Shape Gradient and Classical Gradient of Curvatures: Driving Forces on Micro/Nano Curved Surfaces

YIN Ya-jun; CHEN Chao; LV Cun-jing; ZHENG Quan-shui

doi:10.3879/j.issn.1000-0887.2011.05.001

Volume 32 Issue 5

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YIN Ya-jun, CHEN Chao, LV Cun-jing, ZHENG Quan-shui. Shape Gradient and Classical Gradient of Curvatures: Driving Forces on Micro/Nano Curved Surfaces[J]. Applied Mathematics and Mechanics, 2011, 32(5): 509-521. doi: 10.3879/j.issn.1000-0887.2011.05.001

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Shape Gradient and Classical Gradient of Curvatures: Driving Forces on Micro/Nano Curved Surfaces

doi: 10.3879/j.issn.1000-0887.2011.05.001

Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, P. R. China;
Division of Mechanics, Nanjing University of Technology, Nanjing 211816, P. R. China

Received Date: 2010-11-18
Rev Recd Date: 2011-03-14
Publish Date: 2011-05-15

Abstract

Abstract

Recent experiment and molecule dynamics simulation showed that adhesion droplet on conical surface could move spontaneously and directionally. Besides, this spontaneous and directional motion was independent of the hydrophilicity and hydrophobicity of the conical surface. Aimed at this important phenomenon, a general theoretical explanation was provided from the viewpoint of the geometrization of micro/nano mechanics on curved surfaces. Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard-curved-surface were st udied, and the geometric foundation for the interactions between the particle and the hard-curved-surface were analyzed. The following results are derived: (a) The potential of the particle/hard-curved-surface is of the unified curvature-form (i. e. the potential is always a unified function of the mean curvature and Gauss 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 can be realized; (c) Curvatures and the intrinsic gradients of curvatures form the driving forces on curved spaces; (d) The direction of the driving force is independent of the hydrophilicity and hydroph obicity of the curved surface, which explains the experimental phenomenon of spontaneous and directional motion.
- micro/nano curved surfaces,
- curvatures,
- shape gradient,
- classical gradient,
- driving forces

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References(17)

References

[1]	陈超.水在碳纳米曲面上的湿润和滑移性质的研究[D].清华大学硕士论文. 北京：清华大学, 2010.（CHEN Chao. Study on the wetting and slippy property of water on carbon nano surface[D].M Sc dissertation, Beijing: Tsinghua University, 2010.）
[2]	LV Cun-jing, CHEN Chao, YIN Ya-jun, ZHENG Quan-shui. Surface curvature-induced directional movement of water droplets[J].arXiv: 1011.3689v1, doi: 10.
[3]	YIN Ya-jun, WU Ji-ye. Shaper gradient: a driving force induced by space curvatures[J]. International Journal of Nonlinear Sciences and Numerical Simulations, 2010, 11(4): 259-267.
[4]	殷雅俊.生物膜力学与几何中的对称[J].力学与实践, 2008, 30（2）： 1-10.（YIN Ya-jun. Symmetries in the mechanics and geometry for biomembranes[J]. Mechanics in Engineering, 2008, 30(2): 1-10.(in Chinese)）
[5]	YIN Ya-jun, CHEN Yan-qiu, NI Dong, SHI Hui-ji, FAN Qin-shan. Shape equations and curvature bifurcations induced by inhomogeneous rigidities in cell membranes[J]. Journal of Biomechanics, 2005, 38(7): 1433-1440. doi: 10.1016/j.jbiomech.2004.06.024
[6]	YIN Ya-jun, YIN Jie, LV Cun-jing. Equilibrium theory in 2D Riemann manifold for heterogeneous biomembranes with arbitrary variational modes[J]. Journal of Geometry and Physics, 2008, 58(1): 122-132. doi: 10.1016/j.geomphys.2007.10.001
[7]	YIN Ya-jun, YIN Jie, NI Dong. General mathematical frame for open or closed biomembranes(part Ⅰ): equilibrium theory and geometrically constraint equation[J]. Journal of Mathematical Biology, 2005, 51(4): 403-413. doi: 10.1007/s00285-005-0330-x
[8]	YIN Ya-jun, LV Cun-jing. Equilibrium theory and geometrical constraint equation for two-component lipid bilayer vesicles[J].J Biol Phys, 2008, 34(6): 591-610.
[9]	Israelachvili J N. Intermolecular and Surface Forces[M]. 2nd ed. London: Academic Press, 1991: 27-28.
[10]	黄克智, 夏之熙, 薛明德, 任文敏. 板壳理论[M].北京：清华大学出版社, 1987: 152-154.(HUANG Ke-zhi, XIA Zhi-xi, XUE Ming-de, REN Wen-min. The Theory of Plates and Shells[M]. Beijing: Tsinghua University Press, 1987: 152-154. (in Chinese))
[11]	武际可, 王敏中, 王炜. 弹性力学引论[M]. 北京：北京大学出版社, 2001: 255-257.(WU Ji-ke, WANG Min-zhong, WANG Wei. Introductions to Elasticity[M]. Beijing: Beijing University Press, 2001: 255-257. (in Chinese))
[12]	黄克智, 薛明德, 陆明万. 张量分析[M]. 第二版. 北京：清华大学出版社, 2003: 221-223.(HUANG Ke-zhi, XUE Ming-de, LU Ming-wan. Tensor Analysis[M]. 2nd ed.Beijing: Tsinghua University Press, 2003: 221-223. (in Chinese))
[13]	YIN Ya-jun, WU Ji-ye, FAN Qin-shan, HUANG Ke-zhi. Invariants for parallel mapping[J]. Tsinghua Science & Technology, 2009, 14(5): 646-654.
[14]	Lorenceau E, Quere D. Drops on a conical wire[J]. J Fluid Mech, 2004, 510: 29-45. doi: 10.1017/S0022112004009152
[15]	LIU Jian-lin, XIA Re, LI Bing-wei, FENG Xi-qiao. Directional motion of drops in a conical tube or on a conical fiber[J]. Chin Phys Lett, 2007, 24(11): 3210-3213. doi: 10.1088/0256-307X/24/11/052
[16]	ZHENG Yong-mei, BAI Hao, HUANG Zhong-bing, TIAN Xue-lin, NIE Fu-qiang, ZHAO Yong, ZHAI Jin, JIANG Lei. Directional water collection on wetted spider silk[J]. Nature, 2010, 463(4): 640-643. doi: 10.1038/nature08729
[17]	钱伟长.钱伟长文选[M]. 第四卷.上海：上海大学出版社, 2004, 69.(CHEIN Wei-zang. Selected Works of Chien Wei-zang[M]. Vol.4. Shanghai: Shanghai University Press, 2004, 69. (in Chinese))