COMMENTARY COMMENTARY
Role reversal: Liquid “Cheerios” on a solid sense each other Anand Jagotaa,1
The Cheerios effect (1) refers to the common observation that floating objects (say, little toruses of Cheerios cereal floating on water or milk) tend to agglomerate either with each other or to the wall that contains the liquid. Floating objects distort the liquid–vapor interface due to a combination of wetting and gravity forces. The interfacial deflection due to one particle serves as a gravitational potential energy landscape for others, setting up interesting and sometimes counterintuitive long-range attraction that is potentially useful to direct interfacial self-assembly (2). In PNAS, Karpitschka et al. (3) report interesting experiments and theory in which the roles of the liquid and solid are reversed; that is, they show that liquid drops on a compliant solid surface experience attractive or repulsive interactions, a “reverse Cheerios effect.”
Local Shape at Contact Line Determined by Neumann’s Triangle
A
B
Net horizontal force self-balanced
Liquid Solid Neumann’s Triangle Rotated Due to Other Drop
C
D
Unbalanced Net Horizontal Force Drives Moon
Fig. 1. (A) Surface tension of a liquid drop on a compliant substrate causes significant deformation. The local shape at the contact line (i.e., the angles at which the three interfaces meet) is determined by balance of surface stresses, which is Neumann’s triangle. (B) Diagram of an isolated liquid drop, cutting inside the solid at the lower interface and outside the liquid at the outer interface (Laplace pressure not shown), showing that the net horizontal force is balanced. (C) When two drops approach each other, their surface shape changes, especially in the region between the two drops. (D) This shape change leads to an unbalanced in-plane force in response to which the drops move toward each other.
The Cheerios effect is an example from a class of phenomena driven by liquid–vapor surface tension interacting with solid objects. [The very fact of liquid surface tension has fascinated researchers over a long period (4, 5).] Generally, liquid contacting a solid object can wet it completely, not at all, or partially, in the last case forming an equilibrium contact angle that is governed by the surface energies of the liquid–vapor interface and the two solid–fluid interfaces. Usually, the solid in question is sufficiently stiff that its deformation plays a negligible role. On the other hand, when liquids interact with compliant structures (6), the deformability of the latter can result in large deformations of slender objects: capillary origami (7). When the solid is deformable, its own surface stress can play an important and sometimes dominant role in many types of mechanical deformations. Collectively, these phenomena are being referred to as elastocapillarity (8, 9). Two important (static) problems that involve a contact line are wetting of a compliant solid by drops (two fluids, one solid) and adhesive contact between a particle and a compliant solid (two solids, one fluid). In the contact problem, for small particles, the resistance to deformation is dominated by the solid surface stress instead of bulk elasticity, a breakdown of the standard Johnson–Kendall–Roberts theory of adhesive contact (10–12). In the wetting problem, the liquid–vapor surface tension of a drop can strongly deform the solid surface, and the local shape is often governed by balance of surface stresses (13–17). Elastocapillarity also strongly affects dynamic phenomena, such as durotaxis (drop motion due to gradients in substrate compliance) (18) and contact line motion (19). A common thread running through all these elastocapillary phenomena is that when the solid becomes sufficiently compliant, it behaves quite like a liquid in some ways, in that the resistance to deformation due to bulk elasticity weakens, elevating the role of the solid surface stress. This conflation of liquid- and solid-like character raises many interesting questions about what would happen to known liquid capillarity effects if one were to replace the liquid by a compliant solid.
a
Department of Chemical and Biomolecular Engineering and Bioengineering Program, Lehigh University, Bethlehem, PA 18015 Author contributions: A.J. wrote the paper. The author declares no conflict of interest. See companion article on page 7403. 1 Email: [email protected].
7294–7295 | PNAS | July 5, 2016 | vol. 113 | no. 27
www.pnas.org/cgi/doi/10.1073/pnas.1607893113
Chakrabarti and Chaudhury (20) discovered a new version of the Cheerios effect in which particles placed on a very compliant, but still solid, gel are completely engulfed and come to rest at a vertical position governed by the interplay of gravity, solid elasticity, and surface stress. The deformation field is quite longranged and drives attractive interactions between particles as well as overall motion when gel thickness is graded. In related experiments, it has been shown that cylindrical particles placed on the surface of a compliant gel interact through an effective gravitational potential setup by the elastocapillary deformations of the gel driven by the cylinder’s weight, an “elastic Cheerios effect” (21). Karpitschka et al. (3) complete the role-reversal, reporting experiments and theory on the interaction between liquid drops (ethylene glycol) on a compliant solid surface (a polydimethylsiloxane gel). Because, here, the roles of the solid and liquid are reversed, the authors call it the “inverted Cheerios effect.” The solid surface is now the substrate on which liquid drops are placed. It is sufficiently compliant to be deformed significantly by surface stresses of the liquid–vapor and solid–fluid interfaces, which sets up interactions between drops when more than one is present. By conducting the experiment on drops sliding down a vertical surface, the authors eliminate the effect of gravity, demonstrating that the interaction between drops is governed by elastocapillary deformation of the solid substrate. Using clever in situ calibration of the relationship between the vertical drop velocity and the applied gravitational force, the authors are able to quantify the effective force driving drops together or apart, finding both attraction (thick substrates) and repulsion (thin substrates) between two liquid drops. What drives the motion of the two drops? Fig. 1A shows schematically the shapes of the three interfaces (liquid–vapor, liquid– solid, and solid–vapor) in the vicinity of an isolated drop. In particular, if the solid is compliant, the local shape at the contact line (i.e., the relative angles of the three surface tangents) is determined by balance of the three interfacial stresses, which is Neumann’s triangle. Fig. 1B shows the drop, cut out from the rest of the body, and it is
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
clear by symmetry that the horizontal forces will all balance out so that the drop is happily at equilibrium. Now consider Fig. 1C in which two drops approach sufficiently close that the regions where each one would, by itself, distort the surface profile now overlap. Each drop can be thought of, approximately, as “living” now on a surface that is no longer quite planar but inclined due to the presence of the other drop. Local equilibrium is still maintained by Neumann’s triangle, which determines only the relative angles; the force diagram, as a whole, is free to rotate in response to the other drop. This rotation of the forces now introduces an asymmetry so that, as Fig. 1D shows, there is a net unbalanced horizontal force. In this case, it is an attractive one that drives the drops toward each other. [This argument is slightly different from the argument made by Karpitschka et al. (3), but the net result is the same.] Now, in practice, drop movement is quite slow and quasistatic so that each drop is always in mechanical equilibrium. That is, once drop movement begins, it generates viscous drag, which provides the necessary force at the contact line to balance forces. Why is the interaction repulsive when the substrate is relatively thin? The argument just presented is based on a deformed surface shape that corresponds to a very thick elastic substrate. If the substrate is thin and incompressible, volume conservation demands that a local uplift of the surface be matched by a dip. Coming out of the dip, the slope of the deformed surface is now of opposite sign compared with its slope in Fig. 1. This change in sign of the slope translates into a change in sign of the interaction, rendering it repulsive at larger distances. In broader terms, the effect discovered by Karpitschka et al. (3) is an example of new, sometimes counterintuitive, phenomena emerging from the study of the deformation in soft materials and the significant role played by interfaces and interfacial mechanical properties.
Acknowledgments This work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG0207ER46463.
Vella D, Mahadevan L (2005) The “Cheerios effect”. Am J Phys 73(9):817–825. Furst EM (2011) Directing colloidal assembly at fluid interfaces. Proc Natl Acad Sci USA 108(52):20853–20854. Karpitschka S, et al. (2016) Liquid drops attract or repel by the inverted Cheerios effect. Proc Natl Acad Sci USA 113:7403–7407. deGennes P-G, Brochard-Wyart F, Quere D (2002) Capillarity and Wetting Phenomena. Drops, Bubbles, Pearls, Waves (Springer Science + Business Media, Inc., New York). Rowlinson JS, Widom B (1989) Molecular Theory of Capillarity (Oxford Univ Press, New York). Roman B, Bico J (2010) Elasto-capillarity: Deforming an elastic structure with a liquid droplet. J Phys Condens Matter 22(49):493101. Py C, et al. (2007) Capillary origami: Spontaneous wrapping of a droplet with an elastic sheet. Phys Rev Lett 98(15):156103. Style RW, Jagota A, Hui C-Y, Dufresne ER (2016) Elastocapillarity: Surface tension and the mechanics of soft solids. arXiv:1604.02052. Andreotti B, et al. (2016) Solid capillarity: When and how does surface tension deform soft solids? Soft Matter 12(12):2993–2996. Style RW, Hyland C, Boltyanskiy R, Wettlaufer JS, Dufresne ER (2013) Surface tension and contact with soft elastic solids. Nat Commun 4:2728. Carrillo J-MY, Dobrynin AV (2012) Contact mechanics of nanoparticles. Langmuir 28(29):10881–10890. Hui C-Y, Liu T, Salez T, Raphael E, Jagota A (2015) Indentation of a rigid sphere into an elastic substrate with surface tension and adhesion. Proc Math Phys Eng Sci 471(2175):20140727. Jerison ER, Xu Y, Wilen LA, Dufresne ER (2011) Deformation of an elastic substrate by a three-phase contact line. Phys Rev Lett 106(18):186103. Marchand A, Das S, Snoeijer JH, Andreotti B (2012) Contact angles on a soft solid: From Young’s law to Neumann’s law. Phys Rev Lett 109(23):236101. Hui C-Y, Jagota A (2014) Deformation near a liquid contact line on an elastic substrate. Proc Math Phys Eng Sci 470(2167):20140085. Cao Z, Dobrynin AV (2015) Polymeric droplets on soft surfaces: From Neumann’s triangle to Young’s law. Macromolecules 48(2):443–451. Nadermann N, Hui C-Y, Jagota A (2013) Solid surface tension measured by a liquid drop under a solid film. Proc Natl Acad Sci USA 110(26):10541–10545. Style RW, et al. (2013) Patterning droplets with durotaxis. Proc Natl Acad Sci USA 110(31):12541–12544. Karpitschka S, et al. (2015) Droplets move over viscoelastic substrates by surfing a ridge. Nat Commun 6:7891. Chakrabarti A, Chaudhury MK (2013) Surface folding-induced attraction and motion of particles in a soft elastic gel: Cooperative effects of surface tension, elasticity, and gravity. Langmuir 29(50):15543–15550. Chakrabarti A, Ryan L, Chaudhury MK, Mahadevan L (2015) Elastic Cheerios effect: Self-assembly of cylinders on a soft solid. Europhys Lett 112(5):54001–54005.
Jagota
PNAS | July 5, 2016 | vol. 113 | no. 27 | 7295
Role reversal: Liquid “Cheerios” on a solid sense each other Anand Jagotaa,1
The Cheerios effect (1) refers to the common observation that floating objects (say, little toruses of Cheerios cereal floating on water or milk) tend to agglomerate either with each other or to the wall that contains the liquid. Floating objects distort the liquid–vapor interface due to a combination of wetting and gravity forces. The interfacial deflection due to one particle serves as a gravitational potential energy landscape for others, setting up interesting and sometimes counterintuitive long-range attraction that is potentially useful to direct interfacial self-assembly (2). In PNAS, Karpitschka et al. (3) report interesting experiments and theory in which the roles of the liquid and solid are reversed; that is, they show that liquid drops on a compliant solid surface experience attractive or repulsive interactions, a “reverse Cheerios effect.”
Local Shape at Contact Line Determined by Neumann’s Triangle
A
B
Net horizontal force self-balanced
Liquid Solid Neumann’s Triangle Rotated Due to Other Drop
C
D
Unbalanced Net Horizontal Force Drives Moon
Fig. 1. (A) Surface tension of a liquid drop on a compliant substrate causes significant deformation. The local shape at the contact line (i.e., the angles at which the three interfaces meet) is determined by balance of surface stresses, which is Neumann’s triangle. (B) Diagram of an isolated liquid drop, cutting inside the solid at the lower interface and outside the liquid at the outer interface (Laplace pressure not shown), showing that the net horizontal force is balanced. (C) When two drops approach each other, their surface shape changes, especially in the region between the two drops. (D) This shape change leads to an unbalanced in-plane force in response to which the drops move toward each other.
The Cheerios effect is an example from a class of phenomena driven by liquid–vapor surface tension interacting with solid objects. [The very fact of liquid surface tension has fascinated researchers over a long period (4, 5).] Generally, liquid contacting a solid object can wet it completely, not at all, or partially, in the last case forming an equilibrium contact angle that is governed by the surface energies of the liquid–vapor interface and the two solid–fluid interfaces. Usually, the solid in question is sufficiently stiff that its deformation plays a negligible role. On the other hand, when liquids interact with compliant structures (6), the deformability of the latter can result in large deformations of slender objects: capillary origami (7). When the solid is deformable, its own surface stress can play an important and sometimes dominant role in many types of mechanical deformations. Collectively, these phenomena are being referred to as elastocapillarity (8, 9). Two important (static) problems that involve a contact line are wetting of a compliant solid by drops (two fluids, one solid) and adhesive contact between a particle and a compliant solid (two solids, one fluid). In the contact problem, for small particles, the resistance to deformation is dominated by the solid surface stress instead of bulk elasticity, a breakdown of the standard Johnson–Kendall–Roberts theory of adhesive contact (10–12). In the wetting problem, the liquid–vapor surface tension of a drop can strongly deform the solid surface, and the local shape is often governed by balance of surface stresses (13–17). Elastocapillarity also strongly affects dynamic phenomena, such as durotaxis (drop motion due to gradients in substrate compliance) (18) and contact line motion (19). A common thread running through all these elastocapillary phenomena is that when the solid becomes sufficiently compliant, it behaves quite like a liquid in some ways, in that the resistance to deformation due to bulk elasticity weakens, elevating the role of the solid surface stress. This conflation of liquid- and solid-like character raises many interesting questions about what would happen to known liquid capillarity effects if one were to replace the liquid by a compliant solid.
a
Department of Chemical and Biomolecular Engineering and Bioengineering Program, Lehigh University, Bethlehem, PA 18015 Author contributions: A.J. wrote the paper. The author declares no conflict of interest. See companion article on page 7403. 1 Email: [email protected].
7294–7295 | PNAS | July 5, 2016 | vol. 113 | no. 27
www.pnas.org/cgi/doi/10.1073/pnas.1607893113
Chakrabarti and Chaudhury (20) discovered a new version of the Cheerios effect in which particles placed on a very compliant, but still solid, gel are completely engulfed and come to rest at a vertical position governed by the interplay of gravity, solid elasticity, and surface stress. The deformation field is quite longranged and drives attractive interactions between particles as well as overall motion when gel thickness is graded. In related experiments, it has been shown that cylindrical particles placed on the surface of a compliant gel interact through an effective gravitational potential setup by the elastocapillary deformations of the gel driven by the cylinder’s weight, an “elastic Cheerios effect” (21). Karpitschka et al. (3) complete the role-reversal, reporting experiments and theory on the interaction between liquid drops (ethylene glycol) on a compliant solid surface (a polydimethylsiloxane gel). Because, here, the roles of the solid and liquid are reversed, the authors call it the “inverted Cheerios effect.” The solid surface is now the substrate on which liquid drops are placed. It is sufficiently compliant to be deformed significantly by surface stresses of the liquid–vapor and solid–fluid interfaces, which sets up interactions between drops when more than one is present. By conducting the experiment on drops sliding down a vertical surface, the authors eliminate the effect of gravity, demonstrating that the interaction between drops is governed by elastocapillary deformation of the solid substrate. Using clever in situ calibration of the relationship between the vertical drop velocity and the applied gravitational force, the authors are able to quantify the effective force driving drops together or apart, finding both attraction (thick substrates) and repulsion (thin substrates) between two liquid drops. What drives the motion of the two drops? Fig. 1A shows schematically the shapes of the three interfaces (liquid–vapor, liquid– solid, and solid–vapor) in the vicinity of an isolated drop. In particular, if the solid is compliant, the local shape at the contact line (i.e., the relative angles of the three surface tangents) is determined by balance of the three interfacial stresses, which is Neumann’s triangle. Fig. 1B shows the drop, cut out from the rest of the body, and it is
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
clear by symmetry that the horizontal forces will all balance out so that the drop is happily at equilibrium. Now consider Fig. 1C in which two drops approach sufficiently close that the regions where each one would, by itself, distort the surface profile now overlap. Each drop can be thought of, approximately, as “living” now on a surface that is no longer quite planar but inclined due to the presence of the other drop. Local equilibrium is still maintained by Neumann’s triangle, which determines only the relative angles; the force diagram, as a whole, is free to rotate in response to the other drop. This rotation of the forces now introduces an asymmetry so that, as Fig. 1D shows, there is a net unbalanced horizontal force. In this case, it is an attractive one that drives the drops toward each other. [This argument is slightly different from the argument made by Karpitschka et al. (3), but the net result is the same.] Now, in practice, drop movement is quite slow and quasistatic so that each drop is always in mechanical equilibrium. That is, once drop movement begins, it generates viscous drag, which provides the necessary force at the contact line to balance forces. Why is the interaction repulsive when the substrate is relatively thin? The argument just presented is based on a deformed surface shape that corresponds to a very thick elastic substrate. If the substrate is thin and incompressible, volume conservation demands that a local uplift of the surface be matched by a dip. Coming out of the dip, the slope of the deformed surface is now of opposite sign compared with its slope in Fig. 1. This change in sign of the slope translates into a change in sign of the interaction, rendering it repulsive at larger distances. In broader terms, the effect discovered by Karpitschka et al. (3) is an example of new, sometimes counterintuitive, phenomena emerging from the study of the deformation in soft materials and the significant role played by interfaces and interfacial mechanical properties.
Acknowledgments This work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG0207ER46463.
Vella D, Mahadevan L (2005) The “Cheerios effect”. Am J Phys 73(9):817–825. Furst EM (2011) Directing colloidal assembly at fluid interfaces. Proc Natl Acad Sci USA 108(52):20853–20854. Karpitschka S, et al. (2016) Liquid drops attract or repel by the inverted Cheerios effect. Proc Natl Acad Sci USA 113:7403–7407. deGennes P-G, Brochard-Wyart F, Quere D (2002) Capillarity and Wetting Phenomena. Drops, Bubbles, Pearls, Waves (Springer Science + Business Media, Inc., New York). Rowlinson JS, Widom B (1989) Molecular Theory of Capillarity (Oxford Univ Press, New York). Roman B, Bico J (2010) Elasto-capillarity: Deforming an elastic structure with a liquid droplet. J Phys Condens Matter 22(49):493101. Py C, et al. (2007) Capillary origami: Spontaneous wrapping of a droplet with an elastic sheet. Phys Rev Lett 98(15):156103. Style RW, Jagota A, Hui C-Y, Dufresne ER (2016) Elastocapillarity: Surface tension and the mechanics of soft solids. arXiv:1604.02052. Andreotti B, et al. (2016) Solid capillarity: When and how does surface tension deform soft solids? Soft Matter 12(12):2993–2996. Style RW, Hyland C, Boltyanskiy R, Wettlaufer JS, Dufresne ER (2013) Surface tension and contact with soft elastic solids. Nat Commun 4:2728. Carrillo J-MY, Dobrynin AV (2012) Contact mechanics of nanoparticles. Langmuir 28(29):10881–10890. Hui C-Y, Liu T, Salez T, Raphael E, Jagota A (2015) Indentation of a rigid sphere into an elastic substrate with surface tension and adhesion. Proc Math Phys Eng Sci 471(2175):20140727. Jerison ER, Xu Y, Wilen LA, Dufresne ER (2011) Deformation of an elastic substrate by a three-phase contact line. Phys Rev Lett 106(18):186103. Marchand A, Das S, Snoeijer JH, Andreotti B (2012) Contact angles on a soft solid: From Young’s law to Neumann’s law. Phys Rev Lett 109(23):236101. Hui C-Y, Jagota A (2014) Deformation near a liquid contact line on an elastic substrate. Proc Math Phys Eng Sci 470(2167):20140085. Cao Z, Dobrynin AV (2015) Polymeric droplets on soft surfaces: From Neumann’s triangle to Young’s law. Macromolecules 48(2):443–451. Nadermann N, Hui C-Y, Jagota A (2013) Solid surface tension measured by a liquid drop under a solid film. Proc Natl Acad Sci USA 110(26):10541–10545. Style RW, et al. (2013) Patterning droplets with durotaxis. Proc Natl Acad Sci USA 110(31):12541–12544. Karpitschka S, et al. (2015) Droplets move over viscoelastic substrates by surfing a ridge. Nat Commun 6:7891. Chakrabarti A, Chaudhury MK (2013) Surface folding-induced attraction and motion of particles in a soft elastic gel: Cooperative effects of surface tension, elasticity, and gravity. Langmuir 29(50):15543–15550. Chakrabarti A, Ryan L, Chaudhury MK, Mahadevan L (2015) Elastic Cheerios effect: Self-assembly of cylinders on a soft solid. Europhys Lett 112(5):54001–54005.
Jagota
PNAS | July 5, 2016 | vol. 113 | no. 27 | 7295
