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June  2012, 17(4): 1289-1307. doi: 10.3934/dcdsb.2012.17.1289

Simulating binary fluid-surfactant dynamics by a phase field model

Chun-Hao Teng 1, , I-Liang Chern 2, and  Ming-Chih Lai 1,

1. 

Department of Applied Mathematics, Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30010, Taiwan, Taiwan
2. 

Department of Mathematics, National Center for Theoretical Sciences (Taipei offce), National Taiwan University, Taipei 10617, Taiwan

Received  February 2011 Revised  September 2011 Published  February 2012

In this paper, the dynamics of a binary fluid-surfactant system described by a phenomenological phase field model is investigated through analytical and numerical computations. We first consider the case of one-dimensional planar interface and prove the existence of the equilibrium solution. Then we derive the analytical equilibrium solution for the order parameter and the surfactant concentration in a particular case. The results show that the present phase field formulation qualitatively mimics the surfactant adsorption on the binary fluid interfaces. We further study the time-dependent solutions of the system by numerical computations based on the pseudospectral Fourier computational framework. The present numerical results are in a good agreement with the previous theoretical study in the way that the surfactant favors the creation of interfaces and also stabilizes the formation of phase regions.
Keywords: phase field model, pseudospectral method., binary fluid, Surfactant, Cahn-Hilliard equation.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 65N3.
Citation: Chun-Hao Teng, I-Liang Chern, Ming-Chih Lai. Simulating binary fluid-surfactant dynamics by a phase field model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1289-1307. doi: 10.3934/dcdsb.2012.17.1289
References:
[1]

A. B. Branger and D. M. Eckmann, Accelerated arteriolar gas embolism reabsorption by an exogenous surfactant, Anesthesiology, 96 (2002), 971-979. doi: 10.1097/00000542-200204000-00027.

[2]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[3]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: 10.1063/1.1730447.

[4]

H. Diamant and D. Andelman, Kinetics of surfactant adsorption at fluid-fluid interfaces, J. Phys. Chem., 100 (1996), 13732-13742. doi: 10.1021/jp960377k.

[5]

H. Diamant, G. Ariel and D. Andelman, Kinetics of surfactant adsorption: The free energy approach, Colloids Surf A, 183-185 (2001), 259-276. doi: 10.1016/S0927-7757(01)00553-2.

[6]

C. D. Eggleton, T. M. Tsai and K. J. Stebe, Tip streaming from a drop in the presence of surfactants, Phys. Rev. Lett., 87 (2001), 048302-1-048302-1. doi: 10.1103/PhysRevLett.87.048302.

[7]

I. Fonseca, M. Morini and V. Slastikov, Surfactants in foam stability: A phase-field model, Arch. Rational Mech. Anal., 183 (2007), 411-456. doi: 10.1007/s00205-006-0012-x.

[8]

J. S. Hesthaven, S. Gottlieb and D. Gottlieb, "Spectral Methods for Time-Dependent Problems," Cambridge Monographs on Applied and Computational Mathematics, 21, Cambridge University Press, Cambridge, 2007.

[9]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96-127. doi: 10.1006/jcph.1999.6332.

[10]

Y. T. Hu, D. J. Pine and L. G. Leal, Drop deformation, breakup, and coalescence with compatibilizer, Phys. Fluids, 18 (2000), 484-489. doi: 10.1063/1.870254.

[11]

T. Kawakatsu, K. Kawasaki, M. Furusaka, H. Okabayashi and T. Kanaya, Late stage dynamics of phase separation processes of binary mixtures containing surfactants, J. Chem. Phys., 99 (1993), 8200-8217. doi: 10.1063/1.466213.

[12]

J. Kim, Numerical simulations of phase separation dynamics in a water-oil-surfactant system, J. Colloid Interface Sci., 303 (2006), 272-279. doi: 10.1016/j.jcis.2006.07.032.

[13]

S. Komura and H. Kodama, Two-order-parameter model for an oil-water-surfactant system, Phys. Rew. E, 55 (1997), 1722-1727. doi: 10.1103/PhysRevE.55.1722.

[14]

M. Laradji, H. Gau, M. Grant and M. Zuckermann, The effect of surfactants on the dynamics of phase separation, J. Phys.: Condens. Matter, 4 (1992), 6715-6728. doi: 10.1088/0953-8984/4/32/006.

[15]

G. B. McFadden and A. A. Wheeler, On the Gibbs adsorption equation and diffuse interface models, Proc. R. Soc. Lond. A, 458 (2002), 1129-1149. doi: 10.1098/rspa.2001.0908.

[16]

E. B. Nauman and D. Q. He, Non-linear diffusion and phase separation, Chem. Eng. Sci., 49 (2001), 1999-2018. doi: 10.1016/S0009-2509(01)00005-7.

[17]

D. Raabe, "Computational Materials Science: The Simulation of Materials, Microstructures and Properties," Wiley-VCH, Weinheim, 1998.

[18]

T. Teramoto and F. Yonezawa, Droplet growth dynamics in a water/oil/surfactant system, J. Colloid Interface Sci., 235 (2001), 329-333. doi: 10.1006/jcis.2000.7349.

[19]

R. G. M. van der Sman and S. van der Graaf, Diffuse interface model of surfactant adsorption onto flat and droplet interfaces, Rheol Acta, 46 (2006), 3-11.

show all references

References:
[1]

A. B. Branger and D. M. Eckmann, Accelerated arteriolar gas embolism reabsorption by an exogenous surfactant, Anesthesiology, 96 (2002), 971-979. doi: 10.1097/00000542-200204000-00027.

[2]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[3]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: 10.1063/1.1730447.

[4]

H. Diamant and D. Andelman, Kinetics of surfactant adsorption at fluid-fluid interfaces, J. Phys. Chem., 100 (1996), 13732-13742. doi: 10.1021/jp960377k.

[5]

H. Diamant, G. Ariel and D. Andelman, Kinetics of surfactant adsorption: The free energy approach, Colloids Surf A, 183-185 (2001), 259-276. doi: 10.1016/S0927-7757(01)00553-2.

[6]

C. D. Eggleton, T. M. Tsai and K. J. Stebe, Tip streaming from a drop in the presence of surfactants, Phys. Rev. Lett., 87 (2001), 048302-1-048302-1. doi: 10.1103/PhysRevLett.87.048302.

[7]

I. Fonseca, M. Morini and V. Slastikov, Surfactants in foam stability: A phase-field model, Arch. Rational Mech. Anal., 183 (2007), 411-456. doi: 10.1007/s00205-006-0012-x.

[8]

J. S. Hesthaven, S. Gottlieb and D. Gottlieb, "Spectral Methods for Time-Dependent Problems," Cambridge Monographs on Applied and Computational Mathematics, 21, Cambridge University Press, Cambridge, 2007.

[9]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96-127. doi: 10.1006/jcph.1999.6332.

[10]

Y. T. Hu, D. J. Pine and L. G. Leal, Drop deformation, breakup, and coalescence with compatibilizer, Phys. Fluids, 18 (2000), 484-489. doi: 10.1063/1.870254.

[11]

T. Kawakatsu, K. Kawasaki, M. Furusaka, H. Okabayashi and T. Kanaya, Late stage dynamics of phase separation processes of binary mixtures containing surfactants, J. Chem. Phys., 99 (1993), 8200-8217. doi: 10.1063/1.466213.

[12]

J. Kim, Numerical simulations of phase separation dynamics in a water-oil-surfactant system, J. Colloid Interface Sci., 303 (2006), 272-279. doi: 10.1016/j.jcis.2006.07.032.

[13]

S. Komura and H. Kodama, Two-order-parameter model for an oil-water-surfactant system, Phys. Rew. E, 55 (1997), 1722-1727. doi: 10.1103/PhysRevE.55.1722.

[14]

M. Laradji, H. Gau, M. Grant and M. Zuckermann, The effect of surfactants on the dynamics of phase separation, J. Phys.: Condens. Matter, 4 (1992), 6715-6728. doi: 10.1088/0953-8984/4/32/006.

[15]

G. B. McFadden and A. A. Wheeler, On the Gibbs adsorption equation and diffuse interface models, Proc. R. Soc. Lond. A, 458 (2002), 1129-1149. doi: 10.1098/rspa.2001.0908.

[16]

E. B. Nauman and D. Q. He, Non-linear diffusion and phase separation, Chem. Eng. Sci., 49 (2001), 1999-2018. doi: 10.1016/S0009-2509(01)00005-7.

[17]

D. Raabe, "Computational Materials Science: The Simulation of Materials, Microstructures and Properties," Wiley-VCH, Weinheim, 1998.

[18]

T. Teramoto and F. Yonezawa, Droplet growth dynamics in a water/oil/surfactant system, J. Colloid Interface Sci., 235 (2001), 329-333. doi: 10.1006/jcis.2000.7349.

[19]

R. G. M. van der Sman and S. van der Graaf, Diffuse interface model of surfactant adsorption onto flat and droplet interfaces, Rheol Acta, 46 (2006), 3-11.

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