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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 16 306
Google Scholar
[2] Nisisako T, Torii T 2007 Adv. Mater. 19 1489
Google Scholar
[3] Rodot H 1979 Acta Astronaut. 6 1083
Google Scholar
[4] Strani M, Sabetta F 1984 J. Fluid Mech. 141 233
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[5] Noblin X, Buguin A, Brochard-Wyart F 2009 Eur. Phys. J.Spec. Top. 166 7
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[6] Fujii H, Matsumoto T, Izutani S, Kiguchi S, Nogi K 2006 Acta Mater. 54 1221
Google Scholar
[7] Iwata S, Yamauchi S, Yoshitake Y, Nagumo R, Mori H, Kajiya T 2016 Rev. Sci. Instrum. 87 045106
Google Scholar
[8] Matsumoto T, Nakano T, Fujii H, Kamai M, Nogi K 2002 Phys. Rev. E 65 031201
Google Scholar
[9] Jonas A, Karadag Y, Tasaltin N, Kucukkara I, Kiraz A 2011 Langmuir 27 2150
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[10] Mugele F, Baret J C, Steinhauser D 2006 Appl. Phys. Lett. 88 204106
Google Scholar
[11] Oh J M, Legendre D, Mugele F 2012 Europhys. Lett. 98 34003
Google Scholar
[12] Mugele F 2011 Lab Chip 11 2011
Google Scholar
[13] Mampallil D, van den Ende D, Mugele F 2011 Appl. Phys. Lett. 99 154102
Google 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 41
Google Scholar
[15] Rayleigh L 1879 Proc. R. Soc, London 29 71
Google Scholar
[16] Milne A J, Defez B, Cabrerizo-Vilchez M, Amirfazli A 2014 Adv. Colloid Interface Sci. 203 22
Google Scholar
[17] Vukasinovic B, Smith M K, Glezer A 2007 J. Fluid Mech. 587 395
Google Scholar
[18] Kim H, Yang J, Chung J 2014 Jpn. J. Appl. Phys. 53 05HC03
Google Scholar
[19] Chang C T, Bostwick J B, Steen P H, Daniel S 2013 Phys. Rev. E 88 023015
Google Scholar
[20] McGuiggan P M, Grave D A, Wallace J S, Cheng S, Prosperetti A, Robbins M O 2011 Langmuir 27 11966
Google Scholar
[21] Oh J M, Ko S H, Kang K H 2008 Langmuir 24 8379
Google Scholar
[22] Costalonga M, Brunet P 2020 Phys. Rev. Fluid 5 023601
Google 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 42 125
Google Scholar
[25] SmithwickIII R W, Boulet J A M 1989 J. Colloid Interface Sci. 130 588
Google Scholar
[26] Noblin X, Buguin A, Brochard-Wyart F 2004 Eur. Phys. J. E 14 395
Google Scholar
[27] Lyubimov D V, Lyubimova T P, Shklyaev S V 2006 Phys. Fluids. 18 012101
Google Scholar
[28] Ilyukhina M A, Makov Y N 2009 Acoust. Phys. 55 722
Google Scholar
[29] Brunet P 2011 Eur. Phys. J-Spec. Top. 192 207
Google 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 216801
Google Scholar
Zhou J C, Geng X G, Lin K J, Zhang Y J, Zang D Y 2014 Acta Phys. Sin. 63 216801
Google Scholar
[32] Mettu S, Chaudhury M K 2012 Langmuir 28 14100
Google 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. 163 179
Google Scholar
[35] Li X, Wang Y, Wang R, Wang S, Zang D, Geng X 2018 Adv. Mater. Interfaces 5 1800356
Google Scholar
[36] Ramos S M M 2008 Nucl. Instrum. meth. B 266 3143
Google 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 101602
Google Scholar
[39] Wang R, Li X 2020 Powder Technol. 367 608
Google Scholar
[40] Li X, Wang R, Huang S, Wang Y, Shi H 2018 Soft Matter 14 9877
Google Scholar
[41] Li X 2019 Adv. Colloid Interface Sci. 271 101988
Google Scholar
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图 1 实验装置示意图, 图中玻片凹面处的曲率大小与实际情况相同.
Figure 1. Sketch of experiment set, with the curvature of the concave area equal to that of the real substrate.
图 2 500 μL液滴在不同频率下的振动瞬态图像叠加(接触线位移非常小, 近似认为三相线处为驻点) (a) 30 Hz; (b) 50 Hz; (c) 70 Hz; (d) 100 Hz
Figure 2. Snapshot-superimposed images of an oscillated droplet (500 μL) under different frequencies: (a) 30 Hz; (b) 50 Hz; (c) 70 Hz; (d) 100 Hz.
图 3 接触线固着(左)和移动(右)模式下驻点数分别为6和4的液滴振动示意图
Figure 3. Schematics of droplet oscillations under pinned (left) and moving (right) contact lines, with the quantity of stationary points being 6 and 4, respectively.
图 4 不同体积液滴的幅频关系 (a) 20 μL, (b) 70 μL, (c) 200 μL, (d) 500 μL; 模式区间(e), 共振比振幅(f)与液滴体积的关系
Figure 4. Droplet amplitudes versus oscillation frequencies for different droplet volumes: (a) 20 μL; (b) 70 μL; (c) 200 μL; (d) 500 μL; mode ranges (e) and resonance amplitudes (f) versus droplet volumes.
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
Figure 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.
图 6 普通坐标系(a)和双对数坐标系(b)中液滴3—6阶振动的共振频率与体积的关系曲线
Figure 6. Theoretical and experimental resonance frequencies versus droplet volumes under oscillation mode numbers ranging from 3 to 6: (a) General coordinate system; (b) logarithmic coordinate system.
图 7 超疏水表面液滴表面波的结构示意图
Figure 7. Schematic of the surface wave structure of a droplet on a superhydrophobic surface.
图 8 利用Noblin接触线固着模型(a)和移动模型(b)求得的共振频率的相对误差与体积的关系
Figure 8. Relative errors of resonance frequencies, obtained from the two types of Noblin models, versus droplet volumes: (a) Fixed-contact-line model; (b) mobile-contact-line model.
图 9 包含修正系数
$ \alpha$ 和$\beta $ 的表面波结构示意图Figure 9. Schematic of the surface wave structure with correction coefficients (
$ \alpha\;{\rm{and}}\;\beta $ ) involved.
图 10 (a)修正系数
$ \alpha $ 的拟合曲线; (b)修正前后Noblin模型的相对误差Figure 10. (a) Fitted curves of the correction coefficient
$ \alpha $ ; (b) Relative errors of resonance frequencies from pristine and modified Noblin models.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 
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[1] Vukasinovic B, Smith M K, Glezer A 2004 Phys. Fluids 16 306
Google Scholar
[2] Nisisako T, Torii T 2007 Adv. Mater. 19 1489
Google Scholar
[3] Rodot H 1979 Acta Astronaut. 6 1083
Google Scholar
[4] Strani M, Sabetta F 1984 J. Fluid Mech. 141 233
Google Scholar
[5] Noblin X, Buguin A, Brochard-Wyart F 2009 Eur. Phys. J.Spec. Top. 166 7
Google Scholar
[6] Fujii H, Matsumoto T, Izutani S, Kiguchi S, Nogi K 2006 Acta Mater. 54 1221
Google Scholar
[7] Iwata S, Yamauchi S, Yoshitake Y, Nagumo R, Mori H, Kajiya T 2016 Rev. Sci. Instrum. 87 045106
Google Scholar
[8] Matsumoto T, Nakano T, Fujii H, Kamai M, Nogi K 2002 Phys. Rev. E 65 031201
Google Scholar
[9] Jonas A, Karadag Y, Tasaltin N, Kucukkara I, Kiraz A 2011 Langmuir 27 2150
Google Scholar
[10] Mugele F, Baret J C, Steinhauser D 2006 Appl. Phys. Lett. 88 204106
Google Scholar
[11] Oh J M, Legendre D, Mugele F 2012 Europhys. Lett. 98 34003
Google Scholar
[12] Mugele F 2011 Lab Chip 11 2011
Google Scholar
[13] Mampallil D, van den Ende D, Mugele F 2011 Appl. Phys. Lett. 99 154102
Google 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 41
Google Scholar
[15] Rayleigh L 1879 Proc. R. Soc, London 29 71
Google Scholar
[16] Milne A J, Defez B, Cabrerizo-Vilchez M, Amirfazli A 2014 Adv. Colloid Interface Sci. 203 22
Google Scholar
[17] Vukasinovic B, Smith M K, Glezer A 2007 J. Fluid Mech. 587 395
Google Scholar
[18] Kim H, Yang J, Chung J 2014 Jpn. J. Appl. Phys. 53 05HC03
Google Scholar
[19] Chang C T, Bostwick J B, Steen P H, Daniel S 2013 Phys. Rev. E 88 023015
Google Scholar
[20] McGuiggan P M, Grave D A, Wallace J S, Cheng S, Prosperetti A, Robbins M O 2011 Langmuir 27 11966
Google Scholar
[21] Oh J M, Ko S H, Kang K H 2008 Langmuir 24 8379
Google Scholar
[22] Costalonga M, Brunet P 2020 Phys. Rev. Fluid 5 023601
Google 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 42 125
Google Scholar
[25] SmithwickIII R W, Boulet J A M 1989 J. Colloid Interface Sci. 130 588
Google Scholar
[26] Noblin X, Buguin A, Brochard-Wyart F 2004 Eur. Phys. J. E 14 395
Google Scholar
[27] Lyubimov D V, Lyubimova T P, Shklyaev S V 2006 Phys. Fluids. 18 012101
Google Scholar
[28] Ilyukhina M A, Makov Y N 2009 Acoust. Phys. 55 722
Google Scholar
[29] Brunet P 2011 Eur. Phys. J-Spec. Top. 192 207
Google 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 216801
Google Scholar
Zhou J C, Geng X G, Lin K J, Zhang Y J, Zang D Y 2014 Acta Phys. Sin. 63 216801
Google Scholar
[32] Mettu S, Chaudhury M K 2012 Langmuir 28 14100
Google 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. 163 179
Google Scholar
[35] Li X, Wang Y, Wang R, Wang S, Zang D, Geng X 2018 Adv. Mater. Interfaces 5 1800356
Google Scholar
[36] Ramos S M M 2008 Nucl. Instrum. meth. B 266 3143
Google 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 101602
Google Scholar
[39] Wang R, Li X 2020 Powder Technol. 367 608
Google Scholar
[40] Li X, Wang R, Huang S, Wang Y, Shi H 2018 Soft Matter 14 9877
Google Scholar
[41] Li X 2019 Adv. Colloid Interface Sci. 271 101988
Google Scholar
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