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超疏水表面液滴的振动特性与接触线的移动、液滴体积、基底振幅等因素密切相关. 本文在基底振幅较小且恒定的条件下, 研究了超疏水表面液滴的共振振幅、模式区间、共振频率等振动特性及其与液滴体积(20—500 μL)的关系. 此外, 将基于一般性疏水表面建立的Noblin共振频率计算模型应用于超疏水表面, 并提出“虚驻点”的概念, 借此对模型进行了误差分析和修正. 研究表明: 1)共振时, 液滴高度变化率即比振幅随体积增大而增大, 随阶数增大而减小; 2)各模式区间的起止频率首尾相接, 其范围随体积增大而减小; 3)液滴体积越大, 共振频率越小, 随着阶数增大, 共振频率f与体积V的关系趋于f -V–0.4, 不同于一般性疏水表面上的f -V–0.5; 4)直接应用Noblin模型计算共振频率会产生较大误差, 主要原因在于液滴表面波波段数量统计存在较大偏差, 而修正后的模型可以准确计算超疏水表面大体积液滴的共振频率.In-depth understanding is limited to the oscillation properties of a droplet on a superhydrophobic surface, which are closely related to the contact line movement, droplet volume, and substrate amplitude, to name only a few factors. In the present work, we investigate the characteristics of droplet resonance amplitude, mode range, and resonance frequency, as well as their correlations with droplet volume (from 20 to 500 μL). In particular, the theoretical resonance frequency is mainly concerned and addressed. To this end, a model based on general hydrophobic surfaces proposed by Noblin et al. is employed, with its applicability to superhydrophobic surfaces examined. We propose a concept “virtual stationary point” for analyzing the errors from this model, with which we modify the model through using the correction coefficients. The main results are concluded as follows. 1) Under resonance, the change rate in droplet height rises with the increase of droplet volume and reduces with the increase of oscillation mode number. 2) Each number of oscillation mode corresponds to a frequency range, and the ends of adjacent mode ranges are connected to each other. These frequency ranges decrease with the increase of droplet volume. 3) Resonance frequency, f, decreases with the increase of droplet volume, V, and they are related approximated by f -V–0.4 under high mode numbers, which is different from f -V–0.5 as found on general hydrophobic surfaces. 4) Direct application of Noblin model to a superhydrophobic surface results in nonnegligible errors, because geometric characteristics in this case are different from those on a general hydrophobic surface, which leads to inaccuracy in counting the number of surface wave segments. In contrast, results from modified Noblin model accord well with experimental results.
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Keywords:
- superhydrophobic surface /
- droplet oscillation /
- resonance frequency /
- mode range
[1] Vukasinovic B, Smith M K, Glezer A 2004 Phys. Fluids 16 306Google Scholar
[2] Nisisako T, Torii T 2007 Adv. Mater. 19 1489Google Scholar
[3] Rodot H 1979 Acta Astronaut. 6 1083Google Scholar
[4] Strani M, Sabetta F 1984 J. Fluid Mech. 141 233Google Scholar
[5] Noblin X, Buguin A, Brochard-Wyart F 2009 Eur. Phys. J.Spec. Top. 166 7Google Scholar
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[7] Iwata S, Yamauchi S, Yoshitake Y, Nagumo R, Mori H, Kajiya T 2016 Rev. Sci. Instrum. 87 045106Google Scholar
[8] Matsumoto T, Nakano T, Fujii H, Kamai M, Nogi K 2002 Phys. Rev. E 65 031201Google Scholar
[9] Jonas A, Karadag Y, Tasaltin N, Kucukkara I, Kiraz A 2011 Langmuir 27 2150Google Scholar
[10] Mugele F, Baret J C, Steinhauser D 2006 Appl. Phys. Lett. 88 204106Google Scholar
[11] Oh J M, Legendre D, Mugele F 2012 Europhys. Lett. 98 34003Google Scholar
[12] Mugele F 2011 Lab Chip 11 2011Google Scholar
[13] Mampallil D, van den Ende D, Mugele F 2011 Appl. Phys. Lett. 99 154102Google Scholar
[14] Beckingham L J, Todorovic M, Velasquez J T, Vial M L, Chen M, Ekberg J A K, St John J A 2019 J. Biol. Eng. 13 41Google Scholar
[15] Rayleigh L 1879 Proc. R. Soc, London 29 71Google Scholar
[16] Milne A J, Defez B, Cabrerizo-Vilchez M, Amirfazli A 2014 Adv. Colloid Interface Sci. 203 22Google Scholar
[17] Vukasinovic B, Smith M K, Glezer A 2007 J. Fluid Mech. 587 395Google Scholar
[18] Kim H, Yang J, Chung J 2014 Jpn. J. Appl. Phys. 53 05HC03Google Scholar
[19] Chang C T, Bostwick J B, Steen P H, Daniel S 2013 Phys. Rev. E 88 023015Google Scholar
[20] McGuiggan P M, Grave D A, Wallace J S, Cheng S, Prosperetti A, Robbins M O 2011 Langmuir 27 11966Google Scholar
[21] Oh J M, Ko S H, Kang K H 2008 Langmuir 24 8379Google Scholar
[22] Costalonga M, Brunet P 2020 Phys. Rev. Fluid 5 023601Google Scholar
[23] 王再冉 2018 硕士学位论文 (北京: 北京化工大学)
Wang Z R 2018 M. S. Dissertation (Beijing: Beijing University of Chemical Technology) (in Chinese)
[24] Rahimzadeh A, Khan T, Eslamian M 2019 Eur. Phys. J. E 42 125Google Scholar
[25] SmithwickIII R W, Boulet J A M 1989 J. Colloid Interface Sci. 130 588Google Scholar
[26] Noblin X, Buguin A, Brochard-Wyart F 2004 Eur. Phys. J. E 14 395Google Scholar
[27] Lyubimov D V, Lyubimova T P, Shklyaev S V 2006 Phys. Fluids. 18 012101Google Scholar
[28] Ilyukhina M A, Makov Y N 2009 Acoust. Phys. 55 722Google Scholar
[29] Brunet P 2011 Eur. Phys. J-Spec. Top. 192 207Google Scholar
[30] 田野, 张屹然, 王宏, 朱恂, 陈蓉, 丁玉栋, 廖强 2019 工程热物理学报 40 829
Tian Y, Zhang Y R, Wang H, Zhu X, Chen R, Ding Y D, Liao Q 2019 J. Eng. Thermophys. 40 829
[31] 周建臣, 耿兴国, 林可君, 张永建, 臧渡洋 2014 物理学报 63 216801Google Scholar
Zhou J C, Geng X G, Lin K J, Zhang Y J, Zang D Y 2014 Acta Phys. Sin. 63 216801Google Scholar
[32] Mettu S, Chaudhury M K 2012 Langmuir 28 14100Google Scholar
[33] 陈文 2015 硕士学位论文 (重庆: 重庆大学)
Chen W 2015 M. S. Dissertation (Chongqing: Chongqing University) (in Chinese)
[34] Sanyal A, Basu S 2017 Chem. Eng. Sci. 163 179Google Scholar
[35] Li X, Wang Y, Wang R, Wang S, Zang D, Geng X 2018 Adv. Mater. Interfaces 5 1800356Google Scholar
[36] Ramos S M M 2008 Nucl. Instrum. meth. B 266 3143Google Scholar
[37] Landau L, Lifshitz M 1987 Fluid Mechanics 2 (Oxford: Pergamon) pp247−250
[38] Li X, Wang R, Shi H, Song B 2018 Appl. Phys. Lett. 113 101602Google Scholar
[39] Wang R, Li X 2020 Powder Technol. 367 608Google Scholar
[40] Li X, Wang R, Huang S, Wang Y, Shi H 2018 Soft Matter 14 9877Google Scholar
[41] Li X 2019 Adv. Colloid Interface Sci. 271 101988Google Scholar
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图 5 Noblin模型示意图 (a)接触线固着型; (b)接触线移动型, 其中蓝色区域表示静止液滴, 实线、虚线分别为液滴处于最大和最小高度时的轮廓
Fig. 5. Illustrations of two types of Noblin models: (a) Fixed contact line; (b) mobile contact line, where the blue areas represent the static droplets, the solid and dashed curves represent droplet profiles at the maximum and minimum heights, respectively.
表 1 不同体积及阶数对应的修正系数
$ \alpha $ Table 1. Values of correction coefficient
$ \alpha $ under different droplet volumes and oscillation mode numbers.液滴体积/μL 振动阶数 3 4 5 6 20 0.73573 0.68488 0.69771 0.71784 70 0.47875 0.55128 0.49756 0.32655 200 0.46078 0.22800 0.33208 0.09438 500 0.32044 0.16831 0.33258 0.13569 -
[1] Vukasinovic B, Smith M K, Glezer A 2004 Phys. Fluids 16 306Google Scholar
[2] Nisisako T, Torii T 2007 Adv. Mater. 19 1489Google Scholar
[3] Rodot H 1979 Acta Astronaut. 6 1083Google Scholar
[4] Strani M, Sabetta F 1984 J. Fluid Mech. 141 233Google Scholar
[5] Noblin X, Buguin A, Brochard-Wyart F 2009 Eur. Phys. J.Spec. Top. 166 7Google Scholar
[6] Fujii H, Matsumoto T, Izutani S, Kiguchi S, Nogi K 2006 Acta Mater. 54 1221Google Scholar
[7] Iwata S, Yamauchi S, Yoshitake Y, Nagumo R, Mori H, Kajiya T 2016 Rev. Sci. Instrum. 87 045106Google Scholar
[8] Matsumoto T, Nakano T, Fujii H, Kamai M, Nogi K 2002 Phys. Rev. E 65 031201Google Scholar
[9] Jonas A, Karadag Y, Tasaltin N, Kucukkara I, Kiraz A 2011 Langmuir 27 2150Google Scholar
[10] Mugele F, Baret J C, Steinhauser D 2006 Appl. Phys. Lett. 88 204106Google Scholar
[11] Oh J M, Legendre D, Mugele F 2012 Europhys. Lett. 98 34003Google Scholar
[12] Mugele F 2011 Lab Chip 11 2011Google Scholar
[13] Mampallil D, van den Ende D, Mugele F 2011 Appl. Phys. Lett. 99 154102Google Scholar
[14] Beckingham L J, Todorovic M, Velasquez J T, Vial M L, Chen M, Ekberg J A K, St John J A 2019 J. Biol. Eng. 13 41Google Scholar
[15] Rayleigh L 1879 Proc. R. Soc, London 29 71Google Scholar
[16] Milne A J, Defez B, Cabrerizo-Vilchez M, Amirfazli A 2014 Adv. Colloid Interface Sci. 203 22Google Scholar
[17] Vukasinovic B, Smith M K, Glezer A 2007 J. Fluid Mech. 587 395Google Scholar
[18] Kim H, Yang J, Chung J 2014 Jpn. J. Appl. Phys. 53 05HC03Google Scholar
[19] Chang C T, Bostwick J B, Steen P H, Daniel S 2013 Phys. Rev. E 88 023015Google Scholar
[20] McGuiggan P M, Grave D A, Wallace J S, Cheng S, Prosperetti A, Robbins M O 2011 Langmuir 27 11966Google Scholar
[21] Oh J M, Ko S H, Kang K H 2008 Langmuir 24 8379Google Scholar
[22] Costalonga M, Brunet P 2020 Phys. Rev. Fluid 5 023601Google Scholar
[23] 王再冉 2018 硕士学位论文 (北京: 北京化工大学)
Wang Z R 2018 M. S. Dissertation (Beijing: Beijing University of Chemical Technology) (in Chinese)
[24] Rahimzadeh A, Khan T, Eslamian M 2019 Eur. Phys. J. E 42 125Google Scholar
[25] SmithwickIII R W, Boulet J A M 1989 J. Colloid Interface Sci. 130 588Google Scholar
[26] Noblin X, Buguin A, Brochard-Wyart F 2004 Eur. Phys. J. E 14 395Google Scholar
[27] Lyubimov D V, Lyubimova T P, Shklyaev S V 2006 Phys. Fluids. 18 012101Google Scholar
[28] Ilyukhina M A, Makov Y N 2009 Acoust. Phys. 55 722Google Scholar
[29] Brunet P 2011 Eur. Phys. J-Spec. Top. 192 207Google Scholar
[30] 田野, 张屹然, 王宏, 朱恂, 陈蓉, 丁玉栋, 廖强 2019 工程热物理学报 40 829
Tian Y, Zhang Y R, Wang H, Zhu X, Chen R, Ding Y D, Liao Q 2019 J. Eng. Thermophys. 40 829
[31] 周建臣, 耿兴国, 林可君, 张永建, 臧渡洋 2014 物理学报 63 216801Google Scholar
Zhou J C, Geng X G, Lin K J, Zhang Y J, Zang D Y 2014 Acta Phys. Sin. 63 216801Google Scholar
[32] Mettu S, Chaudhury M K 2012 Langmuir 28 14100Google Scholar
[33] 陈文 2015 硕士学位论文 (重庆: 重庆大学)
Chen W 2015 M. S. Dissertation (Chongqing: Chongqing University) (in Chinese)
[34] Sanyal A, Basu S 2017 Chem. Eng. Sci. 163 179Google Scholar
[35] Li X, Wang Y, Wang R, Wang S, Zang D, Geng X 2018 Adv. Mater. Interfaces 5 1800356Google Scholar
[36] Ramos S M M 2008 Nucl. Instrum. meth. B 266 3143Google Scholar
[37] Landau L, Lifshitz M 1987 Fluid Mechanics 2 (Oxford: Pergamon) pp247−250
[38] Li X, Wang R, Shi H, Song B 2018 Appl. Phys. Lett. 113 101602Google Scholar
[39] Wang R, Li X 2020 Powder Technol. 367 608Google Scholar
[40] Li X, Wang R, Huang S, Wang Y, Shi H 2018 Soft Matter 14 9877Google Scholar
[41] Li X 2019 Adv. Colloid Interface Sci. 271 101988Google Scholar
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