Abstract
Recent experiments and molecule dynamics simulations have shown that adhesion droplets on conical surfaces may move spontaneously and directionally. Besides, this spontaneous and directional motion is independent of the hydrophilicity and hydrophobicity of the conical surfaces. Aimed at this important phenomenon, a general theoretical explanation is provided from the viewpoint of the geometrization of micro/nano mechanics on curved surfaces. In the extrinsic mechanics on micro/nano soft curved surfaces, we disclose that the curvatures and their extrinsic gradients form the driving forces on the curved spaces. This paper focuses on the intrinsic mechanics on micro/nano hard curved surfaces and the experiment on the spontaneous and directional motion. Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard curved surface are studied, and the geometric foundation for the interactions between the particle and the hard curved surface is analyzed. The following results are derived: (a) Whatever the exponents in the pair potentials may be, the potential of the particle/hard curved surface is always of the unified curvature form, i.e., the potential is always a unified function of the mean curvature and the Gaussian curvature of the curved surface. (b) On the basis of the curvature-based potential, the geometrization of the micro/nano mechanics on hard curved surfaces may be realized. (c) Similar to the extrinsic mechanics on micro/nano soft curved surfaces, in the intrinsic mechanics on micro/nano hard curved surfaces, the curvatures and their intrinsic gradients form the driving forces on the curved spaces. In other words, either on soft curved surfaces or hard curved surfaces and either in the extrinsic mechanics or the intrinsic mechanics, the curvatures and their gradients are all essential factors for the driving forces on the curved spaces. (d) The direction of the driving force induced by the hard curved surface is independent of the hydrophilicity and hydrophobicity of the curved surface, explaining the experimental phenomenon of the spontaneous and directional motion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, C. Study on the Wetting and Slippy Property of Water on Carbon Nano Surface (in Chinese), M. Sc. dissertation, Tsinghua University, Beijing (2010)
Lv, C. J., Chen, C., Yin, Y. J., and Zheng, Q. S. Surface curvature-induced directional movement of water droplets. Preprint at <http://cdsweb.cern.ch/record/1307731> (2010)
Yin, Y. J. and Wu, J. Y. Shaper gradient: a driving force induced by space curvatures. International Journal of Nonlinear Sciences and Numerical Simulations, 11(4), 259–267 (2010)
Yin, Y. J. Symmetries in the mechanics and geometry for biomembranes. Mechanics in Engineering, 30(2), 1–10 (2008)
Yin, Y. J., Chen, Y. Q., Ni, D., Shi, H. J., and Fan, Q. S. Shape equations and curvature bifurcations induced by inhomogeneous rigidities in cell membranes. Journal of Biomechanics, 38, 1433–1440 (2005)
Yin, Y. J., Yin, J., and Lv, C. J. Equilibrium theory in 2D Riemann manifold for heterogeneous biomembranes with arbitrary variational modes. Journal of Geometry and Physics, 58, 122–132 (2008)
Yin, Y. J., Yin, J., and Ni, D. General mathematical frame for open or closed biomembranes: equilibrium theory and geometrically constraint equation. Journal of Mathematical Biology, 51, 403–413 (2005)
Yin, Y. J. and Lv, C. J. Equilibrium theory and geometrical constraint equation for two-component lipid bilayer vesicles. Journal of Biological Physics, 34, 591–610 (2008)
Israelachvili, J. N. Intermolecular and Surface Forces (in Chinese), 2nd ed., Academic Press, London (1991)
Huang, K. Z., Xia, Z. X., Xue, M. D., and Ren, W. M. The Theory of Plates and Shells (in Chinese), Tsinghua University Press, Beijing (1987)
Wu, J. K., Wang, M. Z., and Wang, W. Introductions of Elasticity (in Chinese), Beijing University Press, Beijing (2001)
Huang, K. Z., Xue, M. D., and Lu, M. W. Tensor Analysis, Tsinghua University Press, Beijing (2003)
Yin, Y. J., Wu, J. Y., Fan, Q. S., and Huang, K. Z. Invariants for parallel mapping. Tsinghua Science and Technology, 14(3), 646–654 (2009)
Lorenceau, E. and Quere, D. Drops on a conical wire. Journal of Fluid Mechanics, 510, 29–45 (2004)
Liu, J. L., Xia, R., Li, B. W., and Feng, X. Q. Directional motion of drops in a conical tube or on a conical fiber. Chinese Physics Letter, 24(11), 3210–3213 (2007)
Zheng, Y. M., Bai, H., Huang, Z. B., Tian, X. L., Nie, F. Q., Zhao, Y., Zhai, J., and Jiang, L. Directional water collection on wetted spider silk. nature, 463(4), 640–643 (2010)
Chien, W. Z. Selected Works of Chien Wei-Zang (in Chinese), Vol. 4, Shanghai University Press, Shanghai, 69 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contributed by Quan-shui ZHENG
Project supported by the National Natural Science Foundation of China (Nos. 10872114, 10672089, 10832005, and 11072125)
Rights and permissions
About this article
Cite this article
Yin, Yj., Chen, C., Lü, Cj. et al. Shape gradient and classical gradient of curvatures: driving forces on micro/nano curved surfaces. Appl. Math. Mech.-Engl. Ed. 32, 533–550 (2011). https://doi.org/10.1007/s10483-011-1436-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-011-1436-6