{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T15:32:25Z","timestamp":1771860745592,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,6]],"date-time":"2021-06-06T00:00:00Z","timestamp":1622937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["BE 2245\/15-1"],"award-info":[{"award-number":["BE 2245\/15-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper proposes a phase-field model for the lattice Boltzmann method which has discretized symmetrical directions of velocities in a cartesian grid, to simulate the soluble surfactant in a Multicomponent multiphase system. Despite other existing phase-field models following Langmuir relation, the interfacial tension can be calculated analytically in this proposed model. Parameters playing roles in the models and controlling the surfactant\u2019s strength and interaction with other phases are obtained directly from a given initial interfacial tension and bulk surfactant. Consequently, there is no further need for trial-and-error simulations, and a real system, e.g., oil-water-surfactant, can be simulated with given initial parameters. The model is validated with the analytical result for a planar oil\u2013water-surfactant system. Furthermore, the method for reobtaining numerical interfacial tension for five different cases is tested and compared with the given initial values for an oil droplet surrounded by water and surfactant. The results show that the obtained interfacial tension from the method is in good agreement with the given initial interfacial tension. Furthermore, the spurious velocity of the model is calculated and seen that the magnitude of spurious velocities is proportional to interfacial tension.<\/jats:p>","DOI":"10.3390\/sym13061019","type":"journal-article","created":{"date-parts":[[2021,6,7]],"date-time":"2021-06-07T01:56:40Z","timestamp":1623031000000},"page":"1019","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Multiphase Phase-Field Lattice Boltzmann Method for Simulation of Soluble Surfactants"],"prefix":"10.3390","volume":"13","author":[{"given":"Ehsan","family":"Kian Far","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, The University of Manchester, Oxford Rd, Manchester M13 9PL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohsen","family":"Gorakifard","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, ETSEQ, Rovira i Virgili University, Pa\u00efsos Catalans, 26, 43007 Tarragona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ehsan","family":"Fattahi","sequence":"additional","affiliation":[{"name":"School of Life Sciences, Technical University of Munich, Freising, 85354 Munich, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.jcp.2010.09.020","article-title":"A diffuse-interface method for two-phase flows with soluble surfactants","volume":"230","author":"Teigen","year":"2011","journal-title":"J. Comput. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Sjoblom, J. (2005). Emulsions and Emulsion Stability: Surfactant Science Series\/61, CRC Press.","DOI":"10.1201\/9781420028089"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1146\/annurev.fluid.36.050802.122124","article-title":"Engineering flows in small devices: Microfluidics toward a lab-on-a-chip","volume":"36","author":"Stone","year":"2004","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6088","DOI":"10.1021\/la9000472","article-title":"Kinetic aspects of emulsion stabilization by surfactants: A microfluidic analysis","volume":"25","author":"Baret","year":"2009","journal-title":"Langmuir"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1021\/la010993u","article-title":"Theoretical and experimental investigation of the equilibrium oil- water interfacial tensions of solutions containing surfactant mixtures","volume":"18","author":"Mulqueen","year":"2002","journal-title":"Langmuir"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.petrol.2014.11.014","article-title":"Effect of surfactants on the deformation and break-up of an aqueous drop in oils under high electric field strengths","volume":"125","author":"Zhang","year":"2015","journal-title":"J. Pet. Sci. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"879","DOI":"10.4208\/cicp.120712.281212a","article-title":"On diffuse interface modeling and simulation of surfactants in two-phase fluid flow","volume":"14","author":"Engblom","year":"2013","journal-title":"Commun. Comput. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1016\/j.jcp.2005.07.016","article-title":"A level-set method for interfacial flows with surfactant","volume":"212","author":"Xu","year":"2006","journal-title":"J. Comput. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.jcp.2014.10.021","article-title":"A 3D front-tracking approach for simulation of a two-phase fluid with insoluble surfactant","volume":"281","author":"Roma","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.compbiomed.2015.09.011","article-title":"Biomechanical implications of excessive endograft protrusion into the aortic arch after thoracic endovascular repair","volume":"66","author":"Rinaudo","year":"2015","journal-title":"Comput. Biol. Med."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.compbiomed.2018.07.013","article-title":"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","volume":"100","author":"Mendez","year":"2018","journal-title":"Comput. Biol. Med."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"016601","DOI":"10.1088\/0034-4885\/75\/1\/016601","article-title":"Droplet based microfluidics","volume":"75","author":"Seemann","year":"2011","journal-title":"Rep. Prog. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1006\/jcis.2000.7349","article-title":"Droplet growth dynamics in a water\/oil\/surfactant system","volume":"235","author":"Teramoto","year":"2001","journal-title":"J. Colloid Interface Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s100510050920","article-title":"Lattice-Boltzmann study of spontaneous emulsification","volume":"11","author":"Theissen","year":"1999","journal-title":"Eur. Phys. J. B Condens. Matter Complex Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s00397-005-0081-z","article-title":"Diffuse interface model of surfactant adsorption onto flat and droplet interfaces","volume":"46","year":"2006","journal-title":"Rheol. Acta"},{"key":"ref_16","first-page":"1289","article-title":"Simulating binary fluid-surfactant dynamics by a phase field model","volume":"17","author":"Teng","year":"2012","journal-title":"Discret. Contin. Dyn. Syst. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjb\/e2012-30184-1","article-title":"A comparison study of phase-field models for an immiscible binary mixture with surfactant","volume":"85","author":"Li","year":"2012","journal-title":"Eur. Phys. J. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.cpc.2015.10.002","article-title":"Analysis of improved Lattice Boltzmann phase field method for soluble surfactants","volume":"199","author":"Meinders","year":"2016","journal-title":"Comput. Phys. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"023311","DOI":"10.1103\/PhysRevE.96.023311","article-title":"Grid refinement for aeroacoustics in the lattice Boltzmann method: A directional splitting approach","volume":"96","author":"Gendre","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gorakifard, M., Salue\u00f1a, C., Cuesta, I., and Far, E.K. (2021). Analysis of Aeroacoustic Properties of the Local Radial Point Interpolation Cumulant Lattice Boltzmann Method. Energies, 14.","DOI":"10.3390\/en14051443"},{"key":"ref_21","unstructured":"Kian Far, E., and Langer, S. (August, January 30). Analysis of the cumulant lattice Boltzmann method for acoustics problems. Proceedings of the 13th International Conference on Theoretical and Computational Acoustics, Vienna, Austria."},{"key":"ref_22","unstructured":"Gorakifard, M., Salue\u00f1a, C., Cuesta, I., and Kian Far, E. (2019, January 23\u201326). Acoustical analysis of fluid structure interaction using the Cumulant lattice Boltzmann method. Proceedings of the 16th International Conference for Mesoscopic Methods in Engineering and Science, Edinburgh, UK."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.camwa.2021.02.011","article-title":"Acoustic wave propagation and its application to fluid structure interaction using the Cumulant Lattice Boltzmann Method","volume":"87","author":"Gorakifard","year":"2021","journal-title":"Comput. Math. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.compfluid.2018.02.029","article-title":"Regularized lattice Boltzmann multicomponent models for low capillary and Reynolds microfluidics flows","volume":"167","author":"Montessori","year":"2018","journal-title":"Comput. Fluids"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1039\/c6ib00009f","article-title":"Computational fluid dynamics in the microcirculation and microfluidics: What role can the lattice Boltzmann method play?","volume":"8","author":"Day","year":"2016","journal-title":"Integr. Biol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"103547","DOI":"10.1016\/j.elstat.2020.103547","article-title":"A pseudopotential based lattice Boltzmann modeling of electro wetting-on-dielectric for droplet operations","volume":"109","author":"Pravinraj","year":"2021","journal-title":"J. Electrost."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1016\/j.powtec.2020.06.074","article-title":"Lattice Boltzmann simulations of magnetic particles in a three-dimensional microchannel","volume":"373","author":"He","year":"2020","journal-title":"Powder Technol."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kian Far, E., Geier, M., Kutscher, K., and Krafczyk, M. (2017). Implicit Large Eddy Simulation of Flow in a Micro-Orifice with the Cumulant Lattice Boltzmann Method. Computation, 5.","DOI":"10.3390\/computation5020023"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/j.compfluid.2016.10.001","article-title":"Simulation of micro aggregate breakage in turbulent flows by the cumulant lattice Boltzmann method","volume":"140","author":"Geier","year":"2016","journal-title":"Comput. Fluids"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.ijthermalsci.2011.09.001","article-title":"Lattice Boltzmann simulation of natural convection heat transfer in nanofluids","volume":"52","author":"Fattahi","year":"2012","journal-title":"Int. J. Therm. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2353","DOI":"10.1016\/j.ijthermalsci.2010.07.014","article-title":"Lattice Boltzmann simulation of natural convection heat transfer in eccentric annulus","volume":"49","author":"Fattahi","year":"2010","journal-title":"Int. J. Therm. Sci."},{"key":"ref_32","unstructured":"Kian Far, E., and Shirani, E. (2009, January 16). Simulation of natural convection heat transfer using the lattice boltzmann method in enclosures. Proceedings of the 17th Annual Conference of Mechanical Engineering, Tehran, Iran."},{"key":"ref_33","unstructured":"Fattahi, E., Pribec, I., and Becker, T. (2018). A Novel Lattice Boltzmann Method for Deformable Media. Reactive Flows in Deformable, Complex Media. Math. Forsch. Oberwolfach."},{"key":"ref_34","unstructured":"Kian Far, E., Shirani, E., and Geller, S. (2010, January 26\u201328). Fluid structure interaction with using of lattice Boltzmann method. In Proceedings of the 13th Annual and 2nd International Fluid Dynamics Conference, Shiraz, Iran."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Wang, X., Ban, X., He, R., Wu, D., Liu, X., and Xu, Y. (2018). Fluid-solid boundary handling using pairwise interaction model for non-Newtonian fluid. Symmetry, 10.","DOI":"10.3390\/sym10040094"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1016\/S0045-7949(01)00050-5","article-title":"Two-dimensional simulation of fluid\u2013structure interaction using lattice-Boltzmann methods","volume":"79","author":"Krafczyk","year":"2001","journal-title":"Comput. Struct."},{"key":"ref_37","unstructured":"Kian Far, E., Geier, M., and Krafczyk, M. (2018). Simulation of rotating objects in fluids with the cumulant lattice Boltzmann model on sliding meshes. Comput. Math. Appl."},{"key":"ref_38","unstructured":"Kian Far, E. (2016, January 22). A sliding mesh LBM approach for the simulation of the rotating objects. Proceedings of the 13th International Conference for Mesoscopic Methods in Engineering and Science, Hamburg, Germany."},{"key":"ref_39","unstructured":"Kian Far, E. (2010). Simulation of Moving Body in Field Flow and Fluid Structure Interaction with using Lattice Boltzmann Method. [Master\u2019s Thesis, Isfahan University of Technology]."},{"key":"ref_40","unstructured":"Kian Far, E. (2019, January 24). Turbulent flow simulation of dispersion microsystem with Cumulant lattice Boltzmann method. Proceedings of the Formula X, Manchester, UK."},{"key":"ref_41","first-page":"231","article-title":"Distributed cumulant lattice Boltzmann simulation of the dispersion process of ceramic agglomerates","volume":"16","author":"Geier","year":"2016","journal-title":"J. Comput. Methods Sci. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.compfluid.2016.10.007","article-title":"Lattice Boltzmann methods in porous media simulations: From laminar to turbulent flow","volume":"140","author":"Fattahi","year":"2016","journal-title":"Comput. Fluids"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1016\/j.molliq.2015.10.007","article-title":"Numerical simulation of droplet detachment from solid walls under gravity force using lattice Boltzmann method","volume":"212","author":"Fattahi","year":"2015","journal-title":"J. Mol. Liq."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"063314","DOI":"10.1103\/PhysRevE.98.063314","article-title":"Simplified multiphase lattice Boltzmann method for simulating multiphase flows with large density ratios and complex interfaces","volume":"98","author":"Chen","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"103312","DOI":"10.1063\/5.0023639","article-title":"Mesoscopic simulation of three-dimensional pool boiling based on a phase-change cascaded lattice Boltzmann method","volume":"32","author":"Fei","year":"2020","journal-title":"Phys. Fluids"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"9166","DOI":"10.1016\/j.jcp.2010.08.031","article-title":"Phase-field modeling droplet dynamics with soluble surfactants","volume":"229","author":"Liu","year":"2010","journal-title":"J. Comput. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1016\/j.cpc.2007.11.009","article-title":"Emulsion droplet deformation and breakup with lattice Boltzmann model","volume":"178","year":"2008","journal-title":"Comput. Phys. Commun."},{"key":"ref_48","unstructured":"Fard, E.G. (2015). A Cumulant LBM approach for Large Eddy Simulation of Dispersion Microsystems. [Ph.D. Thesis, Technische Universit\u00e4t Braunschweig]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1019\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:11:24Z","timestamp":1760163084000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1019"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,6]]},"references-count":48,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["sym13061019"],"URL":"https:\/\/doi.org\/10.3390\/sym13061019","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,6]]}}}