Computational Mechanics

, Volume 60, Issue 1, pp 143–161 | Cite as

Phoretic motion of soft vesicles and droplets: an XFEM/particle-based numerical solution

  • Tong Shen
  • Franck VernereyEmail author
Original Paper


When immersed in solution, surface-active particles interact with solute molecules and migrate along gradients of solute concentration. Depending on the conditions, this phenomenon could arise from either diffusiophoresis or the Marangoni effect, both of which involve strong interactions between the fluid and the particle surface. We introduce here a numerical approach that can accurately capture these interactions, and thus provide an efficient tool to understand and characterize the phoresis of soft particles. The model is based on a combination of the extended finite element—that enable the consideration of various discontinuities across the particle surface—and the particle-based moving interface method—that is used to measure and update the interface deformation in time. In addition to validating the approach with analytical solutions, the model is used to study the motion of deformable vesicles in solutions with spatial variations in both solute concentration and temperature.


Phoretic motion Diffusiophoresis Marangoni effect Droplets Extended finite element method 



Research reported in this publication was supported by the National Science Foundation under the CAREER award 1350090 and by the National Institute of Arthritis and Musculoskeletal and Skin Diseases of the National Institutes of Health under Award Number 1R01AR065441. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.


  1. 1.
    Haley B, Frenkel E (2008) Nanoparticles for drug delivery in cancer treatment. Urol Oncol Semin Orig Invest 26(1):57–64Google Scholar
  2. 2.
    Krystal H (2015) Integration and self healing: Affect trauma, alexithymia. Routledge, AbingdonGoogle Scholar
  3. 3.
    Ebbens SJ, Howse JR (2010) In pursuit of propulsion at the nanoscale. Soft Matter 6(4):726–738CrossRefGoogle Scholar
  4. 4.
    Jiang S, Chen Q, Tripathy M, Luijten E, Schweizer KS, Granick S (2010) Janus particle synthesis and assembly. Adv Mater 22(10):1060–1071CrossRefGoogle Scholar
  5. 5.
    Young NO, Goldstein JS, Block MJ (1959) The motion of bubbles in a vertical temperature gradient. J Fluid Mech 6(03):350–356CrossRefzbMATHGoogle Scholar
  6. 6.
    Derjaguin BV, Sidorenkov GP, Zubashchenkov EA, Kiseleva EV (1947) Kinetic phenomena in boundary films of liquids. Kolloidn. Zh 9:335–347Google Scholar
  7. 7.
    Lin MMJ, Prieve DC (1983) Electromigration of latex induced by a salt gradient. J Colloid Interface Sci 95(2):327–339CrossRefGoogle Scholar
  8. 8.
    Lechnick WJ, Shaeiwitz JA (1984) Measurement of diffusiophoresis in liquids. J Colloid Interface Sci 102(1):71–87CrossRefGoogle Scholar
  9. 9.
    Anderson JL (1989) Colloid transport by interfacial forces. Ann Rev Fluid Mech 21(1):61–99CrossRefzbMATHGoogle Scholar
  10. 10.
    Brady JF (2011) Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives. J Fluid Mech 667:216–259MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jlicher F, Prost J (2009) Generic theory of colloidal transport. Eur Phys J E 29(1):27–36CrossRefGoogle Scholar
  12. 12.
    Michelin S, Lauga E, Bartolo D (2013) Spontaneous autophoretic motion of isotropic particles. Phys Fluids ( 1994-present) 25(6):061701Google Scholar
  13. 13.
    Keh HJ, Weng JC (2001) Diffusiophoresis of colloidal spheres in nonelectrolyte gradients at small but finite Pclet numbers. Colloid Polym Sci 279(4):305–311CrossRefGoogle Scholar
  14. 14.
    Riske KA, Dimova R (2005) Electro-deformation and poration of giant vesicles viewed with high temporal resolution. Biophys J 88(2):1143–1155CrossRefGoogle Scholar
  15. 15.
    Glaser N, Adams DJ, Böker A, Krausch G (2006) Janus particles at liquid–liquid interfaces. Langmuir 22(12):5227–5229CrossRefGoogle Scholar
  16. 16.
    Shin S, Um E, Sabass B, Ault JT, Rahimi M, Warren PB, Stone HA (2016) Size-dependent control of colloid transport via solute gradients in dead-end channels. PNAS 113:257–261CrossRefGoogle Scholar
  17. 17.
    Benet E, Vernerey FJ (2016) Mechanics and stability of vesicles and droplets in confined spaces. Phys Rev E 94(6):062613CrossRefGoogle Scholar
  18. 18.
    Kreissl P, Holm C, de Graaf J (2016) The efficiency of self-phoretic propulsion mechanisms with surface reaction heterogeneity. J Chem Phys. doi: 10.1063/1.4951699
  19. 19.
    Khair AS (2013) Diffusiophoresis of colloidal particles in neutral solute gradients at finite Peclet number. J Fluid Mech 731:64–94MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gupta S, Sreeja KK, Thakur S (2015) Autonomous movement of a chemically powered vesicle. Phys Rev E 92(4):42703CrossRefGoogle Scholar
  21. 21.
    Ferziger JH, Peric M (2012) Computational methods for fluid dynamics. Springer Science and Business Media, BerlinzbMATHGoogle Scholar
  22. 22.
    Peskin CS (2002) The immersed boundary method. Acta Numer 11:479–517MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Peskin CS (1977) Numerical analysis of blood flow in the heart. J Comput Phys 25(3):220–252MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Peskin CS (1972) Flow patterns around heart valves: a numerical method. J Comput Phys 10(2):252–271MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang L et al (2004) Immersed finite element method. Comput Methods Appl Mech Eng 193.21:2051–2067MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu WK et al (2006) Immersed finite element method and its applications to biological systems. Comput Methods Appl Mech Eng 195.13:1722–1749MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Liu WK, Tang S (2007) Mathematical foundations of the immersed finite element method. Comput Mech 39.3:211–222MathSciNetzbMATHGoogle Scholar
  28. 28.
    Glowinski R, Pan TW, Hesla TI, Joseph DD (1999) A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int J Multiph Flow 25(5):755–794MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Glowinski R, Pan TW, Hesla TI, Joseph DD, Periaux J (2001) A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J Comput Phys 169(2):363–426MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Cottet GH, Maitre E (2006) A level set method for fluid-structure interactions with immersed surfaces. Math Models Methods Appl Sci 16(03):415–438MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hou TY, Lowengrub JS, Shelley MJ (2001) Boundary integral methods for multicomponent fluids and multiphase materials. J Comput Phys 169(2):302–362MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Bazhlekov IB, Anderson PD, Meijer HEH (2004) Nonsingular boundary integral method for deformable drops in viscous flows. Phys Fluids (1994-Present) 16(4):1064–1081MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hyvaluoma J, Harting J (2008) Slip flow over structured surfaces with entrapped microbubbles. Phys Rev Lett 100(24):246001CrossRefGoogle Scholar
  34. 34.
    Debye P, Robert LC (1959) Flow of liquid hydrocarbons in porous Vycor. J Appl Phys 30(6):843–849CrossRefGoogle Scholar
  35. 35.
    Joseph P et al (2006) Slippage of water past superhydrophobic carbon nanotube forests in microchannels. Phys Rev Lett 97.15:156104CrossRefGoogle Scholar
  36. 36.
    Ho TA et al (2011) Liquid water can slip on a hydrophilic surface. Proc Natl Acad Sci 108.39:16170–16175CrossRefGoogle Scholar
  37. 37.
    Foucard L, Vernerey FJ (2016) A particle based moving interface method (PMIM) for modeling the large deformation of boundaries in soft matter systems. Int J Numer Methods Eng. doi: 10.1002/nme.5191
  38. 38.
    Foucard LC, Pellegrino J, Vernerey FJ (2014) Particle-based moving interface method for the study of the interaction between soft colloid particles and immersed fibrous network. Comput Model Eng Sci 98(1):101–127MathSciNetzbMATHGoogle Scholar
  39. 39.
    Vernerey FJ, Farsad M (2011) An Eulerian/XFEM formulation for the large deformation of cortical cell membrane. Comput Methods Biomech Biomed Eng 14(05):433–445CrossRefGoogle Scholar
  40. 40.
    Vernerey FJ, Farsad M (2011) A constrained mixture approach to mechano-sensing and force generation in contractile cells. J Mech Behav Biomed Mater 4(8):1683–1699CrossRefGoogle Scholar
  41. 41.
    Farsad M, Vernerey FJ (2012) An XFEM? Based numerical strategy to model mechanical interactions between biological cells and a deformable substrate. Int J Numer Methods Eng 92(3):238–267MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kabiri Mi, Vernerey FJ (2013) An xfem based multiscale approach to fracture of heterogeneous media. Int J Multiscale Comput Eng 11(6)Google Scholar
  43. 43.
    Vernerey FJ, Farsad M (2014) A mathematical model of the coupled mechanisms of cell adhesion, contraction and spreading. J Math Biol 68(4):989–1022MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Farsad M, Vernerey FJ, Park HS (2010) An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials. Int J Numer Methods Eng 84(12):1466–1489MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Vernerey FJ (2011) A theoretical treatment on the mechanics of interfaces in deformable porous media. Int J Solids Struct 48(22):3129–3141CrossRefGoogle Scholar
  46. 46.
    Young T (1805) An essay on the cohesion of fluids. Philos Trans R Soc Lond 95:65–87CrossRefGoogle Scholar
  47. 47.
    Anderson JL, Prieve DC (1991) Diffusiophoresis caused by gradients of strongly adsorbing solutes. Langmuir 7(2):403–406CrossRefGoogle Scholar
  48. 48.
    Golestanian R, Liverpool TB, Ajdari A (2005) Propulsion of a molecular machine by asymmetric distribution of reaction products. Phys Rev Lett 94(22):220801CrossRefGoogle Scholar
  49. 49.
    Anderson JL, Prieve DC (1984) Diffusiophoresis: migration of colloidal particles in gradients of solute concentration. Sep Purif Methods 13(1):67–103CrossRefGoogle Scholar
  50. 50.
    Fanton X, Cazabat AM (1998) Spreading and instabilities induced by a solutal Marangoni effect. Langmuir 14(9):2554–2561 ChicagoCrossRefGoogle Scholar
  51. 51.
    Moes N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69(7):813–833CrossRefGoogle Scholar
  52. 52.
    Sukumar N, Chopp DL, Moes N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite-element method. Comput Methods Appl Mech Eng 190(46):6183–6200MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Moes N, Bechet E, Tourbier M (2006) Imposing Dirichlet boundary conditions in the extended finite element method. Int J Numer Methods Eng 67(12):1641–1669MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Sauerland H, Fries TP (2013) The stable XFEM for two-phase flows. Comput Fluids 87:41–49MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Gilbert JR, Moler C, Schreiber R (1992) Sparse matrices in MATLAB: design and implementation. SIAM J Matrix Anal Appl 13(1):333–356MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Bathe KJ, Wilson EL (1976) Numerical methods in finite element analysis. Prentice-Hall, EnglewoodGoogle Scholar
  57. 57.
    Fries T-P, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304MathSciNetzbMATHGoogle Scholar
  58. 58.
    Babuska I, Banerjee U (2012) Stable generalized finite element method (SGFEM). Comput Methods Appli Mech Eng 201:91–111MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Bechet E et al (2005) Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int J Numer Methods Eng 64.8:1033–1056CrossRefzbMATHGoogle Scholar
  60. 60.
    Foucard L, Aryal A, Duddu R, Vernerey F (2015) A coupled Eulerian–Lagrangian extended finite element formulation for simulating large deformations in hyperelastic media with moving free boundaries. Comput Methods Appl Mech Eng 283:280–302MathSciNetCrossRefGoogle Scholar
  61. 61.
    Leung S, Lowengrub J, Zhao H (2011) A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion. J Comput Phys 230(7):2540–2561MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Rusanov AI, Prokhorov VA (1996) Interfacial tensiometry, vol 3. Elsevier, LondonCrossRefGoogle Scholar
  63. 63.
    Lamb H (1945) Hydrodynamics, vol 43. Dover, New YorkzbMATHGoogle Scholar
  64. 64.
    Chen PY, Keh HJ (2003) Boundary effects on osmophoresis: motion of a spherical vesicle parallel to two plane walls. Chem Eng Sci 58(19):4449–4464CrossRefGoogle Scholar
  65. 65.
    Michelin S, Lauga E (2010) Efficiency optimization and symmetry-breaking in a model of ciliary locomotion. Phys Fluids (1994-present) 22(11):111901CrossRefGoogle Scholar
  66. 66.
    Akalp U et al (2015) Determination of the polymer-solvent interaction parameter for PEG hydrogels in water: Application of a self learning algorithm. Polymer 66:135–147CrossRefGoogle Scholar
  67. 67.
    Foucard LC, Vernerey FJ (2015) An X-FEM based numerical asymptotic expansion for simulating a stokes flow near a sharp corner. Int J Numer Methods Eng 102(2):79–98MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Linke GT, Lipowsky R, Gruhn T (2006) Osmotically induced passage of vesicles through narrow pores. EPL 74(5):916CrossRefGoogle Scholar
  69. 69.
    Hannig K (1982) New aspects in preparative and analytical continuous free-flow cell electrophoresis. Electrophoresis 3(5):235–243CrossRefGoogle Scholar
  70. 70.
    Nguyen TLA, Abdelbary H, Arguello M, Breitbach C, Leveille S, Diallo JS, Snoulten VE (2008) Chemical targeting of the innate antiviral response by histone deacetylase inhibitors renders refractory cancers sensitive to viral oncolysis. Proc Natl Acad Sci 105(39):14981–14986CrossRefGoogle Scholar
  71. 71.
    Vernerey FJ (2016) A mixture approach to investigate interstitial growth in engineering scaffolds. Biomech Model Mechanobiol 15(2):259–278CrossRefGoogle Scholar
  72. 72.
    Akalp U, Bryant SJ, Vernerey FJ (2016) Tuning tissue growth with scaffold degradation in enzyme-sensitive hydrogels: a mathematical model. Soft Matter 12(36):7505–7520CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mechanical EngineeringUniversity of Colorado BoulderBoulderUSA

Personalised recommendations