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Numerical multiscale modelling of nonlinear elastowetting. (English) Zbl 1406.74100
Summary: We investigate here the static finite deformation of two (or more) three-dimensional nonlinear elastic solids which merge in a third medium, defining a triple line (TL). The total energy, accounting for elastic, surface and possible gravity potentials is then minimized numerically in order to solve two physical problems of interest: (i) a soft incompressible axisymmetric drop at rest on a stiffer substrate and (ii) a stiff drop at rest upon a softer substrate, both in situations of strong substrate surface tension effects. Because of the interplay between surface and bulk properties, the behavior of the TL is related to another smaller length-scale (typically micrometre scale) than the elastic solids length-scale (typically millimetre scale). A unified deformability parameter relating all scales is introduced and an adaptive finite element discretization of the domain is used.
A first example approached by our model is the elasto-wetting problem in which a very soft drop sits on a stiffer elastic solid. The numerical predictions of our model are compared with the analytical results of the linear elastic theory. In particular, our computations show that Laplace pressure, together with the asymmetry of the solid surface tensions (between the wet and dry part of the interface), well account for the rotation of the cusp of the ridge.
In the second example, motivated by indentation problems of a nonlinear elastic substrate by a stiff sphere, we analyse the interplay of the solid surface tensions and elasticity at small-scale. The numerical results also show that gravity might have an important effect and must be included in any micro-scale modelling.
74B20 Nonlinear elasticity
74S05 Finite element methods applied to problems in solid mechanics
Full Text: DOI
[1] Ben Amar, M.; Ciarletta, P., Swelling instability of surface-attached gels as a model of soft tissue growth under geometric constraints, J. Mech. Phys. Solid., 58, 935-954, (2010) · Zbl 1244.74055
[2] Boland, T.; Xu, T.; Damon, B.; Cui, X., Application of inkjet printing to tissue engineering, Biotechnol. J., 1, 910-917, (2006)
[3] Bostwick, J. B.; Shearer, M.; Daniels, K. E., Elastocapillary deformations on partially-wetting substrates: rival contact-line models, Soft Matter, 10, 7361-7369, (2014)
[4] Cao, Z.; Stevens, M. J.; Dobrynin, A. V., Adhesion and wetting of nanoparticles on soft surfaces, Macromolecules, 47, 3203-3209, (2014)
[5] Chung, S.e.a., Inkjet-printed stretchable silver electrode on wave structured elastomeric substrate, Appl. Phys. Lett., 98, 2011-2014, (2011)
[6] Ciarletta, P., Surface instability of a gel disc in swelling, Eur. Phys. J. E, 36, 1-4, (2013)
[7] De Gennes, P. G.; Brochard-Wyart, F.; Quéré, D., Capillarity and wetting phenomena: drops, bubbles, pearls, waves, (2013), Springer Science & Business Media · Zbl 1139.76004
[8] Dervaux, J.; Ben Amar, M., Buckling condensation in constrained growth, J. Mech. Phys. Solid., 59, 538-560, (2011) · Zbl 1270.74086
[9] Dervaux, J.; Couder, Y.; Guedeau-Boudeville, M. A.; Ben Amar, M., Shape transition in artificial tumors: from smooth buckles to singular creases, Phys. Rev. Lett., 107, 018103, (2011)
[10] Dervaux, J.; Limat, L., Contact lines on soft solids with uniform surface tension: analytical solutions and double transition for increasing deformability, (Proc. R. Soc. A, (2015), The Royal Society), 20140813 · Zbl 1371.74087
[11] Jerison, E. R.; Xu, Y.; Wilen, L. A.; Dufresne, E. R., Deformation of an elastic substrate by a three-phase contact line, Phys. Rev. Lett., 106, 186103, (2011)
[12] Le Tallec, P., Numerical methods for nonlinear three-dimensional elasticity, Handb. Numer. Anal., 3, 465-622, (1994) · Zbl 0875.73234
[13] Lester, G., Contact angles of liquids at deformable solid surfaces, J. Colloid Sci., 16, 315-326, (1961)
[14] Limat, L., Straight contact lines on a soft, incompressible solid, Eur. Phys. J. E, 35, 1-13, (2012)
[15] Lubarda, V. A., Mechanics of a liquid drop deposited on a solid substrate, Soft Matter, 8, 10288-10297, (2012)
[16] Lubbers, L. A.; Weijs, J. H.; Botto, L.; Das, S.; Andreotti, B.; Snoeijer, J. H., Drops on soft solids: free energy and double transition of contact angles, J. Fluid Mech., 747, R1, (2014) · Zbl 1371.76146
[17] Madasu, S.; Cairncross, R. A., Static wetting on flexible substrates: a finite element formulation, Int. J. Numer. Meth. Fluid., 45, 301-319, (2004) · Zbl 1079.76582
[18] Marchand, A.; Das, S.; Snoeijer, J. H.; Andreotti, B., Capillary pressure and contact line force on a soft solid, Phys. Rev. Lett., 108, 094301, (2012)
[19] Marchand, A.; Das, S.; Snoeijer, J. H.; Andreotti, B., Contact angles on a soft solid: from youngs law to neumanns law, Phys. Rev. Lett., 109, 236101, (2012)
[20] Mora, S.; Abkarian, M.; Tabuteau, H.; Pomeau, Y., Surface instability of soft solids under strain, Soft Matter, 7, 10612-10619, (2011)
[21] Mora, S.; Maurini, C.; Phou, T.; Fromental, J. M.; Audoly, B.; Pomeau, Y., Solid drops: large capillary deformations of immersed elastic rods, Phys. Rev. Lett., 111, 114301, (2013)
[22] Mora, S.; Phou, T.; Fromental, J. M.; Pismen, L. M.; Pomeau, Y., Capillarity driven instability of a soft solid, Phys. Rev. Lett., 105, 214301, (2010)
[23] Mora, S.; Pomeau, Y., Softening of edges of solids by surface tension, J. Phys. Condens. Matter, 27, 194112, (2015)
[24] Nadermann, N.; Hui, C. Y.; Jagota, A., Solid surface tension measured by a liquid drop under a solid film, Proc. Natl. Acad. Sci. Unit. States Am., 110, 10541-10545, (2013)
[25] Park, S. J.; Weon, B. M.; San Lee, J.; Lee, J.; Kim, J.; Je, J. H., Visualization of asymmetric wetting ridges on soft solids with x-ray microscopy, Nat. Commun., 5, (2014)
[26] Pericet-Camara, R.; Auernhammer, G. K.; Koynov, K.; Lorenzoni, S.; Raiteri, R.; Bonaccurso, E., Solid-supported thin elastomer films deformed by microdrops, Soft Matter, 5, 3611-3617, (2009)
[27] Roman, B.; Bico, J., Elasto-capillarity: deforming an elastic structure with a liquid droplet, J. Phys. Condens. Matter, 22, 493101, (2010)
[28] Rusanov, A., Theory of wetting of elastically deformed bodies. 1. deformation with a finite contact-angle, Colloid J. USSR, 37, 614-622, (1975)
[29] Sauer, R. A., Stabilized finite element formulations for liquid membranes and their application to droplet contact, Int. J. Numer. Meth. Fluid., 75, 519-545, (2014)
[30] Sauer, R. A., A contact theory for surface tension driven systems, Math. Mech. Solid, 21, 305-325, (2016) · Zbl 1370.74121
[31] Shanahan, M., The influence of solid micro-deformation on contact angle equilibrium, J. Phys. Appl. Phys., 20, 945, (1987)
[32] Shanahan, M.; De Gennes, P., Equilibrium of the triple line solid/liquid/fluid of a sessile drop, (Adhesion, vol. 11, (1987), Springer), 71-81
[33] Shanahan, M. E., Equilibrium of liquid drops on thin plates; plate rigidity and stability considerations, J. Adhes., 20, 261-274, (1987)
[34] Shuttleworth, R., The surface tension of solids, Proc. Phys. Soc. A, 63, 444, (1950)
[35] Sokuler, M.e.a., The softer the better: fast condensation on soft surfaces, Langmuir, 26, 1544-1547, (2010)
[36] Steigmann, D.; Ogden, R., Plane deformations of elastic solids with intrinsic boundary elasticity, (Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, (1997), The Royal Society), 853-877 · Zbl 0938.74014
[37] Style, R. W.; Boltyanskiy, R.; Che, Y.; Wettlaufer, J.; Wilen, L. A.; Dufresne, E. R., Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses, Phys. Rev. Lett., 110, 066103, (2013)
[38] Style, R. W.; Dufresne, E. R., Static wetting on deformable substrates, from liquids to soft solids, Soft Matter, 8, 7177-7184, (2012)
[39] Style, R. W.; Hyland, C.; Boltyanskiy, R.; Wettlaufer, J. S.; Dufresne, E. R., Surface tension and contact with soft elastic solids, Nat. Commun., 4, (2013)
[40] Taffetani, M.; Ciarletta, P., Beading instability in soft cylindrical gels with capillary energy: weakly non-linear analysis and numerical simulations, J. Mech. Phys. Solid., 81, 91-120, (2015)
[41] Unger, M. A.; Chou, H. P.; Thorsen, T.; Scherer, A.; Quake, S. R., Monolithic microfabricated valves and pumps by multilayer soft lithography, Science, 80, 113-116, (2000)
[42] Weijs, J. H.; Andreotti, B.; Snoeijer, J. H., Elasto-capillarity at the nanoscale: on the coupling between elasticity and surface energy in soft solids, Soft Matter, 9, 8494-8503, (2013)
[43] White, L. R., The contact angle on an elastic substrate. 1. the role of disjoining pressure in the surface mechanics, J. Colloid Interface Sci., 258, 82-96, (2003)
[44] Xu, X.; Jagota, A.; Hui, C. Y., Effects of surface tension on the adhesive contact of a rigid sphere to a compliant substrate, Soft Matter, 10, 4625-4632, (2014)
[45] Xu, X.; Jagota, A.; Peng, S.; Luo, D.; Wu, M.; Hui, C. Y., Gravity and surface tension effects on the shape change of soft materials, Langmuir, 29, 8665-8674, (2013)
[46] Yu, Y.s., Substrate elastic deformation due to vertical component of liquid-vapor interfacial tension, Appl. Math. Mech., 33, 1095-1114, (2012)
[47] Yu, Y. S.; Zhao, Y. P., Deformation of pdms membrane and microcantilever by a water droplet: comparison between Mooney-Rivlin and linear elastic constitutive models, J. Colloid Interface Sci., 332, 467-476, (2009)
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