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Analyses of droplet spreading and the movement of wetting line on a solid surface

Jiao Yun-Long Liu Xiao-Jun Pang Ming-Hua Liu Kun

Analyses of droplet spreading and the movement of wetting line on a solid surface

Jiao Yun-Long, Liu Xiao-Jun, Pang Ming-Hua, Liu Kun
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  • Droplet spreading behavior on a substrate is closely bound up with the wettability of the substrate, and plays a critical role in many industrial applications, such as lubrication, painting, coating, and mineral flotation. In this paper, a dynamical model of droplet spreading on a smooth substrate is established through a mechanical analysis. According to the lubrication approximation theory and Navier-Stokes equation, a general nonlinear evolution equation or equations are derived, including the momentum equation, the continuity equation, and the evolution equation of film thickness. We adopt numerical methods to solve these equations, and also quantitatively analyze the relation among film thickness, spreading radius, speed of wetting contact line and time in detail. The results show that the droplet spreading process is mainly divided into two phases, namely expansion phase and contraction phase. Moreover, the spreading process is along with mutual transformation among surface energy, kinetic energy, and different kinds of potential energies. In addition, the final spreading radius Rf of droplet is determined by the inherent wettability of solid surface, and the collapse effect, which emerges at t=0.006 s in the spreading process, is related to Laplace pressure difference of curved liquid surface. Finally, by controlling the droplet size, we obtain the scaling law of droplet spreading radius with time, which approximately meets R ~ t1/7. The scaling law is validated both experimentally and numerically. The results of this study are expected to enhance our knowledge of the movement of wetting contact line and also provide some guidance for the wetting theory.
      Corresponding author: Liu Kun, liukun@hfut.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51375132) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120111110026).
    [1]

    Becker J, Grun G 2005 J. Phys.: Condens. Mat. 17 291

    [2]

    Liu X C 2010 Ph. D. Dissertation (Xi'an: Northwest University) (in Chinese) [刘小川 2010 博士学位论文 (西安: 西北大学)]

    [3]

    Yuan Q Z, Zhao Y P 2013 Sci. Rep. 3 1944

    [4]

    Young T 1805 Phil. Trans. 95 65

    [5]

    Wenzel R N 1936 J. Ind. Eng. Chem. 28 988

    [6]

    Cassie A B D, Baxter S 1944 Trans. Faraday. Soc. 40 546

    [7]

    Lafuma A, Qur D 2003 Nat. Mater. 2 457

    [8]

    Mei M F, Yu B M, Luo L, Cai J C 2010 Chin. Phys. Lett. 27 076802

    [9]

    Wang Y X, Chen S 2015 Acta Phys. Sin. 64 054701 (in Chinese) [王宇翔, 陈硕 2015 物理学报 64 054701]

    [10]

    Blake T D 1969 J. Colloid Interf. Sci. 299 1

    [11]

    Oron A, Davis S H, Bankoff S G 1997 Rev. Mod. Phys. 69 931

    [12]

    de Gennes P G 1985 Rev. Mod. Phys. 57 827

    [13]

    Das S, Marchand A, Andreotti B, Snoeijer J H 2011 Phys. Fluids 23 072006

    [14]

    Yu Y S 2010 Ph. D. Dissertation (Beijing: Institute of Mechanics, Chinese Academy of Sciences) (in Chinese) [余迎松 2010 博士学位论文 (北京: 中国科学院力学研究所)]

    [15]

    Wang X D, Peng X F, He J C, Liu T 2002 J. Eng. Thermophys. 23 67 (in Chinese) [王晓东, 彭晓峰, 阂敬春, 刘涛 2002 工程热物理学报 23 67]

    [16]

    Wang X D, Peng X F, Li D Z 2003 Sci. China Ser. E 33 625 (in Chinese) [王晓东, 彭晓峰, 李笃中 2003 中国科学E辑 33 625]

    [17]

    Wang X D, Peng X F, Wang B X 2003 J. Basic Sci. Eng. 11 396 (in Chinese) [王晓东, 彭晓峰, 王补宣 2003 应用基础与工程科学学报 11 396]

    [18]

    Wang X D, Peng X F, Li D Z 2004 J. Chem. Ind. Eng. 55 402 (in Chinese) [王晓东, 彭晓峰, 李笃中 2004 化工学报 55 402]

    [19]

    Cao X P, Jiang Y M 2005 Acta Phys. Sin. 54 2202 (in Chinese) [曹晓平, 蒋亦民 2005 物理学报 54 2202]

    [20]

    Zhou J C, Geng X G, Lin K J, Zhang Y J, Zang D Y 2014 Acta Phys. Sin. 63 216801 (in Chinese) [周建臣, 耿兴国, 林可君, 张永健, 臧渡洋 2014 物理学报 63 216801]

    [21]

    Chen S, Tao Y, Shen S Q, Li D W 2014 Acta Mech. Sin. 46 329 (in Chinese) [陈石, 陶英, 沈胜强, 李德伟 2014 力学学报 46 329]

    [22]

    Barenblatt G I, Beretta E, Bertsch M 1977 Proc. Nat. Acad. Sci. 94 10024

    [23]

    Seong J K, Myoung W M, Kwang R L, Dae Y L, Woung S C, Ho Y K 2011 J. Fluid Mech. 680 477

    [24]

    Yuan Q Z, Zhao Y P 2013 J. Fluid Mech. 716 171

    [25]

    Navier C L M H 1823 Memories. De France VI. 6 389

    [26]

    Yuan Q Z, Zhao Y P 2010 Phys. Rev. Lett. 104 246101

  • [1]

    Becker J, Grun G 2005 J. Phys.: Condens. Mat. 17 291

    [2]

    Liu X C 2010 Ph. D. Dissertation (Xi'an: Northwest University) (in Chinese) [刘小川 2010 博士学位论文 (西安: 西北大学)]

    [3]

    Yuan Q Z, Zhao Y P 2013 Sci. Rep. 3 1944

    [4]

    Young T 1805 Phil. Trans. 95 65

    [5]

    Wenzel R N 1936 J. Ind. Eng. Chem. 28 988

    [6]

    Cassie A B D, Baxter S 1944 Trans. Faraday. Soc. 40 546

    [7]

    Lafuma A, Qur D 2003 Nat. Mater. 2 457

    [8]

    Mei M F, Yu B M, Luo L, Cai J C 2010 Chin. Phys. Lett. 27 076802

    [9]

    Wang Y X, Chen S 2015 Acta Phys. Sin. 64 054701 (in Chinese) [王宇翔, 陈硕 2015 物理学报 64 054701]

    [10]

    Blake T D 1969 J. Colloid Interf. Sci. 299 1

    [11]

    Oron A, Davis S H, Bankoff S G 1997 Rev. Mod. Phys. 69 931

    [12]

    de Gennes P G 1985 Rev. Mod. Phys. 57 827

    [13]

    Das S, Marchand A, Andreotti B, Snoeijer J H 2011 Phys. Fluids 23 072006

    [14]

    Yu Y S 2010 Ph. D. Dissertation (Beijing: Institute of Mechanics, Chinese Academy of Sciences) (in Chinese) [余迎松 2010 博士学位论文 (北京: 中国科学院力学研究所)]

    [15]

    Wang X D, Peng X F, He J C, Liu T 2002 J. Eng. Thermophys. 23 67 (in Chinese) [王晓东, 彭晓峰, 阂敬春, 刘涛 2002 工程热物理学报 23 67]

    [16]

    Wang X D, Peng X F, Li D Z 2003 Sci. China Ser. E 33 625 (in Chinese) [王晓东, 彭晓峰, 李笃中 2003 中国科学E辑 33 625]

    [17]

    Wang X D, Peng X F, Wang B X 2003 J. Basic Sci. Eng. 11 396 (in Chinese) [王晓东, 彭晓峰, 王补宣 2003 应用基础与工程科学学报 11 396]

    [18]

    Wang X D, Peng X F, Li D Z 2004 J. Chem. Ind. Eng. 55 402 (in Chinese) [王晓东, 彭晓峰, 李笃中 2004 化工学报 55 402]

    [19]

    Cao X P, Jiang Y M 2005 Acta Phys. Sin. 54 2202 (in Chinese) [曹晓平, 蒋亦民 2005 物理学报 54 2202]

    [20]

    Zhou J C, Geng X G, Lin K J, Zhang Y J, Zang D Y 2014 Acta Phys. Sin. 63 216801 (in Chinese) [周建臣, 耿兴国, 林可君, 张永健, 臧渡洋 2014 物理学报 63 216801]

    [21]

    Chen S, Tao Y, Shen S Q, Li D W 2014 Acta Mech. Sin. 46 329 (in Chinese) [陈石, 陶英, 沈胜强, 李德伟 2014 力学学报 46 329]

    [22]

    Barenblatt G I, Beretta E, Bertsch M 1977 Proc. Nat. Acad. Sci. 94 10024

    [23]

    Seong J K, Myoung W M, Kwang R L, Dae Y L, Woung S C, Ho Y K 2011 J. Fluid Mech. 680 477

    [24]

    Yuan Q Z, Zhao Y P 2013 J. Fluid Mech. 716 171

    [25]

    Navier C L M H 1823 Memories. De France VI. 6 389

    [26]

    Yuan Q Z, Zhao Y P 2010 Phys. Rev. Lett. 104 246101

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  • Received Date:  28 April 2015
  • Accepted Date:  25 August 2015
  • Published Online:  05 January 2016

Analyses of droplet spreading and the movement of wetting line on a solid surface

    Corresponding author: Liu Kun, liukun@hfut.edu.cn
  • 1. Institute of Tribology, Hefei University of Technology, Hefei 230009, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 51375132) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120111110026).

Abstract: Droplet spreading behavior on a substrate is closely bound up with the wettability of the substrate, and plays a critical role in many industrial applications, such as lubrication, painting, coating, and mineral flotation. In this paper, a dynamical model of droplet spreading on a smooth substrate is established through a mechanical analysis. According to the lubrication approximation theory and Navier-Stokes equation, a general nonlinear evolution equation or equations are derived, including the momentum equation, the continuity equation, and the evolution equation of film thickness. We adopt numerical methods to solve these equations, and also quantitatively analyze the relation among film thickness, spreading radius, speed of wetting contact line and time in detail. The results show that the droplet spreading process is mainly divided into two phases, namely expansion phase and contraction phase. Moreover, the spreading process is along with mutual transformation among surface energy, kinetic energy, and different kinds of potential energies. In addition, the final spreading radius Rf of droplet is determined by the inherent wettability of solid surface, and the collapse effect, which emerges at t=0.006 s in the spreading process, is related to Laplace pressure difference of curved liquid surface. Finally, by controlling the droplet size, we obtain the scaling law of droplet spreading radius with time, which approximately meets R ~ t1/7. The scaling law is validated both experimentally and numerically. The results of this study are expected to enhance our knowledge of the movement of wetting contact line and also provide some guidance for the wetting theory.

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