Encoding quadrupolar capillary information into saddle-shaped objects for self-assembly

Capillary interactions; Complex assembly; Encodings; Interaction potentials; Liquid interface; Particle geometries; Self-assembling structure; Simple++; Spherical particle; Theoretical modeling; Computational Mechanics; Modeling and Simulation; Fluid Flow and Transfer Processes

Abstract :

[en] The “Cheerios effect,” which describes the capillary interaction between spherical particles at a liquid interface, has traditionally been limited to simple self-assembling structures. Our study addresses the need for more complex assemblies by exploring specific particle geometries. We introduce a theoretical model to describe the interaction potential between quadrupolar objects at the mesoscale, which reveals that both side-by-side and end-to-end long-range attractions can be achieved under specific capillary and geometrical conditions—a phenomenon we term the “Pringles effect.” We experimentally validate our model’s predictions by encoding this specific capillary information into 3D-printed curved ellipses, which mimic the surface geometry of Pringles crisps. This approach showcases the potential for complex self-assembling mesoscale structures through precise control over capillary-driven interactions.

Disciplines :

Physics

Author, co-author :

Delens, Megan ^✱; Université de Liège - ULiège > Complex and Entangled Systems from Atoms to Materials (CESAM)

Vandewalle, Nicolas ^✱; Université de Liège - ULiège > Département de physique > Physique statistique

^✱ These authors have contributed equally to this work.

Language :

English

Title :

Encoding quadrupolar capillary information into saddle-shaped objects for self-assembly

Publication date :

14 October 2025

Journal title :

Physical Review Fluids

ISSN :

2469-9918

eISSN :

2469-990X

Publisher :

American Physical Society

Volume :

Issue :

Pages :

1040031 - 10400314

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://link.aps.org/article/10.1103/f3tr-kby1

Funders :

F.R.S.-FNRS - Fonds de la Recherche Scientifique

Funding text :

This work is financially supported by the University of Liège through the CESAM Research Unit and the FNRS CDR Project No. J.0186.23 entitled "Magnetocapillary Interactions for Locomotion at Liquid Interfaces" (MILLI).

Data Set :

Data and STL files supporting article: Encoding Quadrupolar Capillary Information into Saddle- Shaped Objects for Self-Assembly

Available on ORBi :

since 03 April 2026

Statistics

Number of views

66 (1 by ULiège)

Number of downloads

56 (0 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenAlex citations

Bibliography

[1] N. Bowden, A. Terfort, J. Carbeck, and G. M. Whitesides, Self-assembly ofmesoscale objects into ordered two-dimensional arrays, Science 276, 233 (1997).
[2] S. E. Chung, W. Park, S. Shin, S. A. Lee, and S. Kwon, Guided and fluidic self-assembly of microstructures using railed microfluidic channels, Nat. Mater. 7, 581 (2008).
[3] M. Boncheva and G. M. Whitesides, Making things by self-assembly, MRS Bull. 30, 736 (2005).
[4] L. Zhang, J. M. Chan, F. X. Gu, J.-W. Rhee, A. Z.Wang, A. F. Radovic-Moreno, F. Alexis, R. Langer, and O. C. Farokhzad, Self-assembled lipid-polymer hybrid nanoparticles: A robust drug delivery platform, ACS Nano 2, 1696 (2008).
[5] S.-H. Kim, S. Y. Lee, S.-M. Yang, and G.-R. Yi, Self-assembled colloidal structures for photonics, NPG Asia Mater. 3, 25 (2011).
[6] Y. Xia, B. Gates, and Z.-Y. Li, Self-assembly approaches to three-dimensional photonic crystals, Adv. Mater. 13, 409 (2001).
[7] B. A. Grzybowski, H. A. Stone, and G. M. Whitesides, Dynamic self-assembly of magnetized, millimetresized objects rotating at a liquid–air interface, Nature (London) 405, 1033 (2000).
[8] G. M. Whitesides and B. Grzybowski, Self-assembly at all scales, Science 295, 2418 (2002).
[9] J. A. Pelesko, Self Assembly: The Science of Things that Put Themselves Together, 1st ed. (Chapman and Hall/CRC, New York, 2007).
[10] M. M. Nicolson, The interaction between floating particles, Math. Proc. Cambridge Philos. Soc. 45, 288 (1949).
[11] D. Chan, J. Henry, Jr., and L. White, The interaction of colloidal particles collected at fluid interfaces, J. Colloid Interface Sci. 79, 410 (1981).
[12] D. Vella and L. Mahadevan, The “Cheerios effect”, Am. J. Phys. 73, 817 (2005).
[13] M.-J. Dalbe, D. Cosic, M. Berhanu, and A. Kudrolli, Aggregation of frictional particles due to capillary attraction, Phys. Rev. E 83, 051403 (2011).
[14] K. D. Danov and P. A. Kralchevsky, Capillary forces between particles at a liquid interface: General theoretical approach and interactions between capillary multipoles, Adv. Colloid Interface Sci. 154, 91 (2010).
[15] K. Hosokawa, I. Shimoyama, and H. Miura, Two-dimensional micro-self-assembly using the surface tension of water, Sens. Actuators A 57, 117 (1996).
[16] J. Loudet and B. Pouligny, How do mosquito eggs self-assemble on the water surface? Eur. Phys. J. E 34, 76 (2011).
[17] J. Voise, M. Schindler, J. Casas, and E. Raphaël, Capillary-based static self-assembly in higher organisms, J. R. Soc. Interface. 8, 1357 (2011).
[18] P.W. Rothemund, Using lateral capillary forces to compute by self-assembly, Proc. Natl. Acad. Sci. USA 97, 984 (2000).
[19] M. Poty, G. Lumay, and N. Vandewalle, Customizing mesoscale self-assembly with three-dimensional printing, New J. Phys. 16, 023013 (2014).
[20] J. Metzmacher, M. Poty, G. Lumay, and N. Vandewalle, Self-assembly of smart mesoscopic objects, Eur. Phys. J. E 40, 108 (2017).
[21] N. Vandewalle, L. Clermont, D. Terwagne, S. Dorbolo, E. Mersch, and G. Lumay, Symmetry breaking in a few-body system with magnetocapillary interactions, Phys. Rev. E 85, 041402 (2012).
[22] J.-C. Loudet, A. M. Alsayed, J. Zhang, and A. G. Yodh, Capillary interactions between anisotropic colloidal particles, Phys. Rev. Lett. 94, 018301 (2005).
[23] H. Lehle, E. Noruzifar, and M. Oettel, Ellipsoidal particles at fluid interfaces, Eur. Phys. J. E 26, 151 (2008).
[24] B. Madivala, J. Fransaer, and J. Vermant, Self-assembly and rheology of ellipsoidal particles at interfaces, Langmuir 25, 2718 (2009).
[25] E. P. Lewandowski, M. Cavallaro, Jr., L. Botto, J. C. Bernate, V. Garbin, and K. J. Stebe, Orientation and self-assembly of cylindrical particles by anisotropic capillary interactions, Langmuir 26, 15142 (2010).
[26] E. P. Lewandowski, J. A. Bernate, P. C. Searson, and K. J. Stebe, Rotation and alignment of anisotropic particles on nonplanar interfaces, Langmuir 24, 9302 (2008).
[27] D. Ershov, J. Sprakel, J. Appel, M. A. Cohen Stuart, and J. van der Gucht, Capillarity-induced ordering of spherical colloids on an interface with anisotropic curvature, Proc. Natl. Acad. Sci. USA 110, 9220 (2013).
[28] J. Eatson, S. Bauernfeind, B. Midtvedt, A. Ciarlo, J. Menath, G. Pesce, A. B. Schofield, G. Volpe, P. S. Clegg, N. Vogel, D. M. A. Buzza, and M. Rey, Self-assembly of defined core–shell ellipsoidal particles at liquid interfaces, J. Colloid Interface Sci. 683, 435 (2025).
[29] N. Hazra, A. A. Rudov, J. Midya, A. Babenyshev, S. Bochenek, M. Frenken,W. Richtering, G. Gompper, T. Auth, I. I. Potemkin et al., Capillary-driven self-assembly of soft ellipsoidal microgels at the air–water interface, Proc. Natl. Acad. Sci. USA 121, e2403690121 (2024).
[30] P.-G. Gennes, F. Brochard-Wyart, D. Quéré et al., Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, 2004).
[31] K. D. Danov, P. A. Kralchevsky, B. N. Naydenov, and G. Brenn, Interactions between particles with an undulated contact line at a fluid interface: Capillary multipoles of arbitrary order, J. Colloid Interface Sci. 287, 121 (2005).
[32] Á. Domínguez Álvarez, M. Oettel, and S. Dietrich, Force balance of particles trapped at fluid interfaces, J. Chem. Phys. 128, 111904 (2008).
[33] P. A. Kralchevsky and K. Nagayama, Capillary forces between colloidal particles, Langmuir 10, 23 (1994).
[34] P. A. Kralchevsky, N. D. Denkov, and K. D. Danov, Particles with an undulated contact line at a fluid interface: Interaction between capillary quadrupoles and rheology of particulate monolayers, Langmuir 17, 7694 (2001).
[35] A. B. D. Brown, C. G. Smith, and A. R. Rennie, Fabricating colloidal particles with photolithography and their interactions at an air-water interface, Phys. Rev. E 62, 951 (2000).
[36] L. Botto, E. P. Lewandowski, M. Cavallaro, and K. J. Stebe, Capillary interactions between anisotropic particles, Soft Matter 8, 9957 (2012).
[37] D. Stamou, C. Duschl, and D. Johannsmann, Long-range attraction between colloidal spheres at the airwater interface: The consequence of an irregular meniscus, Phys. Rev. E 62, 5263 (2000).
[38] J. Metzmacher, M. Delens, and N. Vandewalle, Ring of capillary actuators as trap, tweezers, and ratchet for floating particles, Sci. Rep. 15, 20617 (2025).
[39] P. A. Kralchevsky and N. D. Denkov, Capillary forces and structuring in layers of colloid particles, Curr. Opin. Colloid Interface Sci. 6, 383 (2001).
[40] P. A. Kralchevsky and K. Nagayama, Capillary interactions between particles bound to interfaces, liquid films and biomembranes, Adv. Colloid Interface Sci. 85, 145 (2000).
[41] N. D. Vassileva, D. van den Ende, F.Mugele, and J. Mellema, Capillary forces between spherical particles floating at a liquid-liquid interface, Langmuir 21, 11190 (2005).
[42] M. Delens, Y. Collard, and N. Vandewalle, Induced capillary dipoles in floating particle assemblies, Phys. Rev. Fluids 8, 074001 (2023).
[43] M. Delens, A. Franckart, D. M. Harris, and N. Vandewalle, 3D-printed spines for programmable liquid topographies and micromanipulation, Nat. Commun. 16, 4348 (2025).
[44] N. Vandewalle, M. Poty, N. Vanesse, J. Caprasse, T. Defize, and C. Jérôme, Switchable self-assembled capillary structures, Soft Matter 16, 10320 (2020).
[45] M. Delens and N. Vandewalle, Data and stl files supporting article: Encoding quadrupolar capillary information into saddle-shaped objects for self-assembly, Zenodo (2024), doi: 10.5281/zenodo.11282790.
[46] F. Moisy, M. Rabaud, and K. Salsac, A synthetic schlieren method for the measurement of the topography of a liquid interface, Exp. Fluids 46, 1021 (2009).
[47] J. Metzmacher, G. Lagubeau,M. Poty, and N. Vandewalle, Double pattern improves the schlieren methods for measuring liquid–air interface topography, Exp. Fluids 63, 120 (2022).
[48] Z. Zhang, P. Pfleiderer, A. B. Schofield, C. Clasen, and J. Vermant, Synthesis and directed self-assembly of patterned anisometric polymeric particles, J. Am. Chem. Soc. 133, 392 (2011).
[49] L. Botto, L. Yao, R. Leheny, and K. Stebe, Capillary bond between rod-like particles and the micromechanics of particle-laden interfaces, Soft Matter 8, 4971 (2012).