Wetting of flexible fibre arrays

Journal name:
Nature
Volume:
482,
Pages:
510–513
Date published:
DOI:
doi:10.1038/nature10779
Received
Accepted
Published online

Fibrous media are functional and versatile materials, as demonstrated by their ubiquity both in natural systems such as feathers1, 2, 3, 4 and adhesive pads5 and in engineered systems from nanotextured surfaces6 to textile products7, where they offer benefits in filtration, insulation, wetting and colouring. The elasticity and high aspect ratios of the fibres allow deformation under capillary forces, which cause mechanical damage8, matting5, 9 self-assembly10, 11 or colour changes12, with many industrial and ecological consequences. Attempts to understand these systems have mostly focused on the wetting of rigid fibres13, 14, 15, 16, 17 or on elastocapillary effects in planar geometries18 and on a fibre brush withdrawn from an infinite bath19. Here we consider the frequently encountered case of a liquid drop deposited on a flexible fibre array and show that flexibility, fibre geometry and drop volume are the crucial parameters that are necessary to understand the various observations referred to above. We identify the conditions required for a drop to remain compact with minimal spreading or to cause a pair of elastic fibres to coalesce. We find that there is a critical volume of liquid, and, hence, a critical drop size, above which this coalescence does not occur. We also identify a drop size that maximizes liquid capture. For both wetting and deformation of the substrates, we present rules that are deduced from the geometric and material properties of the fibres and the volume of the drop. These ideas are applicable to a wide range of fibrous materials, as we illustrate with examples for feathers, beetle tarsi, sprays and microfabricated systems.

At a glance

Figures

  1. Shape transitions of a drop sitting on two parallel fibres.
    Figure 1: Shape transitions of a drop sitting on two parallel fibres.

    a. A drop of perfectly wetting liquid (silicone oil) of volume 2μl deposited on two parallel, rigid glass fibres (radius, r = 0.145mm) adopts three different shapes depending on the distance between the fibres: a ‘bridge’ (d0/r = 2.5), a ‘barrel’ (d0/r = 1.5) and a column (d0/r = 1). dc, critical distance at which the drop-to-column transition occurs. b, Experimental set-up used to investigate the behaviour of a drop deposited on two flexible fibres, which are clamped at one end and free to move at the other. Left: top and side views, recorded simultaneously using a mirror. Right: expected cross-sections for a concave liquid column and a convex drop. The direction of gravity is indicated by g. c, Typical experiment with d0/r = 2.7, V = 1.5μl and L = 4cm. The time between successive images is 25s. When the drop is deposited on flexible fibres, the fibres deflect inward. The drop spontaneously moves towards the free ends of the fibres. At a given location, zs, the drop starts spreading and the fibres are drawn together. The final wet length is denoted Ls.

  2. The three different final states of a drop between two flexible fibres.
    Figure 2: The three different final states of a drop between two flexible fibres.

    ac, Top and side views of the final state obtained for d0/r = 2.6; a fixed volume, V = 2μl; and increasing length, L = 3, 3.5 and 4cm. The final state changes from one of no spreading to one of partial spreading to one of total spreading as L increases. d, Phase diagram of the different regimes for d0/r = 2.1. Depending on L and V, three different regimes are observed: the drop either moves towards the ends without spreading (diamonds; a), spreads until partial depletion of the drop (circles; b) or spreads completely into a liquid column (squares; c). The solid and dashed curves correspond to equations (2) and (3), respectively. The vertical dashed line corresponds to L = Ldry.

  3. Influence of the initial drop volume on the final state.
    Figure 3: Influence of the initial drop volume on the final state.

    a, Transition between total and partial spreading: evolution of the spreading length, Ls, with the volume, V, of the drop for d0/r = 2.6 and L = 3.5cm. We observe an optimum (maximum Ls) at a critical volume of Vc = 1.5μl. b, Transition between spreading and no spreading: evolution of the position of the drop at spreading, zs, with V for d0/r = 1.9 (orange), 2.4 (purple), 2.7 (red) and 4.2 (blue). The position of the drop at spreading is independent of the fibre length and increases with increasing volume, spacing and fibre rigidity (that is, the bending modulus, which is proportional to r4). The solid lines correspond to the theoretical prediction (Supplementary Information, equation (2)).

  4. Aerosol size and fibre matrix properties needed to collect, trap or displace a known volume of liquid.
    Figure 4: Aerosol size and fibre matrix properties needed to collect, trap or displace a known volume of liquid.

    a, Map of the three spreading regimes as a function of the two dimensionless parameters L/Ls,max and V/Vc. The solid curves show three limits for d0/r = 2.4, 2.7 and 4.2 and the points show data for total spreading (asterisks), partial spreading (squares) and no spreading (circles) of silicone oil (total wetting, d0/r = 2.1; blue and purple points) and water (partial wetting, d0/r = 3; black and grey points). Stars correspond to the three situations observed in b. b, Microscope pictures of goose feathers sprayed with oil (smaller drops have volumes of order 10−14–10−13m3), showing no spreading (d0/r = 4.8, L/Ls,max = 2, V/Vc5; blue star in a), total spreading (d0/r = 3.4, L/Ls,max = 1.5 and Ls = Ls,max = 0.8mm; white star in a) and partial spreading (d0/r = 3.5, L/Ls,max = 1.5, V/Vc4; pink star in a) in agreement with our predictions. Scale bars, 500μm.

References

  1. Rijke, A. M. & Jesser, W. A. The feather structure of dippers: water repellency and resistance to water penetration. Wilson J. Ornithol. 122, 563568 (2010)
  2. Rijke, B. Y. A. M. The water repellency and feather structure of cormorants, Phalacrocoracidae. J. Exp. Biol. 48, 185189 (1968)
  3. Dawson, C., Vincent, J., Jeronimidis, G., Rice, G. & Forshaw, P. Heat transfer through penguin feathers. J. Theor. Biol. 199, 291295 (1999)
  4. Zi, J. et al. Coloration strategies in peacock feathers. Proc. Natl Acad. Sci. USA 100, 1257612578 (2003)
  5. Eisner, T. & Aneshansley, D. J. Defense by foot adhesion in a beetle (Hemisphaerota cyanea). Proc. Natl Acad. Sci. USA 97, 65686573 (2000)
  6. Liu, K. & Jiang, L. Bio-inspired design of multiscale structures for function integration. Nano Today 6, 155175 (2011)
  7. Eadie, L. & Ghosh, T. K. Biomimicry in textiles: past, present and potential. An overview. J. R. Soc. Interface 6, 761775 (2011)
  8. Kamo, J., Hiram, T. & Kamada, K. Solvent-induced morphological change of microporous hollow fiber membranes. J. Membr. Sci. 70, 217224 (1992)
  9. O’Hara, P. D. & Morandin, L. A. Effects of sheens associated with offshore oil and gas development on the feather microstructure of pelagic seabirds. Mar. Pollut. Bull. 60, 672678 (2010)
  10. Pokroy, B., Kang, S. H., Mahadevan, L. & Aizenberg, J. Self-organization of a mesoscale bristle into ordered, hierarchical helical assemblies. Science 323, 237240 (2009)
  11. Chandra, D. & Yang, S. Stability of high-aspect-ratio micropillar arrays against adhesive and capillary forces. Acc. Chem. Res. 43, 10801091 (2010)
  12. Chandra, D., Yang, S., Soshinsky, A., a & Gambogi, R. J. Biomimetic ultrathin whitening by capillary-force-induced random clustering of hydrogel micropillar arrays. ACS Appl. Mater. Interfaces 1, 16981704 (2009)
  13. Princen, H. Capillary phenomena in assemblies of parallel cylinders III. Liquid columns between horizontal parallel cylinders. J. Colloid Interface Sci. 34, 171184 (1970)
  14. Wu, X.-F., Bedarkar, A. & Vaynberg, K. A. Droplets wetting on filament rails: surface energy and morphology transition. J. Colloid Interface Sci. 341, 326332 (2010)
  15. Bedarkar, A., Wu, X.-f. & Vaynberg, A. Wetting of liquid droplets on two parallel filaments. Appl. Surf. Sci. 256, 72607264 (2010)
  16. Minor, F. W., Schwartz, M., Wulkow, E., a & Buckles, L. C. Part III: The behavior of liquids on single textile fibers. Text. Res. J. 29, 940949 (1959)
  17. Keis, K., Kornev, K. G., Kamath, Y. K. & Neimark, A. V. in Nanoengineered Nanofibrous Materials (eds Guceri, S., Gogotsi, Y. G & Kuznetsov, V.) 173180 (Kluwer, 2004)
  18. Kwon, H.-M., Kim, H.-Y., Puëll, J. R. M. & Mahadevan, L. Equilibrium of an elastically confined liquid drop. J. Appl. Phys. 103, 093519 (2008)
  19. Roman, B. & Bico, J. Elasto-capillarity: deforming an elastic structure with a liquid droplet. J. Phys. Condens. Matter 22, 493101 (2010)
  20. Hubbe, M. A. Bonding between cellulosic fibers in the absence and presence of dry-strength agents - a review. BioResources 1, 281318 (2006)
  21. Prakash, M., Quéré, D. & Bush, J. W. M. Surface tension transport of prey by feeding shorebirds: the capillary ratchet. Science 320, 931934 (2008)
  22. Py, C., Bastien, R., Bico, J., Roman, B. & Boudaoud, A. 3D aggregation of wet fibers. Europhys. Lett. 77, 44005 (2007)
  23. Hartung, R. Energy metabolism in oil-covered ducks. J. Wildl. Mgmt 31, 798804 (1967)

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Author information

Affiliations

  1. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA

    • C. Duprat,
    • A. Y. Beebe &
    • H. A. Stone
  2. CNRS, UMR 7190, Institut Jean le Rond d’Alembert, F-75005 Paris, France

    • S. Protière
  3. UPMC Université Paris 06, UMR 7190, Institut Jean le Rond d’Alembert, F-75005 Paris, France

    • S. Protière

Contributions

C.D. and S.P. designed the experiments; A.Y.B., C.D. and S.P. carried out the experiments; C.D., S.P. and H.A.S. discussed and interpreted the results; C.D. and H.A.S. developed the models; and C.D., S.P. and H.A.S. wrote the manuscript.

Competing financial interests

The authors declare no competing financial interests.

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Supplementary information

PDF files

  1. Supplementary Information (4.7M)

    This file contains Supplementary Text and Data, Supplementary Figures 1-7 with legends, Supplementary Table 1 and legends for Supplementary Movies 1-3.

Movies

  1. Supplementary Movie 1 (10.3M)

    This movie shows the evolution of a drop of volume V = 2 μL on a rail formed by two fibres of length L =3 cm and separated by a distance d0 = 0.76 mm, viewed simultaneously from the side and the top.

  2. Supplementary Movie 2 (11.5M)

    This movie shows the evolution of a drop of volume V = 2 μL on a rail formed by two fibres of length L =4 cm and separated by a distance d0 = 0.76 mm, viewed simultaneously from the side and the top.

  3. Supplementary Movie 3 (9.4M)

    This movie shows the evolution of a drop of volume V = 2 μL on a rail formed by two fibres of length L =3.5 cm and separated by a distance d0 = 0.76 mm, viewed simultaneously from the side and the top.

Additional data