IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 40 (2007) 5564–5570 doi:10.1088/0022-3727/40/18/009 Hierarchical capillary adhesion of microcantilevers or hairs Jian-Lin Liu, Xi-Qiao Feng 1 , Re Xia and Hong-Ping Zhao Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China E-mail: fengxq@tsinghua.edu.cn Received 18 March 2007, in final form 28 July 2007 Published 30 August 2007 Online at stacks.iop.org/JPhysD/40/5564 Abstract As a result of capillary forces, animal hairs, carbon nanotubes or nanowires of a periodically or randomly distributed array often assemble into hierarchical structures. In this paper, the energy method is adopted to analyse the capillary adhesion of microsized hairs, which are modelled as clamped microcantilevers wetted by liquids. The critical conditions for capillary adhesion of two hairs, three hairs or two bundles of hairs are derived in terms of Young’s contact angle, elastic modulus and geometric sizes of the beams. Then, the hierarchical capillary adhesion of hairs is addressed. It is found that for multiple hairs or microcantilevers, the system tends to take a hierarchical structure as a result of the minimization of the total potential energy of the system. The level number of structural hierarchy increases with the increase in the number of hairs if they are sufficiently long. Additionally, we performed experiments to verify our theoretical solutions for the adhesion of microbeams. (Some figures in this article are in colour only in the electronic version) 1. Introduction A great deal of recent work has been directed towards adhesion at micro and nano scales. On the one hand, adhesion induced by van der Waals force, capillary force and electrostatic force may cause the failure of micro- electromechanical systems (MEMS) [1]. On the other hand, adhesion mechanisms may also be useful in the manipulation and operation of microdevices and contribute to various biological processes of creatures. Structural hierarchy plays a significant role in the physical properties and biological behaviours of various kinds of man-made and natural materials and systems (e.g. the branches and roots of trees, and the hierarchical macro/meso/micro/nano structure of bones, silks and other biomaterials). An interesting example is the hierarchical structures of the specialized adhesive feet of geckos which allow them to manoeuvre on vertical walls and ceilings against their body weight [2, 3]. The striking adhesion ability of a gecko is primarily due to the van der Waals force between the contact surfaces and its feet [4]. Gao et al [5, 6] and Federle [7] used fractal concept to 1 Author to whom any correspondence should be addressed. characterize the self-similar fibrillar structures of gecko feet at multiple levels, and elucidated the importance of the nanometre characteristic size and structural hierarchy in the superior adhesion strength. Adhesion associated with van der Waals force, Casimir force and electrostatic force may exist in micro- or nano-electromechanical systems. In microcontact printing technology, for instance, adhesion caused by van der Waals force leads to stamp deformation and limits the practice and precision of the technology [8–10]. Three kinds of stamp deformation have been observed, namely, roof collapse, buckling and lateral sticking of the fibrillar structures. Van der Waals force may also cause the stiction of high aspect ratio photoresist (SU-8) when preparing photonic crystals [11], adhesion of neighbouring carbon nanotubes (CNTs) [12], adhesion of biological macromolecules, cells or vesicles on a substrate [13–17]. In consideration of the elastic energy and the interaction potential between the vesicles and the substrate, the equilibrium shape of the vesicles may be calculated through analytical or numerical methods. Adhesion may also happen as a result of the capillary force of liquid films/bridges in the fast developing MEMS techniques, in which typical surface separations are in the range 500–2000nm. Water can be trapped in the gaps of the high 0022-3727/07/185564+07$30.00 © 2007 IOP Publishing Ltd Printed in the UK 5564