Diffuse interface modelling of soluble surfactants in two-phase flow

Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are derived from balance equations and an energy inequality so that thermodynamic consistency is guaranteed. Via a formal asymptotic analysis, they are related to sharp interface models. Both cases of dynamic as well as instantaneous adsorption are covered. Flexibility with respect to the choice of bulk and surface free energies allows to realise various isotherms and relations of state between surface tension and surfactant. Some numerical simulations display the effectiveness of the presented approach.

[1]  G. Homsy,et al.  On physical mechanisms in chemical reaction-driven tip-streaming , 2004 .

[2]  Peter W. Bates,et al.  Convergence of the Cahn-Hilliard equation to the Hele-Shaw model , 1994 .

[3]  Paul C. Fife,et al.  Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter , 1995 .

[4]  Krassimir D. Danov,et al.  Chemical Physics of Colloid Systems and Interfaces , 2015 .

[5]  Zhilin Li,et al.  A level-set method for interfacial flows with surfactant , 2006, J. Comput. Phys..

[6]  G. Homsy,et al.  On a new surfactant-driven fingering phenomenon in a Hele-Shaw cell , 2004, Journal of Fluid Mechanics.

[7]  Minh Do-Quang,et al.  On modeling and simulation of surfactants in diffuse interface flow , 2011 .

[8]  Krassimir D. Danov,et al.  Thermodynamics of Ionic Surfactant Adsorption with Account for the Counterion Binding: Effect of Salts of Various Valency , 1999 .

[9]  G. Ariel,et al.  Kinetics of surfactant adsorption: the free energy approach , 2001 .

[10]  P. Hohenberg,et al.  Theory of Dynamic Critical Phenomena , 1977 .

[11]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  J. Eastoe,et al.  Dynamic surface tension and adsorption mechanisms of surfactants at the air-water interface. , 2000, Advances in colloid and interface science.

[13]  J. Lowengrub,et al.  A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant , 2004 .

[14]  David S. Rumschitzki,et al.  On the surfactant mass balance at a deforming fluid interface , 1996 .

[15]  Xiangrong Li,et al.  SOLVING PDES IN COMPLEX GEOMETRIES: A DIFFUSE DOMAIN APPROACH. , 2009, Communications in mathematical sciences.

[16]  G. Homsy,et al.  Chemical reaction-driven tip-streaming phenomena in a pendant drop , 2004 .

[17]  David Kay,et al.  Finite element approximation of a Cahn−Hilliard−Navier−Stokes system , 2008 .

[18]  F. Coutelieris,et al.  Low Peclet mass transport in assemblages of spherical particles for two different adsorption mechanisms. , 2003, Journal of colloid and interface science.

[19]  D. M. Anderson,et al.  DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS , 1997 .

[20]  F. Coutelieris.  The effect of geometry and axial orientation of spheroidal particles on the adsorption rate in a granular porous medium , 2002 .

[21]  Junseok Kim,et al.  A continuous surface tension force formulation for diffuse-interface models , 2005 .

[22]  John W. Barrett,et al.  Finite Element Approximation of a Phase Field Model for Void Electromigration , 2004, SIAM J. Numer. Anal..

[23]  H. Abels,et al.  Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities , 2011, 1104.1336.

[24]  Morton E. Gurtin,et al.  Continuum theory of thermally induced phase transitions based on an order parameter , 1993 .

[25]  V. Zhdanov.  Adsorption–desorption kinetics and chemical potential of adsorbed and gas-phase particles , 2001 .

[26]  Jie Shen,et al.  Variational Approach in Two-Phase Flows of Complex Fluids: Transport and Induced Elastic Stress , 2004 .

[27]  H. Elman,et al.  Efficient preconditioning of the linearized NavierStokes equations for incompressible flow , 2001 .

[28]  Dieter Bothe,et al.  Stability of Equilibria for Two-Phase Flows with Soluble Surfactant , 2010 .

[29]  Peng Song,et al.  A diffuse-interface method for two-phase flows with soluble surfactants , 2011, J. Comput. Phys..

[30]  C. M. Elliott,et al.  Numerical computation of advection and diffusion on evolving diffuse interfaces , 2011 .

[31]  Haihu Liu,et al.  Phase-field modeling droplet dynamics with soluble surfactants , 2010, J. Comput. Phys..

[32]  Harald Garcke,et al.  Second order phase field asymptotics for multi-component systems , 2006 .

[33]  L. Tordai,et al.  Time‐Dependence of Boundary Tensions of Solutions I. The Role of Diffusion in Time‐Effects , 1946 .

[34]  G. Simonett,et al.  Well-posedness of a Two-phase Flow with Soluble Surfactant , 2005 .

[35]  I-Liang Chern,et al.  Simulating binary fluid-surfactant dynamics by a phase field model , 2012 .

[36]  A. Voigt,et al.  PDE's on surfaces---a diffuse interface approach , 2006 .

[37]  David Betounes.  Kinematics of submanifolds and the mean curvature normal , 1986 .

[38]  Metin Muradoglu,et al.  A front-tracking method for computation of interfacial flows with soluble surfactants , 2008, J. Comput. Phys..

[39]  R. V. D. Sman,et al.  Diffuse interface model of surfactant adsorption onto flat and droplet interfaces , 2006 .

[40]  C. Pozrikidis,et al.  A Finite-volume/Boundary-element Method for Flow Past Interfaces in the Presence of Surfactants, with Application to Shear Flow Past a Viscous Drop , 1998 .

[41]  Axel Voigt,et al.  A DIFFUSE-INTERFACE APPROACH FOR MODELING TRANSPORT, DIFFUSION AND ADSORPTION/DESORPTION OF MATERIAL QUANTITIES ON A DEFORMABLE INTERFACE. , 2009, Communications in mathematical sciences.

[42]  F. Coutelieris,et al.  Low to moderate Peclet mass transport in assemblages of spherical particles for a realistic adsorption–reaction–desorption mechanism , 2005 .

[43]  Ashish Kumar,et al.  A phase field model for failure in interconnect lines due to coupled diffusion mechanisms , 2002 .

[44]  J. Kim,et al.  A comparison study of phase-field models for an immiscible binary mixture with surfactant , 2012, The European Physical Journal B.

[45]  H. Diamant,et al.  Kinetics of Surfactant Adsorption at Fluid-Fluid Interfaces , 1996 .

[46]  Michelle Schatzman,et al.  Geometrical evolution of developed interfaces , 1995, Emerging applications in free boundary problems.

[47]  Kunibert G. Siebert,et al.  Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA , 2005, Lecture Notes in Computational Science and Engineering.

[48]  A. Tornberg,et al.  A numerical method for two phase flows with insoluble surfactants , 2011 .

[49]  Gerhard Gompper,et al.  Lattice-Boltzmann study of spontaneous emulsification , 1999 .

[50]  James F. Blowey,et al.  Curvature Dependent Phase Boundary Motion and Parabolic Double Obstacle Problems , 1993 .

[51]  M. Gurtin,et al.  TWO-PHASE BINARY FLUIDS AND IMMISCIBLE FLUIDS DESCRIBED BY AN ORDER PARAMETER , 1995, patt-sol/9506001.

[52]  Ming-Chih Lai,et al.  An immersed boundary method for interfacial flows with insoluble surfactant , 2008, J. Comput. Phys..

[53]  Takashi Teramoto,et al.  Droplet Growth Dynamics in a Water/Oil/Surfactant System. , 2001, Journal of colloid and interface science.

[54]  Jinchao Xu,et al.  Phase Field Model of Thermo-Induced Marangoni Effects in the Mixtures and its Numerical Simulations with Mixed Finite Element Method , 2009 .

[55]  Gunduz Caginalp,et al.  Convergence of the phase field model to its sharp interface limits , 1998, European Journal of Applied Mathematics.