Surfactants play a critical role in the physics of paint and coating formulations, affecting key rheological properties such as viscosity, yield stress, and thixotropy. This paper proposes a new three-dimensional phase-field model that uses the cumulant lattice Boltzmann method (LBM) to simulate soluble surfactants. Although current phase-field models commonly use Langmuir's relationship, they cannot calculate interfacial tension analytically, or the LBM models used are unstable when viscosities are low. However, the proposed method overcomes these limitations through two main features. First, the main parameters for modeling and controlling the surfactant's strength and interaction with other phases are directly obtained from a given initial interfacial tension and bulk surfactant, eliminating the need for trial-and-error simulations. Second, a new equilibrium distribution function in the moment space that includes diagonal and off diagonal elements of the pressure tensor is used to minimize Galilean invariance violation. Additionally, there is no need to use an external force to recover multiphase flows, which could break mass conservation. Furthermore, this method has significant potential for parallelization since only one neighbor's cell is used for discretization. The method shows Langmuir relation behavior and is validated with analytical solutions for various interfacial tensions and surfactant concentrations. Moreover, the paper demonstrates the influence of interfacial tension and surfactants on spurious velocities, indicating the method's stability at low viscosities. The dynamics of droplets in the presence of the surfactants is studied in spinodal decomposition and under various external forces. The method accurately simulates the breaking-up and coalescence for these cases. Furthermore, the method successfully simulates the breakage of a liquid thread at a high viscosity ratio.
REFERENCES
1.
K.
Holmberg
, “
Applications of surfactants in paints
,” in Surfactants in Polymers, Coatings, Inks and Adhesives
(
Blackwell
, 2020
), pp. 152
–179
.2.
M.-R.
Pendar
,
F.
Rodrigues
,
J. C.
Páscoa
, and
R.
Lima
, “
Review of coating and curing processes: Evaluation in automotive industry
,” Phys. Fluids
34
, 101301
(2022
).3.
A. M.
Jiménez-López
and
G. A.
Hincapié-Llanos
, “
Identification of factors affecting the reduction of VOC emissions in the paint industry: Systematic literature review-SLR
,” Prog. Organic Coat.
170
, 106945
(2022
).4.
M.
Mulqueen
and
D.
Blankschtein
, “
Theoretical and experimental investigation of the equilibrium oil- water interfacial tensions of solutions containing surfactant mixtures
,” Langmuir
18
, 365
–376
(2002
).5.
L.
Zhang
,
L.
He
,
M.
Ghadiri
, and
A.
Hassanpour
, “
Effect of surfactants on the deformation and break-up of an aqueous drop in oils under high electric field strengths
,” J. Pet. Sci. Eng.
125
, 38
–47
(2015
).6.
S.
Engblom
,
M.
Do-Quang
,
G.
Amberg
, and
A.-K.
Tornberg
, “
On diffuse interface modeling and simulation of surfactants in two-phase fluid flow
,” Commun. Comput. Phys.
14
, 879
–915
(2013
).7.
J.-J.
Xu
,
Z.
Li
,
J.
Lowengrub
, and
H.
Zhao
, “
A level-set method for interfacial flows with surfactant
,” J. Comput. Phys.
212
, 590
–616
(2006
).8.
Z. Y.
Luo
,
X. L.
Shang
, and
B. F.
Bai
, “
Influence of pressure-dependent surface viscosity on dynamics of surfactant-laden drops in shear flow
,” J. Fluid Mech.
858
, 91
–121
(2019
).9.
W. C.
de Jesus
,
A. M.
Roma
,
M. R.
Pivello
,
M. M.
Villar
, and
A.
da Silveira-Neto
, “
A 3D front-tracking approach for simulation of a two-phase fluid with insoluble surfactant
,” J. Comput. Phys.
281
, 403
–420
(2015
).10.
A.
Alke
and
D.
Bothe
, “
3D numerical modeling of soluble surfactant at fluidic interfaces based on the volume-of-fluid method
,” Fluid Dyn. Mater. Process.
5
, 345
–372
(2009
).11.
S.
Khatri
and
A.-K.
Tornberg
, “
An embedded boundary method for soluble surfactants with interface tracking for two-phase flows
,” J. Comput. Phys.
256
, 768
–790
(2014
).12.
J.-J.
Xu
,
W.
Shi
, and
M.-C.
Lai
, “
A level-set method for two-phase flows with soluble surfactant
,” J. Comput. Phys.
353
, 336
–355
(2018
).13.
J.-J.
Xu
and
W.
Ren
, “
A level-set method for two-phase flows with moving contact line and insoluble surfactant
,” J. Comput. Phys.
263
, 71
–90
(2014
).14.
A.
Rinaudo
,
G. M.
Raffa
,
F.
Scardulla
,
M.
Pilato
,
C.
Scardulla
, and
S.
Pasta
, “
Biomechanical implications of excessive endograft protrusion into the aortic arch after thoracic endovascular repair
,” Comput. Biol. Med.
66
, 235
–241
(2015
).15.
V.
Mendez
,
M.
Di Giuseppe
, and
S.
Pasta
, “
Comparison of hemodynamic and structural indices of ascending thoracic aortic aneurysm as predicted by 2-way FSI, CFD rigid wall simulation and patient-specific displacement-based FEA
,” Comput. Biol. Med.
100
, 221
–229
(2018
).16.
R.
Seemann
,
M.
Brinkmann
,
T.
Pfohl
, and
S.
Herminghaus
, “
Droplet based microfluidics
,” Rep. Prog. Phys.
75
, 016601
(2012
).17.
C.-H.
Teng
,
I.-L.
Chern
, and
M.-C.
Lai
, “
Simulating binary fluid-surfactant dynamics by a phase field model
,” Discrete Contin. Dyn. Syst., B
17
, 1289
(2012
).18.
Y.
Li
and
J.
Kim
, “
A comparison study of phase-field models for an immiscible binary mixture with surfactant
,” Eur. Phys. J. B
85
, 1
–9
(2012
).19.
E. G.
Fard
, “
A cumulant LBM approach for large eddy simulation of dispersion microsystems
,” Ph.D. thesis (
University-Bible
, 2015
).20.
M.
Gorakifard
,
C.
Salueña
,
I.
Cuesta
,
E. K.
Far
, and
T.
Spain
, “
Mfree local weak-form cumulant lattice Boltzmann methods: Aeroacoustic consideration
,” in 53rd Spanish Congress of Acoustics-Techniacoustic
, Tarragona, Spain (2022
).21.
J.
Farhadi
and
V.
Bazargan
, “
Marangoni flow and surfactant transport in evaporating sessile droplets: A lattice Boltzmann study
,” Phys. Fluids
34
, 032115
(2022
).22.
Y.
Zong
,
C.
Zhang
,
H.
Liang
,
L.
Wang
, and
J.
Xu
, “
Modeling surfactant-laden droplet dynamics by lattice Boltzmann method
,” Phys. Fluids
32
, 122105
(2020
).23.
R.
Van der Sman
and
S.
Van der Graaf
, “
Diffuse interface model of surfactant adsorption onto flat and droplet interfaces
,” Rheol. Acta
46
, 3
(2006
).24.
Y.
Shi
,
G.
Tang
,
L.
Cheng
, and
H.
Shuang
, “
An improved phase-field-based lattice Boltzmann model for droplet dynamics with soluble surfactant
,” Comput. Fluids
179
, 508
–520
(2019
).25.
J.
Zhang
,
S.
Shu
,
X.
Guan
, and
N.
Yang
, “
Regime mapping of multiple breakup of droplets in shear flow by phase-field lattice Boltzmann simulation
,” Chem. Eng. Sci.
240
, 116673
(2021
).26.
L.
Al-Tamimi
,
H.
Farhat
, and
W. F.
Hasan
, “
Hybrid quasi-steady thermal lattice Boltzmann model for investigating the effects of thermal, surfactants and contact angle on the flow characteristics of oil in water emulsions between two parallel plates
,” J. Pet. Sci. Eng.
204
, 108572
(2021
).27.
R. H. F.
Hanashmooni
, Multi-Scale Simulations of Waterflooding and Surfactant Flooding in Microfluidic Micromodels
(
Colorado School of Mines
, 2019
).28.
J.
Zhang
,
X.
Zhang
,
W.
Zhao
,
H.
Liu
, and
Y.
Jiang
, “
Effect of surfactants on droplet generation in a microfluidic t-junction: A lattice Boltzmann study
,” Phys. Fluids
34
, 042121
(2022
).29.
T.
Krüger
,
H.
Kusumaatmaja
,
A.
Kuzmin
,
O.
Shardt
,
G.
Silva
, and
E. M.
Viggen
, The Lattice Boltzmann Method
(
Springer International Publishing
, 2017
), Vol.
10
, pp. 4
–15
.30.
E.
Kian Far
,
M.
Gorakifard
, and
E.
Fattahi
, “
Multiphase phase-field lattice boltzmann method for simulation of soluble surfactants
,” Symmetry
13
, 1019
(2021
).31.
E. G.
Fard
,
E.
Shirani
, and
S.
Geller
, “
The fluid structure interaction with using of lattice Boltzmann method
,” in Proceedings of the 13th Annual International Fluid Dynamic Conference
, Shiraz, Iran (2010
), pp. 26
–28
.32.
M.
Gorakifard
,
I.
Cuesta
,
C.
Salueña
, and
E.
Kian Far
, “
Acoustic wave propagation and its application to fluid structure interaction using the cumulant lattice Boltzmann method
,” Comput. Math. Appl.
87
, 91
(2021
).33.
E. K.
Far
,
M.
Geier
, and
M.
Krafczyk
, “
A sliding mesh LBM approach for the simulation of the rotating objects
,” in 13th International Conference for Mesoscopic Methods in Engineering and Science
, 2016
.34.
E.
Nauman
and
N. P.
Balsara
, “
Phase equilibria and the Landau-Ginzburg functional
,” Fluid Phase Equilib.
45
, 229
–250
(1989
).35.
M.
Gorakifard
, “
Meshfree methods: moving beyond the cumulant lattice Boltzmann method
,” Ph.D. thesis (
University-Bible
, 2023
).36.
R.
Zanella
,
G.
Tegze
,
R.
Le Tellier
, and
H.
Henry
, “
Two-and three-dimensional simulations of Rayleigh–Taylor instabilities using a coupled Cahn–Hilliard/Navier–Stokes model
,” Phys. Fluids
32
, 124115
(2020
).37.
A.
Dadvand
,
M.
Bagheri
,
N.
Samkhaniani
,
H.
Marschall
, and
M.
Wörner
, “
Advected phase-field method for bounded solution of the Cahn–Hilliard Navier–Stokes equations
,” Phys. Fluids
33
, 053311
(2021
).38.
N.
Anjum
and
S. A.
Vanapalli
, “
Microfluidic emulsification with a surfactant and a particulate emulsifier: Dripping-to-jetting transitions and drop size scaling
,” Phys. Fluids
34
, 032008
(2022
).39.
J.
Zhang
,
H.
Liu
, and
X.
Zhang
, “
Modeling the deformation of a surfactant-covered droplet under the combined influence of electric field and shear flow
,” Phys. Fluids
33
, 042109
(2021
).40.
M.
Muradoglu
and
G.
Tryggvason
, “
A front-tracking method for computation of interfacial flows with soluble surfactants
,” J. Comput. Phys.
227
, 2238
–2262
(2008
).41.
H.
Liu
and
Y.
Zhang
, “
Phase-field modeling droplet dynamics with soluble surfactants
,” J. Comput. Phys.
229
, 9166
–9187
(2010
).42.
E.
Kian Far
,
M.
Geier
,
K.
Kutscher
, and
M.
Krafczyk
, “
Distributed cumulant lattice Boltzmann simulation of the dispersion process of ceramic agglomerates
,” J. Comput. Methods Sci. Eng.
16
, 231
–252
(2016
).43.
M.
Geier
,
M.
Schönherr
,
A.
Pasquali
, and
M.
Krafczyk
, “
The cumulant lattice boltzmann equation in three dimensions: Theory and validation
,” Comput. Math. Appl.
70
, 507
–547
(2015
).44.
E.
Kian Far
,
M.
Geier
,
K.
Kutscher
, and
M.
Krafczyk
, “
Implicit large eddy simulation of flow in a micro-orifice with the cumulant lattice Boltzmann method
,” Computation
5
, 23
(2017
).45.
K.
Maeyama
,
S.
Ishida
, and
Y.
Imai
, “
Peristaltic transport of a power-law fluid induced by a single wave: A numerical analysis using the cumulant lattice Boltzmann method
,” Phys. Fluids
34
, 111911
(2022
).46.
E.
Kian Far
,
M.
Geier
,
K.
Kutscher
, and
M.
Konstantin
, “
Simulation of micro aggregate breakage in turbulent flows by the cumulant lattice Boltzmann method
,” Comput. Fluids
140
, 222
–231
(2016
).47.
E. K.
Far
and
S.
Langer
, “
Analysis of the cumulant lattice Boltzmann method for acoustics problems
,” in The 13th International Conference on Theoretical and Computational Acoustics
, Vienna, Austria, 2017
.48.
E.
Kian Far
,
M.
Geier
, and
M.
Krafczyk
, “
Simulation of rotating objects in fluids with the cumulant lattice Boltzmann model on sliding meshes
,” Comput. Math. Appl.
79
, 3
(2020
).49.
E. K.
Far
, “
Turbulent flow simulation of dispersion microsystem with cumulant lattice Boltzmann method
,” in Formula X
, 2019
.50.
M.
Gorakifard
,
C.
Salueña
,
I.
Cuesta
, and
E.
Kian Far
, “
The meshless local Petrov–Galerkin cumulant lattice Boltzmann method: Strengths and weaknesses in aeroacoustic analysis
,” Acta Mech.
233
, 1467
–1483
(2022
).51.
M.
Geier
and
K.
Far
, “
E. a sliding grid method for the lattice Boltzmann method using compact interpolation
,” in Proceedings of the First International Workshop on Lattice Boltzmann for Wind Energy, Online Conference
(Swiss wind energy R&D network, 2021
), pp. 25
–26
.52.
M.
Gorakifard
,
C.
Salueña
,
I.
Cuesta
, and
E. K.
Far
, “
Analysis of aeroacoustic properties of the local radial point interpolation cumulant lattice Boltzmann method
,” Energies
14
, 1443
(2021
).53.
M.
Gorakifard
,
C.
Salueña
,
I.
Cuesta
, and
E.
Kian Far
, “
Acoustical analysis of fluid structure interaction using the cumulant lattice Boltzmann method
,” in The 16th International Conference for Mesoscopic Methods in Engineering and Science at: Heriot-Watt University
, Edinburgh, Scotland, 2019
.54.
E.
Kian Far
,
M.
Gorakifard
, and
E.
Fattahi
, “
A pressure approach of cumulant phase-field lattice Boltzmann method for simulating multiphase flows
,” Phys. Fluids
35
, 023314
(2023
).55.
D. G.
Duff
,
S. M.
Ross
, and
D. H.
Vaughan
, “
Adsorption from solution: An experiment to illustrate the Langmuir adsorption isotherm
,” J. Chem. Educ.
65
, 815
(1988
).56.
K.
Feigl
,
D.
Megias-Alguacil
,
P.
Fischer
, and
E. J.
Windhab
, “
Simulation and experiments of droplet deformation and orientation in simple shear flow with surfactants
,” Chem. Eng. Sci.
62
, 3242
–3258
(2007
).57.
J.
Lee
and
C.
Pozrikidis
, “
Effect of surfactants on the deformation of drops and bubbles in Navier–Stokes flow
,” Comput. Fluids
35
, 43
–60
(2006
).58.
Y.
Hu
,
D.
Pine
, and
L. G.
Leal
, “
Drop deformation, breakup, and coalescence with compatibilizer
,” Phys. Fluids
12
, 484
–489
(2000
).59.
S.
Lyu
,
T. D.
Jones
,
F. S.
Bates
, and
C. W.
Macosko
, “
Role of block copolymers on suppression of droplet coalescence
,” Macromolecules
35
, 7845
–7855
(2002
).60.
J.-J.
Xu
,
Z.
Li
,
J.
Lowengrub
, and
H.
Zhao
, “
Numerical study of surfactant-laden drop-drop interactions
,” Commun. Comput. Phys.
10
, 453
–473
(2011
).61.
Y.
Tian
,
L.
Di
,
W.
Lai
,
Y.
Guan
,
W.
Deng
, and
Y.
Huang
, “
Numerical investigation of air cushioning in the impact of micro-droplet under electrostatic fields
,” Phys. Fluids
35
, 013339
(2023
).62.
P.
Zhu
,
H.
Ma
,
F.
Shu
,
X.
Wang
,
Y.
Wang
,
X.
Diao
, and
Y.
Shi
, “
Magnet-actuated loading of magnetic conductive high-viscosity droplets
,” Precis. Eng.
79
, 164
–172
(2023
).63.
J.
Plateau
, Experimental and Theoretical Statics of Liquids Subject to Molecular Forces Only
(
Gauthier-Villars
,
Paris
, 1873
), Vol.
4
.64.
L.
Rayleigh
, “
On the instability of jets
,” Proc. London Math. Soc.
s1–10
, 4
–13
(1878
).65.
P.
Lafrance
, “
Nonlinear breakup of a laminar liquid jet
,” Phys. Fluids
18
, 428
–432
(1975
).66.
H.
Liu
,
L.
Wu
,
Y.
Ba
,
G.
Xi
, and
Y.
Zhang
, “
A lattice boltzmann method for axisymmetric multicomponent flows with high viscosity ratio
,” J. Comput. Phys.
327
, 873
–893
(2016
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.