COMMENTARY COMMENTARY Role reversal: “Cheerios” on a solid sense each other

Anand Jagotaa,1

The Cheerios effect (1) refers to the common observa- The Cheerios effect is an example from a class of tion that floating objects (say, little toruses of Cheerios phenomena driven by liquid–vapor cereal floating on water or milk) tend to agglomerate interacting with solid objects. [The very fact of liquid either with each other or to the wall that contains the surface tension has fascinated researchers over a long liquid. Floating objects distort the liquid–vapor inter- period (4, 5).] Generally, liquid contacting a solid object face due to a combination of and gravity forces. can wet it completely, not at all, or partially, in the last The interfacial deflection due to one particle serves as case forming an equilibrium that is gov- a gravitational potential energy landscape for others, erned by the surface energies of the liquid–vapor in- setting up interesting and sometimes counterintuitive terface and the two solid–fluid interfaces. Usually, the long-range attraction that is potentially useful to direct solid in question is sufficiently stiff that its deformation interfacial self-assembly (2). In PNAS, Karpitschka et al. plays a negligible role. On the other hand, when (3) report interesting experiments and theory in which interact with compliant structures (6), the deformability the roles of the liquid and solid are reversed; that is, of the latter can result in large deformations of slender they show that liquid drops on a compliant solid sur- objects: capillary origami (7). face experience attractive or repulsive interactions, a When the solid is deformable, its own surface stress “reverse Cheerios effect.” can play an important and sometimes dominant role in many types of mechanical deformations. Collec- tively, these phenomena are being referred to as elas- Local Shape at Contact tocapillarity (8, 9). Two important (static) problems that ABLine Determined by Net horizontal force involve a contact line are wetting of a compliant solid by self-balanced Neumann’s Triangle drops (two fluids, one solid) and adhesive contact be- tween a particle and a compliant solid (two solids, one fluid). In the contact problem, for small particles, the Liquid resistance to deformation is dominated by the solid Solid surface stress instead of bulk elasticity, a breakdown of Neumann’s Triangle the standard Johnson–Kendall–Roberts theory of ad- C Rotated Due to Other Drop hesive contact (10–12). In the wetting problem, the liquid–vapor surface tension of a drop can strongly de- form the solid surface, and the local shape is often governed by balance of surface stresses (13–17). Elasto-

D Unbalanced Net capillarity also strongly affects dynamic phenomena, such Horizontal Force Drives as durotaxis (drop motion due to gradients in substrate Moon compliance) (18) and contact line motion (19). A common thread running through all these elastocapillary phenom- ena is that when the solid becomes sufficiently compliant, 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 it behaves quite like a liquid in some ways, in that the which the three interfaces meet) is determined by balance of surface stresses, resistance to deformation due to bulk elasticity weakens, which is Neumann’s triangle. (B) Diagram of an isolated liquid drop, cutting inside elevating the role of the solid surface stress. This confla- the solid at the lower interface and outside the liquid at the outer interface tion of liquid- and solid-like character raises many inter- (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 esting questions about what would happen to known theregionbetweenthetwodrops.(D) This shape change leads to an unbalanced liquid capillarity effects if one were to replace the liquid in-plane force in response to which the drops move toward each other. by a compliant solid.

aDepartment 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. 1Email: [email protected].

7294–7295 | PNAS | July 5, 2016 | vol. 113 | no. 27 www.pnas.org/cgi/doi/10.1073/pnas.1607893113 Downloaded by guest on September 26, 2021 Chakrabarti and Chaudhury (20) discovered a new version of clear by symmetry that the horizontal forces will all balance out so the Cheerios effect in which particles placed on a very compliant, that the drop is happily at equilibrium. Now consider Fig. 1C in but still solid, gel are completely engulfed and come to rest at a which two drops approach sufficiently close that the regions where vertical position governed by the interplay of gravity, solid elas- each one would, by itself, distort the surface profile now overlap. ticity, and surface stress. The deformation field is quite long- Each drop can be thought of, approximately, as “living” now on a ranged and drives attractive interactions between particles as well surface that is no longer quite planar but inclined due to the pres- as overall motion when gel thickness is graded. In related ex- ence of the other drop. Local equilibrium is still maintained by Neu- periments, it has been shown that cylindrical particles placed mann’s triangle, which determines only the relative angles; the force on the surface of a compliant gel interact through an effective diagram, as a whole, is free to rotate in response to the other drop. gravitational potential setup by the elastocapillary deformations This rotation of the forces now introduces an asymmetry so that, as of the gel driven by the cylinder’sweight,an“elastic Cheerios Fig. 1D shows, there is a net unbalanced horizontal force. In this ” effect (21). case, it is an attractive one that drives the drops toward each Karpitschka et al. (3) complete the role-reversal, reporting ex- other. [This argument is slightly different from the argument made periments and theory on the interaction between liquid drops by Karpitschka et al. (3), but the net result is the same.] Now, in (ethylene glycol) on a compliant solid surface (a polydimethylsi- practice, drop movement is quite slow and quasistatic so that each loxane gel). Because, here, the roles of the solid and liquid are drop is always in mechanical equilibrium. That is, once drop move- “ ” reversed, the authors call it the inverted Cheerios effect. The ment begins, it generates viscous drag, which provides the necessary solid surface is now the substrate on which liquid drops are force at the contact line to balance forces. placed. It is sufficiently compliant to be deformed significantly Why is the interaction repulsive when the substrate is relatively – – by surface stresses of the liquid vapor and solid fluid interfaces, thin? The argument just presented is based on a deformed surface which sets up interactions between drops when more than one is shape that corresponds to a very thick elastic substrate. If the present. By conducting the experiment on drops sliding down a substrateisthinandincompressible, volume conservation demands vertical surface, the authors eliminate the effect of gravity, dem- that a local uplift of the surface be matched by a dip. Coming out of onstrating that the interaction between drops is governed by the dip, the slope of the deformed surface is now of opposite sign elastocapillary deformation of the solid substrate. Using clever compared with its slope in Fig. 1. This change in sign of the slope in situ calibration of the relationship between the vertical drop translates into a change in sign of the interaction, rendering it repul- velocity and the applied gravitational force, the authors are able sive at larger distances. to quantify the effective force driving drops together or apart, In broader terms, the effect discovered by Karpitschka et al. (3) finding both attraction (thick substrates) and repulsion (thin sub- is an example of new, sometimes counterintuitive, phenomena strates) between two liquid drops. emerging from the study of the deformation in soft materials and What drives the motion of the two drops? Fig. 1A shows sche- the significant role played by interfaces and interfacial mechanical matically the shapes of the three interfaces (liquid–vapor, liquid– properties. 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 Acknowledgments relative angles of the three surface tangents) is determined by This work was supported by the US Department of Energy, Office of Basic Energy ’ balance of the three interfacial stresses, which is Neumann striangle. Sciences, Division of Materials Sciences and Engineering under Award DE-FG02- Fig. 1B shows the drop, cut out from the rest of the body, and it is 07ER46463.

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