Mixed finite element method for electrowetting on dielectric with contact line pinning

Interfaces and Free Boundaries

Volumn 12, Issue 1, 2010, Pages 85-119

  Walker, Shawn W ^a   Bonito, Andrea ^b   Nochetto, Ricardo H ^c  

^a NEW YORK UNIVERSITY   (United States)

^b TEXAS A AND M UNIVERSITY   (United States)

^c University of Maryland   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77954143299     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/228     Document Type: Article

References (59)

1. ADAMS, R. A., & FOURNIER, J. J. F. Sobolev Spaces. 2nd ed., Pure Appl. Math. 140, Elsevier (2003). Zbl 1098.46001 MR 2424078.
2. ARNOLD, D. N., BREZZI, F., & FORTIN, M. A stable finite element for the Stokes equations. Calcolo 21 (1984), 337-344. Zbl 0593.76039 MR 0799997.
3. BÄNSCH, E. Finite element discretization of the Navier-Stokes equations with a free capillary surface. Numer. Math.. 88 (2001), 203-235. Zbl 0985.35060 MR 1826211.
4. BATCHELOR, G. K. An Introduction to Fluid Dynamics. Cambridge Univ. Press, New York (1967). Zbl 0152.44402 MR 1744638.
5. BEDEKAR, A. S., JENKINS, J. W., & SUNDARAM, S. A computational model for the design of electrowetting on dielectric (ewod) systems. In: NSTI Nanotechnology Conference & Trade Show (Anaheim, CA, 2005), Session Presentation.
6. BERGE, B. Électrocapillarité et mouillage de films isolants par l'eau. C. R. Acad. Sci. Paris Sér. II 317 (1993), 157-163.
7. BONITO, A., NOCHETTO, R. H., & PAULETTI, M. S. Geometrically consistent mesh modification. In preparation.
8. BRAESS, D. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. 2nd ed., Cambridge Univ. Press (2001). Zbl 0976.65099 MR 1827293.
9. BRENNER, S. C., & SCOTT, L. R. The Mathematical Theory of Finite Element Methods. 2nd ed., Springer, New York (2002). Zbl 1012.65115 MR 1894376.
10. BREZZI, F., & FORTIN, M. Mixed and Hybrid Finite Element Methods. Springer, New York (1991). Zbl 0788.73002 MR 1115205.
11. BREZZI, F., HAGER, W. W., & RAVIART, P. A. Error estimates for the finite element solution of variational inequalities: Part II. Mixed methods. Numer. Math. 31 (1978), 1-16. Zbl 0427.65077 MR 0508584.
12. BURDON, R. S. Surface Tension and the Spreading of Liquids. Cambridge Univ. Press, Cambridge (1949). Zbl 0028.04102.
13. CHENG, C. H. A., COUTAND, D., & SHKOLLER, S. Navier-Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 39 (2007), 742-800. Zbl 1138.74022 MR 2349865.
14. CHO, S. K., & KIM, C.-J. Particle separation and concentration control for digital microfluidic systems. In: The 16th Annual IEEE International Conference on MEMS (Kyoto, 2003), 686-689.
15. CHO, S. K., MOON, H., & KIM, C.-J. Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits. J. Microelectromech. Systems 12 (2003), 70-80.
16. CĹEMENT, P. Approximation by finite element functions using local regularization. R.A.I.R.O. Anal. Numér. 9 (1975), 77-84. Zbl 0368.65008 MR 0400739.
17. DARHUBER, A. A., DAVIS, J. M., TROIAN, S. M., & REISNER, W. W. Thermocapillary actuation of liquid flow on chemically patterned surfaces. Phys. Fluids 15 (2003), 10150-10153.
18. DE GENNES, P.-G., BROCHARD-WYART, F., & QÚERÉ, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York (2004). Zbl 1139.76004.
19. DO CARMO, M. P. Differential Geometry of Curves and Surfaces. Prentice Hall, Upper Saddle River, NJ (1976). Zbl 0326.53001 MR 0394451.
20. DRELICH, E. J., LASKOWSKI, J. S., & MITTAL, K. L. (eds.) Apparent and Microscopic Contact Angles. VSP, Zeist (2000).
21. DUSSAN, E. B. On the ability of drops or bubbles to stick to non-horizontal surfaces of solids. Part 2. Small drops or bubbles having contact angles of arbitrary size. J. Fluid Mech. 151 (1985), 1-20. Zbl 0574.76102.
22. DUVAUT, G., & LIONS, J.-L. Inequalities in Mechanics and Physics. Springer, New York (1976). Zbl 0331.35002 MR 0521262.
23. DZIUK, G. An algorithm for evolutionary surfaces. Numer. Math. 58 (1990), 603-611. Zbl 0714.65092 MR 1083523.
24. ECK, C., FONTELOS, M., GRÜN, G., KLINGBEIL, F., & VANTZOS, O. On a phase-field model for electrowetting. Interfaces Free Bound. 11 (2009), 259-290. Zbl 1167.35553 MR 2511642.
25. EVANS, L. C. Partial Differential Equations. Amer. Math. Soc., Providence, RI (1998). Zbl 0902.35002 MR 1625845.
26. FORTNER, N., SHAPIRO, B., & HIGHTOWER, A. Modeling and passive control of channel filling for micro-fluidic networks with thousands of channels. In: American Institute of Aeronautics and Astronautics (AIAA) 33rd AIAA Fluid Dynamics Conference and Exhibit (Orlando, FL, 2003).
27. FRIEDMAN, A. Variational Principles and Free-Boundary Problems. Wiley (1982). Zbl 0564.49002 MR 0679313.
28. GALDI, G. P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol.I. Springer Tracts in Natural Philosophy 38, Springer, New York (1994). Zbl 0949.35004 MR 1284205.
29. GIRAULT, V., & RAVIART, P. A. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin (1986). Zbl 0585.65077 MR 0851383.
30. GLOWINSKI, R. Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984). Zbl 0536.65054 MR 0737005.
31. GONG, J., FAN, S. K., & KIM, C. J. Portable digital microfluidics platform with active but disposable lab-on-chip. In: 17th IEEE International Conference on Micro Electro Mechanical Systems (MEMS) (Maastricht, 2004), IEEE Press, 355-358.
32. HE, B., & LEE, J. Dynamic wettability switching by surface roughness effect. In: The 16th Annual IEEE International Conference on MEMS (Kyoto, 2003), 120-123.
33. HELE-SHAW, H. S. The flow of water. Nature (London) 58 (1898), 34-36.
34. HENNING, A., STAUDINGER, U., EICHHOM, K.-J., SAHRE, K., ROGALLI, M., STAMM, M., NEUMANN, A. W., & GRUNDKE, K. Contact angle hysteresis: study by dynamic cycling contact angle measurements and variable angle spectroscopic ellipsometry on polyimide. Langmuir 20 (2004), 6685-6691.
35. JOANNY, J. F., & ROBBINS, M. O. Motion of a contact line on a heterogeneous surface. J. Chem. Phys. 92 (1990), 3206-3212.
36. LANG, S. Real and Functional Analysis. 3rd ed., Grad. Texts in Math. 142, Springer (1993). Zbl 0831.46001 MR 1216137.
37. LAX, P. D. Functional Analysis. Wiley-Interscience (2002). Zbl 1009.47001 MR 1892228.
38. LENOIR, M. Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986), 562-580. Zbl 0605.65071 MR 0842644.
39. LIENEMANN, J., GREINER, A., & KORVINK, J. G. Modeling, simulation, and optimization of electrowetting. IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems 25 (2006), 234-247.
40. LIONS, J.-L., & MAGENES, E. Non-Homogeneous Boundary Value Problems. Vol.1, Springer (1972). Zbl 0223.35039 MR 0350177.
41. LIONS, J.-L., & STAMPACCHIA, G. Variational inequalities. Comm. Pure Appl. Math. 20 (1967), 493-519. Zbl 0152.34601 MR 0216344.
42. LU, H.-W., GLASNER, K., BERTOZZI, A. L., & KIM, C.-J. A diffuse interface model for electrowetting drops in a Hele-Shaw cell. J. Fluid Mech. 590 (2007), 411-435. Zbl 1141.76482. (Pubitemid 350066517)
43. MINNEMA, L., BARNEVELD, H. A., & RINKEL, P. D. An investigation into the mechanism of water treeing in polyethylene high voltage cables. IEEE Trans. Electrical Insulators 15, 461-472.
44. MOHSENI, K., & DOLATABADI, A. An electrowetting microvalve: Numerical simulation. Ann. New York Acad. Sci. 1077 (2006), 415-425.
45. MOON, H., WHEELER, A. R., GARRELL, R. L., LOO, J. A., & KIM, C.-J. On-chip sample preparation by electrowetting-on-dielectric digital microfluidics for matrix assisted laser desorption/ionization mass spectrometry. In: Proceedings of IEEE MEMS (Miami, FL, 2005), 859-862.
46. ROBBINS, M. O., & JOANNY, J. F. Contact angle hysteresis on random surfaces. Europhys. Lett. 3 (1987), 729-735.
47. SATOH, W., LOUGHRAN, M., & SUZUKI, H. Microfluidic transport based on direct electrowetting. J. Appl. Phys. 96 (2004), 835-841.
48. SCHAFFER, E., & WONG, P. Z. Dynamics of contact line pinning in capillary rise and fall. Phys. Rev. Lett. 80 (1998), 3069-3072.
49. SCOTT, L. R., & ZHANG, S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990), 483-493. Zbl 0696.65007 MR 1011446.
50. SHAPIRO, B., MOON, H., GARRELL, R., & KIM, C.-J. Equilibrium behavior of sessile drops under surface tension, applied external fields, and material variations. J. Appl. Phys. 93 (2003), 5794-5811.
51. SLIMANE, L., BENDALI, A., & LABORDE, P. Mixed formulations for a class of variational inequalities. Math. Modelling Numer. Anal. 38 (2004), 177-201. Zbl 1100.65059 MR 2073936.
52. TEMAM, R. Navier-Stokes Equations. AMS Chelsea Publ., Providence, RI (2001). Zbl 0568.35002 MR 0769654.
53. VERFÜRTH, R. Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition. Numer. Math. 50 (1987), 697-721. Zbl 0596.76031 MR 0884296.
54. WALKER, S. W. Modeling, simulating, and controlling the fluid dynamics of electro-wetting on dielectric. PhD thesis, Univ. of Maryland, College Park (2007).
55. WALKER, S. W., & SHAPIRO, B. A control method for steering individual particles inside liquid droplets actuated by electrowetting. Lab on a Chip 5 (2005), 1404-1407.
56. WALKER, S. W., & SHAPIRO, B. Modeling the fluid dynamics of electrowetting on dielectric (ewod). J. Microelectromech. Systems 15 (2006), 986-1000.
57. WALKER, S. W., SHAPIRO, B., & NOCHETTO, R. H. Electrowetting with contact line pinning: Computational modeling and comparisons with experiments. Phys. Fluids 21 (2009), no.10, 102103.
58. WHEELER, A. R., MOON, H., KIM, C.-J., LOO, J. A., & GARRELL, R. L. Electrowetting-based microfluidics for analysis of peptides and proteins by matrix-assisted laser desorption/ionization mass spectrometry. Analytical Chemistry 76 (2004), 4833-4838.
59. WOLFRAM, E., & FAUST, R. Wetting, Spreading and Adhesion. Chapter "Liquid Drops on a Tilted Plate, Contact Angle Hysteresis and the Young Contact Angle". Academic Press, Leicestershire (1978).