simple shear flow. Part 2. The effects of additives on the orthokinetic coalescence efficiency. Colloids Surf. A 60, 3951.
Nandi, A., Mehra, A. & Khakhar, D. V. 1999 Suppression of coalescence in surfactant stabilized emulsions by shear flow. Phys. Rev. Lett. 83, 24612464.
Nguyen, N. & Werely, S. T. 2002 Fundamentals and Applications of Microfluidics. Artech House.
Park, C. C., Baldessari, F. & Leal, L. G. 2003 Study of molecular weight effects on coalescence: interface slip layer. J. Rheol. 47, 911942.
Pawar, Y. & Stebe, K. J. 1996 Marangoni effects on drop deformation in an extensional flow: the role of surfactant physical chemistry. Part I. Insoluble surfactants. Phys. Fluids 8, 1738.
Porter, M. R. (Ed.) 1990 Recent Developments in the Technology of Surfactants. Elsevier Applied Science.
Pozrikidis, C. 1994 Effects of surface viscosity on the finite deformation of a liquid drop and the rheology of dilute emulsions in simple shearing flow. J. Non-Newtonian. Fluid Mech. 51, 161178.
Pozrikidis, C. 1994 Effects of surface viscosity on the finite deformation of a liquid drop and the rheology of dilute emulsions in simple shearing flow. J. Non-Newtonian. Fluid Mech. 51, 161178.
Rother, M. A. 2009 Effects of incompressible surfactant on thermocapillary interactions of spherical drops. Intl J. Multiphase Flow 35, 417426.
Rother, M. A. & Davis, R. H. 1999 The effects of slight deformation on thermocapillary-driven droplet coalescence and growth. J. Colloid Interface Sci. 214, 297318.
Rother, M. A., Zinchenko, A. Z. & Davis, R. H. 1997 Buoyancy-driven coalescence of slightly deformable drops. J. Fluid Mech. 346, 117148.
Rother, M. A., Zinchenko, A. Z. & Davis, R. H. 2006 Surfactant effects on buoyancy-driven viscous interactions of deformable drops. Colloids Surf. A 282–283, 5060.
Sadhal, S. S. & Johnson, R. E. 1983 Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film - exact solution. J. Fluid Mech. 126, 237250.
Saville, D. 1973 The effect of interfacial tension gradients on droplet behaviour. Chem. Engng J. 5, 251259.
Shen, A. Q., Gleason, B., McKinley, G. H. & Stone, H. A. 2002 Fiber coating with surfactant solutions. Phys. Fluids 14, 40554068.
Stone, H. A. & Leal, L. G. 1990 The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161186.
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics towards a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381411.
Subramanian, R. S. & Balasubramaniam, R. 2001. The Motion of Bubbles and Drops in Reduced Gravity. Cambridge University Press.
Takemura, F. 2005 Adsorption of surfactants onto the surface of a spherical rising bubble and its effect on the terminal velocity of the bubble. Phys. Fluids 17, 048104.
Valkovska, D. S., Danov, K. D. & Ivanov, I. B. 1999 Surfactants role on the deformation of colliding small bubbles. Colloids Surf. A 156, 547566.
Valkovska, D. S., Danov, K. D. & Ivanov, I. B. 2000 Effect of surfactants on the stability of films between two colliding small bubbles. Colloids Surf. A 175, 179192.
Vlahovska, P., Bławzdziewicz, J. & Loewenberg, M. 2002 Nonlinear rheology of a dilute emulsion of surfactant-covered spherical drops in time-dependent flows. J. Fluid Mech. 463, 124.
Wasserman, M. & Slattery, J. 1969 Creeping flow past a fluid globule when a trace of surfactant is present. AIChE J. 15, 533541.
Yamamoto, T. & Ishii, T. 1987 Effect of surface active materials on the drag coefficients and shapes of single large gas bubbles. Chem. Engng Sci. 42, 12971303.
Yiantsios, S. G. & Davis, R. H. 1990 On the buoyancy-driven motion of a drop towards a rigid surface or a deformable interface. J. Fluid Mech. 217, 547573.
Yiantsios, S. G. & Davis, R. H. 1991 Close approach and deformation of two viscous drops due to gravity and van der Waals forces. J. Colloid Interface Sci. 144, 412433.
Yeo, L. Y., Matar, O. K., Perez de Ortiz, E. S. & Hewitt, G. F. 2001 The dynamics of Marangoni-driven local film drainage between two drops. J. Colloid Interface Sci. 241, 233247.
Zhang, X. & Davis, R. H. 1991 The rate of collisions of small drops due to Brownian or gravitational motion. J. Fluid Mech. 230, 479504.
Zhang, Y. & Finch, J. A. 2001 A note on single bubble motion in surfactant solutions. J. Fluid Mech. 429, 6366.
Zinchenko, A. Z. 1978 Calculation of hydrodynamic interaction between drops at low Reynolds numbers. J. Appl. Math. Mech. 42, 10461051.
Zinchenko, A. Z. 1982 Calculations of the effectiveness of gravitational coagulation of drops with allowance for internal circulation. J. Appl. Math. Mech. 46, 5865.
Zinchenko, A. Z. 1994 An efficient algorithm for calculating multiparticle thermal interaction in a concentrated dispersion of spheres. J. Comput. Phys. 111, 120135.
Zinchenko, A. Z. & Davis, R. H. 2000 An efficient algorithm for hydrodynamical interaction of many deformable drops. J. Comput. Phys. 157, 539587.
Zinchenko, A. Z. & Davis, R. H. 2005 A multipole-accelerated algorithm for close interaction of slightly deformable drops. J. Comput. Phys. 207, 695735.
Zinchenko, A. Z. & Davis, R. H. 2008 Algorithm for direct numerical simulation of emulsion flow through a granular material. J. Comput. Phys. 227, 78417888.
Zinchenko, A. Z., Rother, M. A. & Davis, R. H. 1999 Cusping, capture, and breakup of interacting drops by a curvatureless boundary-integral algorithm. J. Fluid Mech. 391, 249292.
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