Elastocapillarity: when Surface Tension Deforms Elastic Solids

Although negligible at large scales, capillary forces may become dominant for submillimetric objects. Surface tension is usually associated with the spherical shape of small droplets and bubbles, wetting phenomena, imbibition, or the motion of insects at the surface of water. However, beyond liquid interfaces, capillary forces can also deform solid bodies in their bulk, as observed in recent experiments with very soft gels. Capillary interactions, which are responsible for the cohesion of sandcastles, can also bend slender structures and induce the bundling of arrays of fibers. Thin sheets can spontaneously wrap liquid droplets within the limit of the constraints dictated by differential geometry. This review aims to describe the different scaling parameters and characteristic lengths involved in elastocapillarity. We focus on three main configurations, each characterized by a specific dimension: three-dimensional (3D), deformations induced in bulk solids; 1D, bending and bundling of rod-like structures; ...

[1]  P. Král,et al.  Nanodroplet activated and guided folding of graphene nanostructures. , 2009, Nano letters.

[2]  M. Dunn,et al.  The role of van der Waals forces in adhesion of micromachined surfaces , 2005, Nature materials.

[3]  R. Legtenberg,et al.  Stiction in surface micromachining , 1996 .

[4]  Zhen Cao,et al.  Molecular Dynamics Simulations of the Effect of Elastocapillarity on Reinforcement of Soft Polymeric Materials by Liquid Inclusions , 2016 .

[5]  Robert W Style,et al.  Patterning droplets with durotaxis , 2013, Proceedings of the National Academy of Sciences.

[6]  C. Majidi,et al.  A simplified formulation of adhesion problems with elastic plates , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[7]  R. Hayward,et al.  Measuring the Elastic Modulus of Thin Polymer Sheets by Elastocapillary Bending. , 2015, ACS applied materials & interfaces.

[8]  D. Vella,et al.  Multiple equilibria in a simple elastocapillary system , 2012, Journal of Fluid Mechanics.

[9]  Joanna Aizenberg,et al.  Control of shape and size of nanopillar assembly by adhesion-mediated elastocapillary interaction. , 2010, ACS nano.

[10]  M. Shanahan.  The spreading dynamics of a liquid drop on a viscoelastic solid , 1988 .

[11]  R Stanley Williams,et al.  Gold nanofingers for molecule trapping and detection. , 2010, Journal of the American Chemical Society.

[12]  John R. Lister,et al.  A fluid-mechanical model of elastocapillary coalescence , 2013, Journal of Fluid Mechanics.

[13]  Chung-Yuen Hui,et al.  Effect of stamp deformation on the quality of microcontact printing: theory and experiment. , 2004, Langmuir : the ACS journal of surfaces and colloids.

[14]  P. Avouris,et al.  Ring Formation in Single-Wall Carbon Nanotubes , 1999 .

[15]  Thomas Salez,et al.  Elastocapillary bending of microfibers around liquid droplets. , 2016, Soft matter.

[16]  Robert W Style,et al.  Solid capillarity: when and how does surface tension deform soft solids? , 2015, Soft matter.

[17]  H. Stone,et al.  Dynamics of elastocapillary rise , 2010, Journal of Fluid Mechanics.

[18]  Howard A. Stone,et al.  Fluid-Structure Interactions in Low-Reynolds-Number Flows , 2015 .

[19]  Y. Pomeau,et al.  Solid drops: large capillary deformations of immersed elastic rods. , 2013, Physical review letters.

[20]  Y. Pomeau,et al.  Surface instability of soft solids under strain , 2011 .

[21]  Sébastien Neukirch,et al.  In-drop capillary spooling of spider capture thread inspires hybrid fibers with mixed solid–liquid mechanical properties , 2016, Proceedings of the National Academy of Sciences.

[22]  R. Fernandes,et al.  Self-folding polymeric containers for encapsulation and delivery of drugs. , 2012, Advanced drug delivery reviews.

[23]  George M. Whitesides,et al.  Making Things by Self-Assembly , 2005 .

[24]  Ho-Young Kim,et al.  Evaporation-driven clustering of microscale pillars and lamellae , 2016 .

[25]  A. Boudaoud,et al.  The liquid blister test , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  Jiawen Li,et al.  Laser printing hierarchical structures with the aid of controlled capillary-driven self-assembly , 2015, Proceedings of the National Academy of Sciences.

[27]  Nicholas D. Brubaker,et al.  Capillary-induced deformations of a thin elastic sheet , 2016, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  Elena Martínez,et al.  Stability of microfabricated high aspect ratio structures in poly(dimethylsiloxane). , 2005, Langmuir : the ACS journal of surfaces and colloids.

[29]  A. Hart,et al.  Precise control of elastocapillary densification of nanostructures via low-pressure condensation , 2014 .

[30]  L. Mahadevan,et al.  Capillary rise between elastic sheets , 2005, Journal of Fluid Mechanics.

[31]  M. D. De Volder,et al.  Engineering Hierarchical Nanostructures by Elastocapillary Self-assembly Angewandte Reviews , 2022 .

[32]  A. Thie,et al.  Fabrication and Biocompatibility of Carbon Nanotube-Based 3D Networks as Scaffolds for Cell Seeding and Growth , 2004 .

[33]  H. Stone,et al.  Capillary bridges between soft substrates. , 2014, Physical review letters.

[34]  Robert W Style,et al.  It's Harder to Splash on Soft Solids. , 2015, Physical review letters.

[35]  S. Tawfick,et al.  Three-dimensional lithography by elasto-capillary engineering of filamentary materials , 2016 .

[36]  Toyoichi Tanaka,et al.  Patterns in shrinking gels , 1992, Nature.

[37]  Alejandro Rico-Guevara,et al.  The hummingbird tongue is a fluid trap, not a capillary tube , 2011, Proceedings of the National Academy of Sciences.

[38]  H. Caps,et al.  Surface tension profiles in vertical soap films. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Xavier Chateau,et al.  Coupling of elasticity to capillarity in soft aerated materials. , 2013, Soft matter.

[40]  Miguel Pineirua,et al.  Capillary buckling of a floating annulus , 2013 .

[41]  Bernard Nysten,et al.  Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy , 2004 .

[42]  L. Abelmann,et al.  Let's twist again: elasto-capillary assembly of parallel ribbons. , 2016, Soft matter.

[43]  C. Liao,et al.  Densification effects of the carbon nanotube pillar array on field-emission properties , 2016 .

[44]  M. D. De Volder,et al.  Corrugated carbon nanotube microstructures with geometrically tunable compliance. , 2011, ACS nano.

[45]  P. Ajayan,et al.  Capillarity-driven assembly of two-dimensional cellular carbon nanotube foams , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[46]  Eric R Dufresne,et al.  Deformation of an elastic substrate by a three-phase contact line. , 2011, Physical review letters.

[47]  Huigao Duan,et al.  Directed self-assembly at the 10 nm scale by using capillary force-induced nanocohesion. , 2010, Nano letters.

[48]  J. Bico,et al.  Elasto-capillarity: deforming an elastic structure with a liquid droplet , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[49]  L. Mahadevan,et al.  Self-Organization of a Mesoscale Bristle into Ordered, Hierarchical Helical Assemblies , 2009, Science.

[50]  Paul B. Reverdy,et al.  Energy criteria for elasto-capillary wrapping , 2010 .

[51]  B. Andreotti,et al.  Elastic deformation due to tangential capillary forces , 2011, 1103.0782.

[52]  Y. Pomeau,et al.  Softening of edges of solids by surface tension , 2014, Journal of physics. Condensed matter : an Institute of Physics journal.

[53]  Anand Jagota,et al.  Surface-tension-induced flattening of a nearly plane elastic solid. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  K. Nagayama,et al.  Capillary interactions between particles bound to interfaces, liquid films and biomembranes. , 2000, Advances in colloid and interface science.

[55]  John S. Wettlaufer,et al.  Surface tension and contact with soft elastic solids , 2013, Nature Communications.

[56]  Yun Zhu,et al.  Influence of substrate elasticity on droplet impact dynamics. , 2013, Langmuir : the ACS journal of surfaces and colloids.

[57]  George M. Whitesides,et al.  Surface tension-powered self-assembly of microstructures - the state-of-the-art , 2003 .

[58]  Thomas Salez,et al.  From adhesion to wetting of a soft particle , 2013, 1309.1755.

[59]  L. Leibler,et al.  Peristaltic instability of cylindrical gels , 1996 .

[60]  Nathan B. Crane,et al.  Fluidic assembly at the microscale: progress and prospects , 2012, Microfluidics and Nanofluidics.

[61]  David H Gracias,et al.  Three-dimensional fabrication at small size scales. , 2010, Small.

[62]  A. Jagota,et al.  Elastocapillarity: Surface Tension and the Mechanics of Soft Solids , 2016, 1604.02052.

[63]  B. Davidovitch,et al.  Capillary deformations of bendable films. , 2013, Physical review letters.

[64]  Miguel Pineirua,et al.  Capillary origami controlled by an electric field , 2010 .

[65]  Eric Falcon,et al.  Instability of the origami of a ferrofluid drop in a magnetic field. , 2011, Physical review letters.

[66]  S. Dietrich,et al.  Complete wetting of elastically responsive substrates. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[67]  B. Davidovitch,et al.  Geometry-Driven Folding of a Floating Annular Sheet. , 2016, Physical review letters.

[68]  A. Boudaoud,et al.  Adhesion: Elastocapillary coalescence in wet hair , 2004, Nature.

[69]  A. I. Rusanov.  On the thermodynamics of deformable solid surfaces , 1978 .

[70]  B. Andreotti,et al.  Droplets move over viscoelastic substrates by surfing a ridge , 2014, Nature Communications.

[71]  Thomas P. Russell,et al.  A drop on a floating sheet: boundary conditions, topography and formation of wrinkles , 2013 .

[72]  J. Biggins,et al.  Finite-wavelength surface-tension-driven instabilities in soft solids, including instability in a cylindrical channel through an elastic solid. , 2015, Physical review. E.

[73]  Elastocapillary instability under partial wetting conditions: bending versus buckling. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[74]  E. Dufresne,et al.  Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses. , 2012, Physical review letters.

[75]  Camille Duprat,et al.  Capillary stretching of fibers , 2015 .

[76]  Zhigang Li,et al.  Critical droplet volume for spontaneous capillary wrapping , 2010 .

[77]  J. Olives.  Capillarity and elasticity. The example of the thin plate , 1993 .

[78]  J. Grotberg,et al.  BIOFLUID MECHANICS IN FLEXIBLE TUBES , 2001 .

[79]  A. Jagota,et al.  Solid surface tension measured by a liquid drop under a solid film , 2013, Proceedings of the National Academy of Sciences.

[80]  Nicholas D. Brubaker,et al.  Two-dimensional capillary origami , 2016 .

[81]  J. Lewis,et al.  Two- and three-dimensional folding of thin film single-crystalline silicon for photovoltaic power applications , 2009, Proceedings of the National Academy of Sciences.

[82]  Wei Lu,et al.  Diverse 3D Microarchitectures Made by Capillary Forming of Carbon Nanotubes , 2010, Advanced materials.

[83]  H. Stone,et al.  Wetting of flexible fibre arrays , 2012, Nature.

[84]  Gareth H. McKinley,et al.  Superhydrophobic Carbon Nanotube Forests , 2003 .

[85]  K. Dalnoki-Veress,et al.  Liquid Droplets on a Highly Deformable Membrane. , 2015, Physical review letters.

[86]  B. Davidovitch,et al.  Optimal wrapping of liquid droplets with ultrathin sheets. , 2015, Nature materials.

[87]  Lei Jiang,et al.  A bio-inspired flexible fiber array with an open radial geometry for highly efficient liquid transfer , 2014 .

[88]  L. Mahadevan,et al.  Kinks, rings, and rackets in filamentous structures , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[89]  Arezki Boudaoud,et al.  3D aggregation of wet fibers , 2007 .

[90]  Extrand,et al.  Contact Angles and Hysteresis on Soft Surfaces , 1996, Journal of colloid and interface science.

[91]  P. Ciarletta,et al.  Elastocapillarity can control the formation and the morphology of beads-on-string structures in solid fibers. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[92]  Dinesh Chandra,et al.  Capillary-force-induced clustering of micropillar arrays: is it caused by isolated capillary bridges or by the lateral capillary meniscus interaction force? , 2009, Langmuir : the ACS journal of surfaces and colloids.

[93]  J. Harting,et al.  Soft particles at a fluid interface. , 2015, Soft matter.

[94]  J. Berenschot,et al.  Controllable elastocapillary folding of three-dimensional micro-objects by through-wafer filling , 2014 .

[95]  Katia Bertoldi,et al.  Modeling of elasto-capillary phenomena. , 2014, Soft matter.

[96]  José Bico,et al.  Wrapping an adhesive sphere with an elastic sheet. , 2011, Physical review letters.

[97]  H. Butt,et al.  The softer the better: fast condensation on soft surfaces. , 2010, Langmuir : the ACS journal of surfaces and colloids.

[98]  Fritz Vollrath,et al.  Modulation of the mechanical properties of spider silk by coating with water , 1989, Nature.

[99]  Tianshu Liu,et al.  Interaction of Droplets Separated by an Elastic Film. , 2017, Langmuir : the ACS journal of surfaces and colloids.

[100]  Yiping Zhao,et al.  Clusters of bundled nanorods in nanocarpet effect , 2006 .

[101]  Sébastien Neukirch,et al.  Instant fabrication and selection of folded structures using drop impact , 2011, Proceedings of the National Academy of Sciences.

[102]  Hunter King,et al.  Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities , 2012, Proceedings of the National Academy of Sciences.

[103]  A. Hazel,et al.  Reopening modes of a collapsed elasto-rigid channel , 2016, Journal of Fluid Mechanics.

[104]  Chung-Yuen Hui,et al.  Adhesive contact of cylindrical lens and a flat sheet , 1996 .

[105]  Ty Phou,et al.  Capillarity driven instability of a soft solid. , 2010, Physical review letters.

[106]  M. L. Reed,et al.  Control of liquid bridging induced stiction of micromechanical structures , 1996 .

[107]  A. Sen,et al.  Capillary flow enhancement in rectangular polymer microchannels with a deformable wall. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[108]  Paul B. Reverdy,et al.  Capillary origami: spontaneous wrapping of a droplet with an elastic sheet. , 2006, Physical review letters.

[109]  Donald E Ingber,et al.  Quantifying cell-generated mechanical forces within living embryonic tissues , 2013, Nature Methods.

[110]  A. Boudaoud,et al.  Elastocapillary coalescence: aggregation and fragmentation with a maximal size. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[111]  Brian L. Wardle,et al.  Fabrication of composite microstructures by capillarity-driven wetting of aligned carbon nanotubes with polymers , 2007 .

[112]  B. Andreotti,et al.  Capillarity of soft amorphous solids: a microscopic model for surface stress. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[113]  Benny Davidovitch,et al.  Prototypical model for tensional wrinkling in thin sheets , 2011, Proceedings of the National Academy of Sciences.

[114]  Todd Emrick,et al.  Capillary Wrinkling of Floating Thin Polymer Films , 2007, Science.

[115]  S. Neukirch,et al.  Instabilities in a drop-strip system: a simplified model , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[116]  L. Mahadevan,et al.  Continuum dynamics of elastocapillary coalescence and arrest , 2014 .

[117]  A. Hazel,et al.  Three-dimensional airway reopening: the steady propagation of a semi-infinite bubble into a buckled elastic tube , 2003, Journal of Fluid Mechanics.

[118]  A. Jagota,et al.  Flattening of a patterned compliant solid by surface stress. , 2014, Soft matter.

[119]  P. Gennes,et al.  Capillarity and Wetting Phenomena , 2004 .

[120]  Z. Suo,et al.  Surface energy as a barrier to creasing of elastomer films: an elastic analogy to classical nucleation. , 2012, Physical review letters.

[121]  B. Andreotti,et al.  Contact angles on a soft solid: from Young's law to Neumann's law. , 2012, Physical review letters.

[122]  José Bico,et al.  Capillary rise between flexible walls , 2011 .

[123]  Ludwik Leibler,et al.  Static and Dynamic Wetting Properties of Thin Rubber Films , 1996 .

[124]  Kiran Singh,et al.  Evaporation effects in elastocapillary aggregation , 2015, Journal of Fluid Mechanics.

[125]  Jean-Philippe M. Péraud,et al.  Geometry and wetting of capillary folding. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[126]  Hans-Jürgen Butt,et al.  Effect of capillary pressure and surface tension on the deformation of elastic surfaces by sessile liquid microdrops: an experimental investigation. , 2008, Langmuir : the ACS journal of surfaces and colloids.

[127]  Eitan Grinspun,et al.  Supplemental : A Multi-Scale Model for Simulating Liquid-Hair Interactions , 2017 .

[128]  H. Butt,et al.  Microdrops on atomic force microscope cantilevers: evaporation of water and spring constant calibration. , 2005, The journal of physical chemistry. B.

[129]  J. Snoeijer.  Analogies between elastic and capillary interfaces , 2016 .

[130]  Chung-Yuen Hui,et al.  Constraints on Microcontact Printing Imposed by Stamp Deformation , 2002 .

[131]  W. Federle,et al.  Elasto-capillarity in insect fibrillar adhesion , 2016, Journal of The Royal Society Interface.

[132]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[133]  Eric R. Dufresne,et al.  Static wetting on deformable substrates, from liquids to soft solids , 2012, 1203.1654.

[134]  B. Andreotti,et al.  Soft wetting and the Shuttleworth effect, at the crossroads between thermodynamics and mechanics , 2016, 1909.00463.

[135]  L. G. Leal,et al.  Microfabricated deflection tensiometers for insoluble surfactants , 2010 .

[136]  E. Dufresne,et al.  Surface tension and the mechanics of liquid inclusions in compliant solids. , 2014, Soft matter.

[137]  B. Davidovitch,et al.  Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets , 2016, Proceedings of the National Academy of Sciences.

[138]  Benny Davidovitch,et al.  Smooth cascade of wrinkles at the edge of a floating elastic film. , 2009, Physical review letters.

[139]  L. Botto,et al.  Liquid drops attract or repel by the inverted Cheerios effect , 2016, Proceedings of the National Academy of Sciences.

[140]  J. Ashley Taylor,et al.  Bio-inspired artificial iridophores based on capillary origami: Fabrication and device characterization , 2011 .

[141]  David H Gracias,et al.  Hierarchical self-assembly of complex polyhedral microcontainers. , 2009, Journal of micromechanics and microengineering : structures, devices, and systems.

[142]  J. Bico,et al.  Piercing an interface with a brush: Collaborative stiffening , 2010, 1006.2116.

[143]  M. Shanahan.  The influence of solid micro-deformation on contact angle equilibrium , 1987 .

[144]  R. Morris,et al.  Capillary origami and superhydrophobic membrane surfaces , 2013 .

[145]  C Van Hoof,et al.  Self-assembly from milli- to nanoscales: methods and applications , 2009, Journal of micromechanics and microengineering : structures, devices, and systems.

[146]  P. Müller,et al.  Deformation of an elastic thin solid induced by a liquid droplet , 1992 .

[147]  Sébastien Neukirch,et al.  Piercing a liquid surface with an elastic rod: Buckling under capillary forces , 2007 .

[148]  R. Howe,et al.  Critical Review: Adhesion in surface micromechanical structures , 1997 .

[149]  M. M. Nicholson.  Surface tension in ionic crystals , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[150]  G. R. Lester.  Contact angles of liquids at deformable solid surfaces , 1961 .

[151]  Martin E. R. Shanahan,et al.  Viscoelastic effects in the spreading of liquids , 1996, Nature.

[152]  Jonathan T. Pham,et al.  Highly Stretchable Nanoparticle Helices Through Geometric Asymmetry and Surface Forces , 2013, Advanced materials.

[153]  Jeong-Hyun Cho,et al.  Self-assembly of lithographically patterned nanoparticles. , 2009, Nano letters.

[154]  Sébastien Neukirch,et al.  Elastocapillary snapping: capillarity induces snap-through instabilities in small elastic beams. , 2013, Physical review letters.

[155]  Joost H. Weijs,et al.  Elasto-capillarity at the nanoscale: on the coupling between elasticity and surface energy in soft solids , 2013, 1303.6779.

[156]  Shu Yang,et al.  Stability of high-aspect-ratio micropillar arrays against adhesive and capillary forces. , 2010, Accounts of chemical research.

[157]  Laurent Limat,et al.  Straight contact lines on a soft, incompressible solid , 2012, The European physical journal. E, Soft matter.

[158]  Tetsuharu Narita,et al.  Advancing liquid contact line on visco-elastic gel substrates: stick-slip vs. continuous motions , 2013 .