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Multiple equilibria in a simple elastocapillary system

Published online by Cambridge University Press:  28 September 2012

Michele Taroni  and
Dominic Vella
Michele Taroni
Affiliation:
Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, UK
Dominic Vella*
Affiliation:
Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK
*
†Email address for correspondence: dominic.vella@cantab.net
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Abstract

We consider the elastocapillary interaction of a liquid drop placed between two elastic beams, which are both clamped at one end to a rigid substrate. This is a simple model system relevant to the problem of surface-tension-induced collapse of flexible micro-channels that has been observed in the manufacture of microelectromechanical systems (MEMS). We determine the conditions under which the beams remain separated, touch at a point, or stick along a portion of their length. Surprisingly, we show that in many circumstances multiple equilibrium states are possible. We develop a lubrication-type model for the flow of liquid out of equilibrium and thereby investigate the stability of the multiple equilibria. We demonstrate that for given material properties two stable equilibria may exist, and show via numerical solutions of the dynamic model that it is the initial state of the system that determines which stable equilibrium is ultimately reached.

JFM classification

Interfacial Flows (free surface): Capillary flows Low-Reynolds-number flows: Lubrication theory Micro-/Nano-fluid dynamics: MEMS/NEMS
Type
Papers
Information
Journal of Fluid Mechanics , Volume 712 , 10 December 2012 , pp. 273 - 294
Copyright
©2012 Cambridge University Press

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References

Andreotti, B., Marchand, A., Das, S. & Snoeijer, J. H. 2011 Elastocapillary instability under partial wetting conditions: bending versus buckling. Phys. Rev. E 84, 061601.CrossRefGoogle ScholarPubMed
Aristoff, J. M., Duprat, C. & Stone, H. A. 2011 Elastocapillary imbibition. Intl J. Non Linear Mech. 46 (4), 648–656.Google Scholar
Berkowski, K. L., Plunkett, K. N., Yu, Q. & Moore, J. S. 2005 Introduction to photolithography: preparation of microscale polymer silhouettes. J. Chem. Ed. 82, 1365–1369.Google Scholar
Bico, J., Roman, B., Moulin, L. & Boudaoud, A. 2004 Elastocapillary coalescence in wet hair. Nature 432, 690.Google Scholar
Boudaoud, A., Bico, J. & Roman, B. 2007 Elastocapillary coalescence: aggregation and fragmentation with a maximal size. Phys. Rev. E 76, 060102(R).CrossRefGoogle ScholarPubMed
Chandra, D. & Yang, S. 2009 Capillary-force-induced clustering of micropillar arrays: is it caused by isolated capillary bridges or by the lateral capillary meniscus interaction force? Langmuir 25 (18), 10430–10434.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle Scholar
Das, S., Marchand, A., Andreotti, B. & Snoeijer, J. H. 2011 Elastic deformation due to tangential capillary forces. Phys. Fluids 23, 072006.Google Scholar
Delamarche, E., Schmid, H., Michel, B. & Biebuyck, H. 1997 Stability of molded polydimethylsiloxane microstructures. Adv. Mater. 9, 741–746.Google Scholar
Duprat, C., Aristoff, J. M. & Stone, H. A. 2011 Dynamics of elastocapillary rise. J. Fluid Mech. 679, 641–654.Google Scholar
Farshid Chini, S. & Amirfazli, A. 2010 Understanding pattern collapse in photolithography process due to capillary forces. Langmuir 26 (16), 13707–13714.CrossRefGoogle Scholar
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2003 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Halpern, D. & Grotberg, J. B. 1992 Fluid-elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244, 615–632.Google Scholar
van Honschoten, J. W., Escalante, M., Tas, N. R., Jansen, H. V. & Elwenspoek, M. 2007 Elastocapillary filling of deformable nanochannels. J. Appl. Phys. 101, 094310.CrossRefGoogle Scholar
Huang, J., Juszkiewicz, M., de Jeu, W. H., Cerda, E., Emrick, T., Menon, N. & Russell, T. P. 2007 Capillary wrinkling of floating thin polymer films. Science 317, 650–653.Google Scholar
Hure, J., Roman, B. & Bico, J. 2011 Wrapping an adhesive sphere with an elastic sheet. Phys. Rev. Lett. 106, 174301.CrossRefGoogle ScholarPubMed
Jensen, O. E. 1997 The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 331, 373–403.Google Scholar
Kim, H.-Y. & Mahadevan, L. 2006 Capillary rise between elastic sheets. J. Fluid Mech. 548, 141–150.Google Scholar
Kwon, H.-M., Kim, H.-Y., Puell, J. & Mahadevan, L. 2008 Equilibrium of an elastically confined liquid drop. J. Appl. Phys. 103, 093519.Google Scholar
Landau, L. D. & Lifschitz, E. M. 1959 Theory of Elasticity. Pergamon.Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena. Cambridge University Press.Google Scholar
Lee, H.-J., Park, J.-T., Yoo, J., An, I. & Oh, H.-K. 2002 Resist pattern collapse with top rounding resist profile. Japan J. Appl. Phys. 42, 3922–3927.Google Scholar
Majidi, C. 2007 Remarks on formulating an adhesion problem using Euler’s elastica. Mech. Res. Commun. 34, 85–90.Google Scholar
Mansfield, E. H., Sepangi, H. R. & Eastwood, E. A. 1997 Equilibrium and mutual attraction or repulsion of objects supported by surface tension. Phil. Trans. R. Soc. Lond. A 355, 869–919.Google Scholar
Marchand, A., Das, S., Snoeijer, J. H. & Andreotti, B. 2012 Capillary pressure and contact line force on a soft solid. Phys. Rev. Lett. 108, 094301.Google Scholar
Mastrangelo, C. H & Hsu, H. 1993a Mechanical stability and adhesion of microstructures under capillary forces. Part 1. Basic theory. J. Microelectromech. Syst. 2, 33–43.Google Scholar
Mastrangelo, C. H & Hsu, H. 1993b Mechanical stability and adhesion of microstructures under capillary forces. Part 2. Experiments. J. Microelectromech. Syst. 2, 44–55.Google Scholar
Matar, O. K. & Kumar, S. 2004 Rupture of a surfactant-covered thin liquid film on a flexible wall. SIAM J. Appl. Math. 64, 2144–2166.Google Scholar
Pokroy, B., Kang, S. G., Mahadevan, L. & Aizenberg, J. 2009 Self-organization of a mesoscale bristle into ordered, hierarchical helical assemblies. Science 323, 237–240.Google Scholar
Py, C., Reverdy, P., Doppler, L., Bico, J., Roman, B. & Baroud, C. N. 2007 Capillary origami: spontaneous wrapping of a droplet with an elastic sheet. Phys. Rev. Lett. 98, 156103.CrossRefGoogle ScholarPubMed
Raccurt, O., Tardif, F., Arnaud d’Avitaya, F. & Vareine, T. 2004 Influence of liquid surface tension on stiction of SOI MEMS. J. Micromech. Microengng 14, 1083–1090.Google Scholar
Roman, B. & Bico, J. 2010 Elasto-capillarity: deforming an elastic structure with a liquid droplet. J. Phys.: Condens Mater 22, 493101.Google Scholar
Roy, R. V. & Schwartz, L. W. 1999 On the stability of liquid ridges. J. Fluid Mech. 391, 293–318.Google Scholar
Shampine, L. F. 2007 Accurate numerical derivatives in MATLAB. ACM Trans. Math. Softw. 33, 26.Google Scholar
Shanahan, M. E. R. 1985 Contact angle equilibrium on thin elastic solids. J. Adhes. 18, 247–267.Google Scholar
van Spengen, W. M., Puers, R. & de Wolf, I. 2002 A physical model to predict stiction in MEMS. J. Micromech. Microengng 12, 702–713.Google Scholar
Tanaka, T., Morigami, M. & Atoda, N. 1993 Mechanism of resist pattern collapse during development process. Japan J. Appl. Phys. 31, 6059–6064.Google Scholar
Vella, D., Adda-Bedia, M. & Cerda, E. 2010 Capillary wrinkling of elastic membranes. Soft Matt. 6, 5778–5782.CrossRefGoogle Scholar
Weinstein, S. J., Dussan, V. E. B. & Ungar, L. H. 1990 A theoretical study of two-phase flow through a narrow gap with a moving contact line: viscous fingering in a Hele-Shaw cell. J. Fluid Mech. 221, 53–76.Google Scholar
Xia, Y. & Whitesides, G. M. 1998 Soft lithography. Annu. Rev. Mater. Sci. 28, 153–184.Google Scholar
Young, T. 1805 An essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. 95, 65–87.Google Scholar