Effects of Proplet on the Deformation of Elastic Gradient Thin Substrate

YANG Yonglin; WANG Xu; LI Xing

doi:10.21656/1000-0887.410175

Volume 42 Issue 1

Jan. 2021

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2021 > 42(1): 58-70

YANG Yonglin, WANG Xu, LI Xing. Effects of Proplet on the Deformation of Elastic Gradient Thin Substrate[J]. Applied Mathematics and Mechanics, 2021, 42(1): 58-70. doi: 10.21656/1000-0887.410175

Citation:

YANG Yonglin, WANG Xu, LI Xing. Effects of Proplet on the Deformation of Elastic Gradient Thin Substrate[J]. Applied Mathematics and Mechanics, 2021, 42(1): 58-70. doi: 10.21656/1000-0887.410175

Citation:

YANG Yonglin, WANG Xu, LI Xing. Effects of Proplet on the Deformation of Elastic Gradient Thin Substrate[J]. Applied Mathematics and Mechanics, 2021, 42(1): 58-70. doi: 10.21656/1000-0887.410175

PDF( 2270 KB)

Effects of Proplet on the Deformation of Elastic Gradient Thin Substrate

doi: 10.21656/1000-0887.410175

School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, P.R.China

Funds: The National Natural Science Foundation of China（11762017）

Received Date: 2020-06-15
Rev Recd Date: 2020-12-17
Publish Date: 2021-01-01

Abstract

Abstract

The phenomenon of droplet wetting has potential significance in cell deformation research and design as well as fabrication of soft devices. In view of the linear tension at the 3-phase contact lines, the gradient thin substrate deformation caused by liquid droplets was studied. Firstly, the constitutive equations of the substrate deformation were solved with the integral transformation method, and the normal displacement expression of the deformation was given. Secondly, the substrate deformation was discussed with different types of elastic moduli of no gradient, the exponential gradient and the power gradient. Finally, the variations of the substrate displacement with the droplet size, the elastic modulus, the linear tension and the gradient index were given. The numerical results show that, with the increases of the elastic modulus and the gradient index, the wetting ridge will go higher and the deformation larger. The smaller the linear tension and the characteristic depth are, the higher the peak displacement value and the larger the deformation will be. When the droplet radius is smaller, the symmetry of the wetting ridge will be better.
- substrate deformation,
- linear tension,
- gradient modulus,
- integral transformation,
- numerical integration of the double Bessel function

FullText(HTML)

References(30)

References

[1]	BUTTH J, GRAF K, KAPPL M. Physics and Chemistry of Interfaces [M]. Wiley-VCH, 2003.
[2]	KUMAR G, PRABHU K N. Review of non-reactive and reactive wetting of liquids on surfaces[J]. Advances in Colloid and Interface Science,2007,133: 61-89.
[3]	QUERE D. Wetting and roughness[J]. Annual Review of Materials Research,2008,38: 71-99.
[4]	LIU M J, WANG S T, JIANG L. Nature-inspired superwettability systems[J]. Nature Reviews Materials,2017,2(7): 17036.
[5]	DE GENNES P G, BROCHARD-WYART F, QUERE D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves [M]. Berlin: Springer, 2010.
[6]	SHULL K R. Contact mechanics and the adhesion of soft solids[J]. Materials Science and Engineering R: Reports,2002,36(1): 1-45.
[7]	ROMAN B, BICO J. Elasto-capillarity: deforming an elastic structure with a liquid droplet[J]. Journal of Physics Condensed Matte r, 2010,22(49): 493101.
[8]	ISRAELACHVILI J N. Intermolecular and Surface Forces [M]. London: Academic Press, 1992.
[9]	WANG H X, ZHOU H, NIU H T. Dual-layer superamphi-phobic/superhydrophobic-oleophilic nanofibrous membranes with unidirectional oil-transport ability and strengthened oil-water separation performance[J]. Advanced Materials Interfaces,2015,2(4): 1400506.
[10]	YOUNG T. An essay on the cohesion of fluids[J]. Philosophical Transactions of the Royal Society of London,1805,95: 65-87.
[11]	LESTER G. Contact angles of liquids at deformable solid surfaces[J]. Journal of Colloid Science,1961,16(4): 315-326.
[12]	TWOHIG T, MAY S, CROLL A B. Microscopic details of a fluid/thin film triple line[J]. Soft Matter,2018,14: 7492-7499.
[13]	CAO Z, DOBRYNIN A V. Polymeric droplets on soft surfaces: from Neumann’s triangle to Young’s law[J]. Macromolecules,2015,48(2):443-451.
[14]	VAN GORCUM M, KARPITSCHKA S, ANDREOTTI B, et al. Spreading on viscoelastic solids: are contact angles selected by Neumann’s law[J].Soft Matter,2020,16: 1306-1322.
[15]	JERISON E R, XU Y, WILEN L A, et al. Deformation of an elastic substrate by a three-phase contact line[J]. Physical Review Letters,2011,106: 186103.
[16]	STYLE R W, DUFRESNE E R. Static wetting on deformable substrates, from liquids to soft solids[J]. Soft Matter,2012,8(27)〖STHZ〗: 7177-7184.
[17]	王宏. 梯度表面能材料上液滴运动及滴状凝结换热[D]. 博士学位论文. 重庆: 重庆大学, 2008.(WANG Hong. Motion of droplets and dropwise condensation on the gradient surface[D]. PhD Thesis. Chongqing: Chongqing University, 2008.(in Chinese))
[18]	BARDALLA, CHEN S Y, DANIELS K E, et al. Gradient-induced droplet motion over soft solids[J]. IMA Journal of Applied Mathematics,2019,85(3): 09413.
[19]	吕存景, 殷雅俊, 郑泉水. 线张力作用下微纳米尺度液滴的非线性粘附[J]. 应用数学和力学, 2008,29(10): 1135-1146.(L Cunjing, YIN Yajun, ZHENG Quanshui. Nonlinear effects of line tension in adhesion of small droplets[J]. Applied Mathematics and Mechanics,2008,29(10): 1135-1146.(in Chinese))
[20]	IRGENS F. Continuum Mechanics [M]. Berlin: Springer-Verlag, 2008.
[21]	XU Y, ENGL W C, JERISON E R, et al. Imaging in-plane and normal stresses near an interface crack using traction force microscopy[J].Proc Natl Acad Sci USA,2010,107(34): 14964-14967.
[22]	LONG D, AJDARI A, LEIBLER L. Static and dynamic wetting properties of thin rubber films[J]. Langmuir,1996,12(21): 5221-5230.
[23]	赵亚溥. 表面与界面物理力学[M]. 北京: 科学出版社, 2012: 177.（ZHAO Yapu. Physical Mechanics of Surface and Interface [M]. Beijing: Science Press, 2012: 177.(in Chinese)）
[24]	BOSTWICK J B, SHEARER M, DANIELS K E. Elastocapillary deformations on partially-wetting substrates: rival contact-line models[J]. Soft Matter,2014,10: 7361-7369.
[25]	华军, 蒋延生, 汪文秉. 双重贝塞尔函数积分的数值计算[J]. 煤田地质与勘探, 2001,29(3): 58-62.(HUA Jun, JIANG Yansheng, WANG Wenbing. The numerical integration of dual Hankel transformation[J]. Coal Geology & Exploration,2001,29(3): 58-62.(in Chinese))
[26]	LI M, HE W, LI Q, et al. The numerical integration algorithm of dual Bessel function and its application[J]. International Journal of Applied Electromagnetics and Mechanics,2010,33: 727-734.
[27]	GINNAKOPOULOS A E, SURESH S. Indentation of solids with gradients in elastic properties, part Ⅱ: axisymmetric indentors[J]. International Journal of Solids and Structures,1997,34(19): 2393-2428.
[28]	GIBSON R E. Some results concerning displacements and stresses in a non-homogeneous elastic half-space[J]. Geotechnique,1967,17: 58-67.
[29]	AMIRFAZLI A, NEUMANN A W. Status of the three-phase line tension: a reviwe[J]. Advances in Colloid & Interface Science,2004,110(3): 121-141.
[30]	DEL ALAMO J C, MEILI R, ALONSO-LATORRE B, et al. Spatio-temporal analysis of eukaryotic cell motility by improved force cytometry[J]. Proceedings of the National Academy of Sciences of the United States of America,2007,104: 13343-13348.