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Maximum spread of droplet impacting onto solid surfaces with different wettabilities: adopting a rim-lamella shape Fujun Wang, Lei Yang, Libing Wang, Yong Zhu, and Tiegang Fang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03748 • Publication Date (Web): 28 Jan 2019 Downloaded from http://pubs.acs.org on January 31, 2019
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Langmuir
1
Maximum spread of droplet impacting onto solid surfaces with different wettabilities: adopting a rim-lamella shape
Fujun Wang1, Lei Yang2, Libing Wang1, Yong Zhu1 and Tiegang Fang1* 1
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA 2
College of Civil Engineering, Shenzhen University, Shenzhen 518060, P. R. China
Experimental and theoretical investigations are presented for the maximum spread factor (ߚ ) of an impacting droplet onto solid surfaces with contact angle hysteresis (CAH). Experiments were conducted with deionized water on six surfaces with different wettabilities. The examined Weber number (ܹ݁) falls between ͳͲିଵ and ͳͲଷ . A new energetic model adopting a rim-lamella shape is proposed in order to better represent the droplet shape at the maximum spread. The dynamic contact angle at the maximum spread (ߠఉ ) is introduced in the model to account for the curvature of the surrounding rim induced by the surface wettabilities. A lamella/rim thickness ratio ߢ̱ି ܹ݁ܣ (ܣǡ ܤ Ͳ) is utilized successfully to depict the droplet shape at different ܹ݁ in a unifying manner. Comprehensive evaluations of the model demonstrate that the theoretical prediction can well recover the features of the experimental observations. The L2-error analysis demonstrates the improvement of the proposed model in predicting ߚ for a wide range of ܹ݁ ൌ ͳͲିଵ ̱ͳͲଷ : the calculated errors are smaller than 8% for all the six surfaces. Moreover, the proposed model can also be applied to predict energy conversion/dissipation during droplet spreading process and the effects of surface wettability on ߚ in a reasonable manner. The variation of the percentage of the surface energy and viscous dissipation is consistent with that in previous simulations. The weakness of the current model for predicting ߚ at extremely low Weber number (ܹ݁ ൏ ͳ) is also explained.
* Email address for correspondence:
[email protected] * Mailing address for correspondence: 3246 Engineering Building III, 911 Oval Drive – Campus Box 7910, Raleigh, NC 27695-7910
Acknowledgements This research was supported in part by the Research and Innovation (RISF) program at the North Carolina State University.
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2 Introduction Knowing the maximum spread ( ܦ ) of an impacting drop (with an initial diameter ܦ ) is important in characterizing the underlying droplet impact dynamics on solid surfaces, as discussed in several comprehensive reviews.1,2 It is also closely related to the performance of inkjet printing and spray coating, heat transfer efficiency in spray/droplet cooling applications, size of the bloodstains formed by blood spattering or dripping in forensic science, etc. Predicting the dimensionless maximum spread factor (ߚ ൌ ܦ Ȁܦ ) can be quite difficult as it requires a combination of the knowledge about the conversion among the surface energy (ܧ௦ ), potential energy (ܧ ), kinetic energy (ܧ ) and corresponding dissipation mechanisms (ܹௗ௦௦ ). Moreover, other factors can be involved when analyzing ߚ for droplets impacting onto spherical,3 structured4,5 and porous6 surfaces or for droplets experiencing a magnetic field.7 In order to tackle this problem, different approaches have been developed including energy balance, momentum balance and scaling analysis. In the momentum balance method,8-11 the equation of motion of the rim/blob surrounding the central lamella is first derived. ߚ can then be obtained through the displacement of the moving rim/blob. The scaling method12-14 is more empirical but very effective in predicting ߚ . The key point in this method is to seek appropriate scaling relationships regarding energy terms with respect to the liquid and surface properties. Moreover, the scaling method is useful for obtaining a universal scaling of ߚ for both the viscous and capillary regimes.13,14 Different from the scaling method, the energy balance method seeks a more accurate depiction of energy conversion and dissipation terms. Early works based on the energy consideration include Refs 15 and 16. Table 1 summarizes several representative energetic models which will be used for comparison in this paper. Similar comparison was made by Dong et al..17 Here we add the new models after 2003 to show more recent efforts in the model improvement. Generally, it is shown that major corrections over the years are made in terms of the viscous dissipation and surface energy. A common assumption used by these models is that the kinetic energy at the maximum spread is zero. Although some swirling motion can exist in the rim at the maximum spread,12 the kinetic energy is only less than 5% of the total energy as simulated by Lee et al..18 Here in the table, we did not include the correction in the gravitational energy term because of the relatively limited amount of work on it.
Improvement in the description of viscous dissipation For the viscous dissipation, it can be approximated as the following:19 ௧
௩௦ ൌ ஐ Ԅɘ ൎ Ȱȳݐ ,
(1)
where Ȱ is the averaged viscous dissipation function, ȳ is the dissipated volume where viscous dissipation occurs, and denotes the elapsed time taken for a droplet to reach its maximum spread. As suggested by Pasandideh̺Fard et al.,20 Ȱ is further approximated using the characteristic velocity (ܷ ) and characteristic length (ܮ ) as follows:
ଶ
ȍ ൎ ȣ ቀ ቁ .
(2)
In most of the research as summarized in Table 1, the droplet initial velocity (ܷ ) is taken as ܷ for its simplicity. The characteristic length was simply considered as the droplet height (ܪ ) at the maximum spread in an early publication19 ଶ ܪ ȀͶ). Later in 1996, a model of an axisymmetric stagnation point flow was proposed to calculate Ȱ and ȳ (ȳ ൌ ߨܦ
to represent the liquid motion in the droplet.20
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3 Maximum spread models Ref. Chandra and Avedisian (19) PasandidehFard et al., (20)
Mao et al.,
Viscous dissipation
Equation
ࢃࢋ ࢼ ሺ െ ࢉ࢙ࣂሻࢼ ࡾࢋ െ ൬ ࢃࢋ ൰ ൌ
velocity
ࢃࢋ ࣈ ൌ ඨ ሺ െ ࢉ࢙ࣂࢇ ሻ ሺࢃࢋȀξࡾࢋሻ ࢃࢋǤૡ ቈ ሺ െ ࢉ࢙ࣂሻ Ǥ ࢼ ࡾࢋǤ
(21)
Park et al.,
ࢃࢋ ξࡾࢋ
െ ࢉ࢙ࢻ െ ࢉ࢙ࣂ ൬ ൰൱ ࢼ ܖܑܛ ࢻ ࢃࢋ ઢࡱࡿࡼ െെ ࣊ࢊ ࢽࡸࢂ
(22)
Stagnation point flow
Stagnation point flow
ࢃࢋ െ൬ ൰ ࢼ ൌ
൭Ǥ
ષ
࢚ࢉ
ࢃ࢙
ࣂ
Shape
࣊ࡰ ࢎ
ࡰ ࢁ
Not
Young’s
Cylindrical
included
CA (ࣂࢅ )
disk*
࣊ࡰ ࢾ
ૡࡰ ࢁ
Not
Advancing
Cylindrical
included
CA (ࣂࢇ )
disk*
ૡࡰ ࢁ
Not
Young’s
Cylindrical
included
CA (ࣂࢅ )
disk
Young’s
Spherical
CA (ࣂࢅ )
cap
Linear gradient
Stagnation point flow
Surface energy
࣊ࡰ ࢾȀ or ࣊ࡰ ࢎȀ
࣊ࡰ ࢾȀ or ࣊ࡰ ࢎȀ
ૡࡰ ࢁ
Included
ൌ Ukiwe and
ሺࢃࢋ ሻࢼ ൌ ૡ ࢼ ሾ ൬
Kwok
(28) Li and
൰ െࢉ࢙ࣂࢇ
ࢃࢋ
point flow
ሿ ξࡾࢋ
Chandra
ࡴ࢙ ࢃࢋ න ࢊ ࢊࢎቇ ቆ െ െ ξࡾࢋ ࡴ
(26)
ൌ ࡿ ࡼሺࡰࢋ ሻ െ ࡼሺࡰ ሻ
Gao and Li (32)
ࢃࢋ
࢘ࢉ
ࡾࢉ
ࢉ ൫ࡾ ࢉ െ ൌ ቂෞ ቃ ࣂࢇ ࢘ෝࢉ ࡾ
Stagnation
ࢃࢋ
ࢉ ࢘ෝࢉ ࢙ࣂࢇ ൯ ሺ ࢘ෝࢉ ࢙ࣂࢇ ൯ ൫ࡾ
ξࡾࢋ
െ ࢉ࢙ࣂࢇ ሻ
Stagnation point flow Stagnation point flow
࣊ࡰ ࢾ
ૡࡰ ࢁ
Not
Advancing
Cylindrical
included
CA (ࣂࢇ )
disk
࣊ࡰ ࢾ
െ
Included
Young’s
Spherical
CA (ࣂࢅ )
cap
ૡࡰ ࢁ
Not
Advancing
Ring-like
included
CA (ࣂࢇ )
shape
Not
maximum
Cylindrical
included
spread
disk
࣊ሺࡾࢉ ࢘ࢉ ࢙ࣂ̴ሻ ࢾ࢙
CA at Lee et al.,
࣋ࢂ ࡰ ࢽࡸࢂ
(23)
ൌ ડࢼ ૡࢽࡸࢂ
Yonemoto and Kunugi (24)
Stagnation Ȁ ඥ࢈Ȁࢉ࣋ࢂ ࡰ ࢼ ࢼ ξࡾࢋ
point flow
࣊ࡰ ࢾ
ࡰ ࢁ
(ࣂࢼ )
ࢃࢋ ૠ ࢘ ࢃࢋ െ ࢼ ࢎ ࡾࢋ
CA at
ࢎ ഥ ሻࢼ ࢼ ࢙ࣂ ഥ െሺ െ ࢉ࢙ࣂ ࢘
Stagnation point flow
࣊ࡰ
ࡰ ࢁ
Not
maximum
Deformed
included
spread
shape
ࡿࢊࢋࢌ ࣋ ࢍ࢘ ࢎ െ ൌ ࣌ࢍ ࣊࢘
(ࣂࢼ )
ࢃࢋכ ࢃࢋ ൬ ൰ ࢼ ξࡾࢋ ξࡾࢋ כ Huang and Chen (25)
ሺ െ ࢉ࢙ࣂࢇ ሻࢼ
CA at
െሺࢃࢋ ሻࢼ ૡ ൌ ǡ ࢂ ൏ ࢂכ
Stagnation
࣊ࡰ ࢾ
ࢃࢋ כࢋࡾ כ ࢃࢋ ൬ ൰ ࢼ ξࡾࢋ ξࡾࢋࢋࡾ כ
point flow
ࡰ ࢁ
Included
maximum
Cylindrical
spread
disk
(ࣂࢼ )
ሺ െ ࢉ࢙ࣂࢇ ሻࢼ െሺࢃࢋ ሻࢼ ૡ ൌ ǡ ࢂ ࢂכ
Table 1 Selection of theoretical models for predicting the maximum spread factor based on different corrections to the viscous dissipation and surface energy terms. The resultant boundary layer thickness (ߜ) as below was suggested to replace ܪ as the characteristic length in the ଶ ߜȀͶሻ: calculation of Ȱ (Ȱ ൌ ߤሺܷ Ȁߜሻଶ ) and ȳሺȳ ൌ ߨܦ
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4 ߜൌʹ
బ ξோ
.
(3)
Furthermore, it was pointed out by Mao et al.21 and Park et al.22 that for some high-viscosity liquids, where ߜ ܪ , viscous dissipation can occur within the whole droplet volume. Therefore, the dissipated volume should stick to ȳ ൌ ଶ ܪ ȀͶ. ߨܦ
The critical time ݐ is another important parameter in characterizing ܹ௩௦ . From the very beginning, ݐ ൌ ܦ Ȁܷ was substituted into Eq. (1).19 Later by Pasandideh-Fard et al.,20 ݐ ൌ ͺܦ Ȁ͵ܷ was derived by examining the temporal change of ߚሺݐሻ. A doubt on the above description of ݐ was first raised by Lee et al..23 Comparisons with experimental data showed significant deviations for water, ethanol and glycerol. As a result ݐ ൌ ܦ Ȁܷ was suggested instead. Two more recent publications24,25 adopted a similar critical time scale as ݐ ൌ ܦ Ȁʹܷ . However, the calculated ݐ still deviated a lot from the experimental data as shown by Huang and Chen.25 The last correction to the viscous dissipation reviewed here is the inclusion of the spontaneous spreading process. It was proposed by Park et al.22 that a certain portion of energy could be spontaneously dissipated during the early stage of the droplet impact process. Li et al.26 utilized a dissipation coefficient ܥௌ to calculate the spontaneous dissipation (ܹ௦ ). It was found that neglecting ܹ௦ would lead to larger ߚ . Moreover, Huang and Chen25 further adopted a critical velocity (ܷ ) כto calculate ܹ௦ , which is regarded as the extra dissipation during “interface relaxation” process.
Review of the treatment of the surface energy Unlike the viscous dissipation, corrections to the surface energy term (ܧ௦ ) have received less attention so far. However, it has been pointed out that both the surface energy and the viscous dissipation are important in predicting ߚ .18 We here attempt to review the depiction of ܧ௦ from available literature and organize them according to the selection of the contact angle (ߠ) and the model of droplet shapes. The “surface energy” should consist of both the droplet surface energy and the solid surface energy. This will introduce the contact angle (ߠ) through Young’s equation.27 The contact angle derived in this way is uniquely related to the surface tension and should inherently be the theoretical Young’s contact angle (ߠ ).21,28 In practice, ߠ was usually approximated by the equilibrium contact angle (ߠ ). It was measured at the end of each experiment after the droplet was at rest. For an ideal surface, ߠ is unique. However, for “real” surfaces, the advancing contact angle (ߠ ) can be different from the receding contact angle (ߠ ). A stable ߠ is much harder to obtain. ߠ was first suggested to replace ߠ .20 It was further pointed out that ߠ can be a good approximation of ߠ 28 if viewing the surface as a specific model of a heterogeneous vertical strip surface.29 Another sort of ߠ used in some works18,24-25 is called a hydrodynamic contact angle at the maximum spread (ߠఉ ). Different from ߠ , ߠఉ can involve hydrodynamic effects, which cannot represent solely the surface energetic effect, as criticized.28 However, it is shown in Refs18,23 that ߠఉ generally keeps the same at different impact velocities, meaning that it mainly depends on the nature of the fluid and solid surface. Moreover, it was claimed by Huang and Chen25 that such a ߠఉ can account for the dissipative work at the contact line, which was ignored by almost all the existing models. Details of how ߠఉ can account for the extra dissipation mechanism were not presented though. Other selections of ߠ may also include the plateau contact angle.30 The
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5 importance of selecting an appropriate ߠ in the model can also been seen from the work by Vadillo et al.30 for the prediction of the maximum spread. The droplet shape is another important factor affecting the calculation of ܧ௦ at the maximum spread. From Table 1, it is shown that most models use the cylindrical disk assumption. The total surface energy (ܧ௦ǡఉ ) at the maximum spread can be written as follows: ܧ௦ǡఉ ൌ ܧ௦ǡ ܧ௦ǡௌ ,
(4)
where ܧ௦ǡ denotes the interfacial energy of the liquid-vapor interface and ܧ௦ǡௌ is for the covered solid area by the liquid droplet. For the cylindrical disk model, ܧ௦ǡ consists of the energy of the upper circular disk and the surrounding area of the cylinder. For the early publications19,20 in Table 1 marked with asterisk, the surrounding area was even neglected. This is wrong for low- and medium-Weber number (ܹ݁) conditions, as the size of surrounding area can be comparable to that of the circular disk. The shape of a spherical cap was assumed at the maximum spread in the work by Park et al.22 and Li et al.,26 especially for low-ܹ݁ impact conditions. However, it is not sufficient to describe the actual droplet shape at medium- or higher-ܹ݁ conditions, where a rimmed-disk shape (surrounding rim + central lamella/film) dominates.31 Gao and Li32 constructed a ring-like shape to approximate the actual droplet shape at the maximum spread. In such a droplet structure, the surrounding rim was assumed to be semicircular and the volume of the central liquid film was neglected. Moreover, one recent publication by Yonemoto and Kunugi24 used the harmonic average of the droplet surface of a spherical cap and of a cylindrical disk to approximate the droplet shape at its maximum spread.
Summary Although a certain amount of research exists, debates are still ongoing about appropriate assumption of droplet shape at its maximum spread. Neither the cylindrical disk nor the spherical cap model can well describe the actual droplet shape at the maximum spread for a wide range of Weber numbers (i.e., ܹ݁ ൌ ͳͲ ̱ͳͲଷ ). The selection of the contact angle in the available models is also confusing. In this work we will focus on the above two unresolved issues related to description of the surface energy term. We will revisit the energy approach to derive a new energetic model for predicting ߚ of an impacting droplet. This new model will be validated by the experimental results on six wellprepared, smooth surfaces with different wettabilities and certain contact angle hysteresis (ȟ ܪൌ ߠ െ ߠ ). A more reasonable, unifying lamella/rim model approximating the actual droplet shape at the maximum spread will be incorporated. Remarks on the selection of ߠ will also be given in the end.
Experimental setup and material Visualization method and fluid properties Experimental setup consists of a high-speed imaging system for capturing drop impact process and a lowspeed/high-resolution imaging system for measuring contact angles. The high-speed visualization part is similar to what was used by Wang and Fang.33 The imaging resolution is around 22ߤ݉Ȁ ݈݁ݔ݅for ܹ݁ ൏ ͳͲͲ and Ͷʹߤ݉Ȁ݈݁ݔ݅ for ܹ݁ ͳͲͲ due to equipment limitation: for high ܹ݁ with large ߚ , the object is less magnified in order to capture the full droplet profile, thus resulting in a lower resolution. Different from the Ref.,33 the deionized (DI) water droplet
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6 is generated from a commercial, stainless steel needle of gauge 24 instead of the previous piezoelectric droplet generator. In the current experiment, DI water has a density of ߩ ൌ ͻͻͺ݇݃Ȁ݉ଷ , viscosity of ߤ ൌ ͺǤͻ ൈ ͳͲିସ ܲܽǤ ݏ, and surface tension of ߪ ൌ ͳǤͻ݀݊ݕȀܿ݉. The generated droplet has a diameter of ʹǤͷ േ ͲǤͳ݉݉ and impacts onto target surfaces from 10 different heights varying between ͵݉݉ and ͳͲͲ݉݉. Therefore, the impact velocity can change from ͲǤͲ݉Ȁ ݏto around ͶǤ݉Ȁݏ. In the contact angle measurement system, the high-speed camera and the macro lens are replaced with a Canon 70D DSLR camera and an Infinity® long-distance microscope, respectively. A final resolution of 4.4ߤ݉Ȁ ݈݁ݔ݅is reached for measuring contact angles. The advancing (ߠ ) and receding (ߠ ) contact angles are measured using the sessile drop method.27 The contact angle measurement setup is sketched in Fig. 1. A certain volume (̱Ǥߤ )ܮof DI water is pre-deposited on the tested surface. A syringe needle of gauge 30 is immersed in the droplet very close to the surface top. A computer-controlled push-pull syringe pump adds or removes liquid from the pre-deposited drop at a constant flow rate of 0.1 ݉ܮȀ݉݅݊. The low flow rate is intended to eliminate any induced inertial effects during the contact angle measurement.
Fig. 1 Sketch of the setup for measuring contact angles using sessile drop technique.
Characterization of the substrates The surfaces used in the experiments are microscope glass slide, acrylic sheet, silicon wafer, Teflon sheet, and polydimethylsiloxane (PDMS) substrates. Prior to each set of experiments, the surfaces were rinsed with ethanol and DI water and then dried with compressed air. The PDMS samples were fabricated by spin-coating the Dow Corning Sylgard 184 Silicone Elastomer mixture with two different base/curing agent ratios: 10:1 and 40:1 on silicon wafers and curing at 120Υ in a vacuum oven.34,35 The resulting thickness of the PDMS substrates were measured as ̱ͳͲͲߤ݉. The roughness of the six surfaces (Glass, Acrylic, Silcion, Teflon, PDMS101, PDMS401) were measured using a stylus profilometer (Mitutoyo Sj-310). The wettability of these surfaces was characterized by the sessile drop technique described above in terms of ߠ and ߠ . The contact angle measurement was conducted for both sides of the sessile drop and repeated three times for each substrate. On surfaces such as Glass, Acrylic, Silicon, Teflon, and PDMS101, with the least interference of the inertial effect, liquid generally advances (increasing ߚ) with a stable ߠ and recedes (decreasing ߚ) with a constant ߠ , which is smaller than ߠ . “Stick-slip” phenomenon36 was observed for PDMS 401 surface (see Section S1 of the Supporting Information). During the receding process, once the contact
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Langmuir
7 angle is smaller than a certain value, the three-phase contact line (TPCL) will suddenly retract. In this case, we consider the constant lower limit as the receding contact angle (ߠ ). We also want to briefly list two more contact angles that are commonly used in the modeling of ߚ of an impacting droplet. The first one is the equilibrium Young contact angle (ߠ ), which is calculated as follows for a real surface with contact angle hysteresis37: ߠ ൌ
ିଵ ሺ
ಲ ௦ఏೌ ାೃ ௦ఏೝ ಲ ାೃ
(5)
ሻ
where Ȟ ൌ ቀ
ୱ୧୬య ఏೌ ଶିଷ௦ఏೌ ାୡ୭ୱయ ఏೌ
ଵȀଷ
ቁ
, Ȟோ ൌ ቀ
ୱ୧୬య ఏೝ ଶିଷ௦ఏೝ ାୡ୭ୱయ ఏೝ
ଵȀଷ
ቁ
.
(6)
This correlation is true for smooth surfaces used in the current study. The second is the averaged dynamic contact angle at the maximum spread ߠఉҧ . ߠఉҧ is an averaged value from different ܹ݁ and does not vary much with ܹ݁ (see section S2 of the Supporting Information). It should be noted that the measurement of ߠఉ are conducted using the high-speed imaging setup. Only the data for ܹ݁ up to 60 have been used in the contact angle calculation. For ܹ݁ Ͳ, the resolution is relatively low as 41.92ߤ݉Ȁ ݈݁ݔ݅and the trailing rim of the impacting droplet is quite thin, which makes it difficult to get accurate measurement of the contact angles. Finally, Table 2 summarizes all the available contact angles for the six substrates which will be used in the energetic models. The roughness measurement of the six surfaces is also included in Table 2. Static Quasi-static Dynamic Roughness ܴܽሺߤ݉ሻ ߠఉҧ ߠ ߠ ߠ Glass 0.0037 ͵Ͷι Ͷι േ ʹι ʹͳι േ ʹι ͷι േ ͵ι Acrylic 0.0061 Ͳι ͳι േ ͵ι Ͷͻι േ ͳι Ͳι േ ͵ι Silicon 0.0042 ͺʹι ͻʹι േ ʹι Ͷι േ ʹι ͻͳι േ ʹι Teflon 0.35 ͻͷι ͳͲι േ ʹι ͺͷι േ ʹι ͳͲͺι േ ͵ι PDMS101 0.029 ͳͲͲι ͳͳͺι േ ʹι ͺͺι േ ʹι ͳͳͷι േ ͵ι PDMS401 0.44 ͷͺι ͳ͵ͺι േ ʹι ʹͶι േ ʹι ͳʹͺι േ ͵ι Table 2 Summary of different contact angles (static, quasi-static, dynamic) and measurement of roughness for the six substrates. Substrates
Derivation of the rim-lamella model This part is dedicated to developing a new energetic model for the prediction of maximum spread factor. Figure 2 demonstrates an overview of the whole post-impact droplet developing process. The evolution of droplet morphology, oscillation mode, energy and dissipation terms are shown along the timeline. Since the maximum spread is the focus of the present work, only process 0-1 will be considered. The droplet energy at the initial state, right before impact, is the sum of kinetic energy (ܧǡ ) and surface energy (ܧ௦ǡ ): ଵ
ଵ
గ
ଶ
ଵଶ
ܧǡ ܧ௦ǡ ൌ ቀ ߩܷଶ ቁ ቀ ߨܦଷ ቁ ߨܦଶ ߪ ൌ
ܦଷ ߩܷଶ ߨܦଶ ߪ.
(7)
The term ߪ in the above sections is by default to represent the surface tension between liquid and vapor phases (ߪ ). At state 1 when the maximum spread is achieved, it is a common practice to assign a zero kinetic energy term (ܧǡଵ ൌ Ͳ) as suggested in the introduction. During the process from state 0 to state 1, three kinds of dissipation mechanisms
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Page 8 of 22
8 0
2
3
Contact Line Getting Pinned
Final State
1
Initial State
Maximum Spread
Timeline
Morphology
PCL-Oscillation
MCL-Oscillation
Oscillation Mode
ࡰ ǡ ࢂ
ࡱǡ ࡱ࢙ǡ
ࡰࢌ ǡ ࣂ ࢌ
Viscous Dissipation
ࢃ࢙࢜
ࢃ࢙
Friction Dissipation
ࢃࢉ
Interface Relaxation
Interface Relaxation
ࢃ࢙
ࡱǡ ࡱ࢙ǡ ࡱǡ ൎ
ࡱǡ ࡱ࢙ǡ
ࡰ ࢌ ǡ ࣂࢊ
ࡰ ǡ ࣂࢇ
ࡱ࢙ǡ
Geometric parameter Energy Dissipation
ࡱǡ ࡱ࢙ǡ
ࡱǡ ࡱ࢙ǡ ࡱǡ ൎ
ࡱǡ ൌ
Energy Term
ࡱ࢙ǡ
ࡱ࢙ǡ
Fig. 2 Overview of the whole post-impact droplet developing process. exist: viscous dissipation (ܹ௩௦ ) within the boundary layer, spontaneous dissipation (ܹ௦ ) associated with “interface relaxation”25 and contact line dissipation (ܹ ) due to moving TPCL.4,5,38-43 Based on the energy balance, the following equation is obtained: ܧǡ ܧ௦ǡ ൌ ܹ௩௦ ܹ௦ ܹ ܧ௦ǡଵ .
(8)
viscous dissipation Viscous dissipation is estimated from Eq. (1) and (2). In the proposed model, we adhere to the ଶ stagnation point flow model.20 The choices of the regarding terms are as follows: ܮ ൌ ߜ, ܷ ൌ ܷ , ȳ ൌ ߨܦ ߜȀͶ,
and ݐ ൌ ܦ Ȁʹܷ . It finally yields the following estimated viscous dissipation:25 ܹ௩௦ ൌ
య గ ఘబమ
ଵ
ξோ
.
(9)
Spontaneous dissipation As is suggested by Huang and Chen25, the spontaneous dissipation is related to the relaxation of interface and adopts the following description: ܹ௦ ൌ ߙ
య గ ఘమ
ଵ ξோ
,
(10)
where ܷ is the critical velocity and ܴ݁ is the associated critical Reynolds number, which are obtained through the energetic analysis of the spontaneous spreading of a droplet on the surface. ߙ is the coefficient to denote the portion of the whole spontaneous dissipation it can take until the maximum spread. ߙ ൌ ͳ for ܷ ൏ ܷ , while ߙ ൌ ܷ Ȁܷ for ܷ ܷ . Contact line dissipation The equilibrium contact angle of an droplet on an ideal surface (smooth, homogeneous, and rigid) can be obtained from the Young’s equation:27 ߪௌ െ ߪௌ ൌ ߪ ܿߠݏ .
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(11)
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Langmuir
9 ߪௌ , ߪௌ , and ߪ are the surface tensions for solid/vapor, solid/liquid, and liquid/vapor interfaces, respectively. For drop impact on real surfaces with contact angle hysteresis, extra contact line dissipation can exist as discussed in Refs.4,5,38-43 As are suggested by Refs,44,45 the difference between the quasi-static contact angles (ߠ for advancing process and ߠ for receding process) and the Young contact angle (ߠ ) can be quantified by a friction force acting on the TPCL as illustrated in Fig. 3.
TPCL during receding
TPCL at equilibrium
ߪ
ߪ
ߠ
ߠ
ߠ
ߪௌ
TPCL during advancing
ߪௌ ݂
ߪௌ
ߪௌ
ߪ
݂ ߪௌ
ߪௌ
Fig. 3 Force balance at the three-phase contact line (TPCL) during receding, equilibrium and advancing phases. When TPCL advances with a high advancing contact angle, it experiences a friction force ݂ that can be calculated as below: (12)
݂ ൌ ߪௌ െ ߪௌ െ ߪ
ߠ ൌ ߪሺ
ߠ െ
ߠ ሻ.
When TPCL recedes with a lower receding contact angle, the friction force ݂ changes its direction to prevent TPCL from moving. Force balance analysis gives the following expression for ݂ : (13)
݂ ൌ ߪௌ െ ߪௌ ߪ
ߠ ൌ ߪሺ
ߠ െ
ߠ ሻ. 4,5,38-42
From the above analysis, the friction dissipation at TPCL during process 0-1 can be calculated as follows Ȁଶ
ܹ ൌ ߪሺ
ߠ െ
ߠ ሻ
గ
ଶ ʹߨ ݎ݀ݎൌ ߪܦ ሺ
ߠ െ
ߠ ሻ. ସ
: (14)
Surface energy: rim-lamella model A unifying rim-lamella model with a variable lamella thickness for different impact Weber numbers and a variable contact angle of the rim for substrates with different wettabilities is proposed to represent the droplet shape at its maximum spread. The structure of the droplet shape is sketched in Fig. 4 with a cartesian coordinates system centered at the base of the droplet on the surface. The droplet at the maximum spread (denoted as ܦ ) consists of a surrounding rim and a central lamella. A similar structure has been proposed and utilized in some publications to study the rim dynamics and energy balance during drop spreading.8-11,31 Here, we would like to incorporate the effects of surface wettability on the maximum spread factor (ߚ ) into the rim-lamella model by applying the dynamic contact angle at the maximum spread (ߠఉ ) to the surrounding rim as shown in Fig. 4. Moreover, the dimensionless number ߢ denotes the ratio of the thickness of the central lamella to the surrounding rim. In the proposed model, the cross-section of the surrounding rim adopts the shape of a spherical cap with a radius of ݎ . The height of the rim is denoted as ܪ and the width of the central lamella can be represented as ʹܴ . Based on the droplet structure, two geometric relations between the rim and the lamella can be obtained as follows: ݎ ൌ
ுೝ ଵି௦ఏഁ
,
(15a)
and ଵ
௦ఏഁ
ଶ
ଵି௦ఏഁ
ܴ ൌ ܦ െ ܪ
.
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(15b)
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Page 10 of 22
10
y
Rc Hr
κHr
θβm
rc
θβm
x o
Dm Fig. 4 Sketch of the droplet structure at its maximum spread The volume of the structure in Fig. 4 can be calculated by adding the volume of the surrounding rim and the central lamella: ଶ ଶ
ு
ܸି ൌ ೝ ߨ ቈܴ ටݎଶ െ ൫ ݕ ݎ ܿߠݏఉ ൯ ݀ ݕെ ߨܴଶ ܪ ߨܴଶ ሺߢܪ ሻ.
(16)
By applying the volume of conservation to the impacting droplet (ܸ ൌ ߨܦଷ Ȁ), the following auxiliary equation can be obtained through the substitution of Eq. (15) and (16): ቈ
ଵ
൫ଵି௦ఏഁ ൯
మ
െ
య ଵ ଵିୡ୭ୱ ఏഁ
య
ଷ ൫ଵି௦ఏഁ ൯
ߦଷ
ሺఏഁ ିଵȀଶ௦ଶఏഁ ሻ ଵ ൫ଵି௦ఏഁ ൯
మ
൬ ߚ െ ߦ ଶ
௦ఏഁ ଵି௦ఏഁ
ଵ
௦ఏഁ
ଶ
ଵି௦ఏഁ
൰ ߦଶ ߢߦ ൬ ߚ െ ߦ
ଶ
ଵ
൰ ൌ . (17)
In Eq. (17), ߚ ൌ ܦ Ȁܦ is the maximum spread factor, and ߦ ൌ ܪ Ȁܦ is the corresponding dimensionless rim thickness at the maximum spread. Combining Eq. (12), the surface energy of the solid-liquid-vapor system can be written as follows: (18)
ܧ௦ ൌ ܵ ߪ ܵௌ ሺߪௌ െ ߪௌ ሻ ൌ ߪሺܵ െ ܵௌ
ߠ ሻ.
At the maximum spread, the area of the liquid-vapor interface should include the area of both the surrounding rim and the central lamella: ଶ
ு
ܵ ൌ ೝ ʹߨ ቆܴ ටݎଶ െ ൫ ݕ ݎ ܿߠݏఉ ൯ ቇ
మ ටమ ି൫௬ା ௦ఏഁ ൯
݀ ݕ ʹߨܴ ܪ ሺȁͳ െ ߢȁሻ ߨܴଶ .
(19)
The solid-liquid interface is covered by the bottom of the droplet. Therefore, its area can be simply calculated as: ଵ
ଶ . ܵௌ ൌ ߨܦ
(20)
ସ
By substituting Eq. (15), (19) and (20) into (18), the surface energy at the drop maximum spread can be written as follows: గ
ுೝమ
ସ
൫ଵି௦ఏഁ ൯
ଶ ߪሺͳ െ ܿߠݏ ሻ ߨߪሼ ܧ௦ǡଵ ൌ ܦ
మ
ቂଶ ߠఉ െ
ுೝ
ߠ݊݅ݏఉ ൫ͳ െ ܿߠݏఉ ൯ ʹሺͳ െ ܿߠݏఉ ሻቃ ʹܪ ሺ ܪ
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௦ఏഁ ଵି௦ఏഁ
ሻሺȁͳ െ ߢȁ
ఏഁ ଵି௦ఏഁ
ଶ
െ
ሻሽ. (21)
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Langmuir
11 Substituting Eq. (7), (9), (10), (14) and (21) into (8) and normalizing the equation with ܦ yield the dimensionless form of the energy balance equation: ଷ
ܹ݁ ͳʹ ൌ ൬
ௐ
ସ ξோ
ߙ
ௐ ඥோ
ଷ ଶ ൰ ߚ ͵ሺͳ െ ܿߠݏ ሻߚ ͳʹሼ
కೝమ ൫ଵି௦ఏഁ ൯ ఉ
ʹሺͳ െ ܿߠݏఉ ሻቃ ʹߦ ሺ
ଶ
మ
ቂଶ ߠఉ െ
െ ߦ
௦ఏഁ ଵି௦ఏഁ
ఉ కೝ
ߠ݊݅ݏఉ ൫ͳ െ ܿߠݏఉ ൯
ሻሺȁͳ െ ߢȁ
ఏഁ ଵି௦ఏഁ
ሻሽ. (22)
Equations (17) and (22) are the two governing equations for the new model. It should be noted that ߢ is also an unknown at the current stage and it can be quite different at different impact conditions. One can imagine that at a relatively low Weber number, the lamella thickness can be comparable to the rim height (large ߢ). However, at high Weber number, a very thin lamella will occur, which gives a small value of ߢ. Therefore, ߢ is then assumed to be correlated with the impact Weber number as follows: ߢ ൌ ݂ሺܹ݁ሻ.
(23)
With the knowledge of ߢ as derived in the next, the spread factor and the dimensionless rim height can be obtained by solving these non-linear equations simultaneously.
Results and discussion In the following discussion, the investigation of ߢ will first be demonstrated. Then, a comprehensive evaluation of our new model will be provided. Comparison with other recent models will be included. Brief discussions will be given on the weakness of this new model. Finally, the proposed model will also be extended to study the energy conversion/dissipation during drop impact and the effects of the wettability (ߠ ) on the maximum spread factor. Scaling of the lamella/rim thickness ratio ૂ To verify the above proposed relationship between ߢ and ܹ݁, we have to first examine the droplet shape from experimental observations. Figure 5 summarizes representative droplet shapes at the maximum spread for different Weber numbers. These different deformation modes as a function of Weber number have also been found by Wildeman et al.46 from simulations when studying drop impact on no-slip and free-slip surfaces. At an extremely small Weber number (ܹ݁ ൌ ͲǤͳ), the droplet basically remains its spherical shape. When ܹ݁ goes a little higher to ܹ݁ ൌ ͳǤͷ, flat surface occurs around the top of the droplet, indicating a surrounding rim is formed. In this case, ߢ should be larger than 1, corresponding to the existence of the spherical top above the rim. When the Weber number increases to ܹ݁ ൌ ͷǤͶ, the spherical top flattens to a similar height to the rim. Moreover, no hollow structure is observed beneath the rim top. Therefore, at such a Weber number, the lamella thickness is roughly the same as the rim height (ߢ ൎ ͳ). When the Weber number goes even higher to ܹ݁ ൌ ͳ͵ǤͶ, a concave shape can be roughly observed located below the rim top. In this case, a certain distance still exists between the concave bottom and the solid surface (Ͳ ൏ ߢ ൏ ͳ). However, at a very high Weber number (ܹ݁ ൌ ͶʹǤͳ), the central lamella can be very thin compared to the surrounding rim. Subsequently, a hollow ring-like shape can be a good approximation (ߢ ൎ Ͳ).
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Page 12 of 22
12 It is seen that the droplet shape at the maximum spread cannot be represented solely by a spherical cap, a cylinder or a hollow ring-like shape for the whole range of Weber numbers. The proposed variable lamella/rim model can depict those different deformation modes with a variable ߢ as shown in Fig. 5. The parameter ߢ should physically
ࢃࢋ ൌ Ǥ ࣄ ࢃࢋ ൌ Ǥ
ࢃࢋ ൌ Ǥ
ࣄൎ
ࢃࢋ ൌ Ǥ
൏ࣄ൏
ࢃࢋ ൌ ૠ
ࣄൎ
Actual droplet shape
Simplified model
Fig. 5 Actual droplet shape at maximum spread and its corresponding simplified model at representative Weber numbers. satisfy the inequality: ߢ Ͳ. Two special cases of ߢ ൌ ͳ and ߢ ൌ Ͳ correspond to the cylindrical shape and hollow ring-like shape, respectively. The lamella thickness ܪ is not available via current visualization setup. Experimental measurement of the lamella thickness could be fulfilled by certain 3D measurement techniques, i.e., Digital Inline Holography (DIH), which should be conducted for further validation of the current model in the future. Some numerical works9,47 were conducted previously to study the flow in the lamella. But the investigation of ܪ was only presented for some selected Weber numbers. The lamella thickness was experimentally measured by Lagubeau et al.,48 but the comparison with the rim height was not presented. In the current work, only the maximum spread factor ߚ and the rim height at maximum spread ܪ are experimentally available. The experimental values of ߢ based on the proposed droplet structure can be then calculated from Eq. (17). For the current study, ߢ is assumed to be only a function of the single variable ܹ݁ as aforementioned. Therefore, a regression analysis is conducted to find a semiempirical correlation for ߢ with respect to ܹ݁ in the following form: ߢ ൌ ି ܹ݁ܣ ሺܣǡ ܤ Ͳሻ.
(24)
The equation above can inherently satisfy the trend of ߢ as summarized in Fig. 5. Figure 6 plots the variation of the calculated ߢ with the Weber number for the six surfaces. At large ܹ݁, it is shown that ߢ does approach an asymptotic value around zero. For ܹ݁ ͷͲ, the variation of ߢ for all the six surfaces generally collapses to a single curve. Differences do exist for different surfaces, especially at low and medium ܹ݁. At extremely low Weber number (ܹ݁ ൌ ͲǤͳ), an abrupt decrease of ߢ was noticed in our calculation. For hydrophobic surfaces, such as Teflon, PDMS101 and PDMS401, ߢ becomes even smaller than zero, which contradicts the physical limit of ߢ for the proposed model. This is because at extremely low Weber number, the detection of the rim height can be very difficult, as the droplet tends to be more like a spherical cap. Therefore, we took the height of the droplet as the rim height in
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Langmuir
13 this case, which inherently overestimates the actual rim height. This indicates that the proposed model may not be suitable for extremely low-ܹ݁ cases on hydrophobic surfaces. This can also be shown from the evaluation of the proposed model in the next. For the regression analysis in current study, we eliminate the data points for extremely low-ܹ݁ cases (ܹ݁ ൌ ͲǤͳ) and only demonstrate the calculated values of ߢ for ܹ݁ ͲǤ in Fig. 6. The dashed line in Fig. 6 denotes the empirical correlation that can best fit the averaged experimental ߢ of the six surfaces. The optimal values of ܣand ܤturn out to be 1.07 and 0.17, respectively. The resultant R-squared value is 0.85 for the data from all the surfaces.
ࣄ ൌ Ǥ ૠࢃࢋିǤૠ
Fig. 6 Variation of the lamella/rim thickness ratio ߢ with different Weber numbers. The dashed line demonstrates the best fitting to correlate the experimental data with R-square = 0.85.
Evaluation of the new model Up to this point, all the governing equations required for solving ߚ have been derived. The solution of ߚ can be obtained through solving Eq. (17), (22), and (24) simultaneously. Efforts in this part will first focus on the evaluation of the new model for the investigated ܹ݁. Variation of the portion of different energy and dissipation terms will be demonstrated. Effects of the advancing contact angle (ߠ ) on ߚ will be analyzed in the end. Nine selected models from Table 1 are first compared with the current experimental data (see Section S3 of the Supporting Information for the comparison). The model by Li et al.26 is not included here because the coefficients of ܥ௦ and ܥ are still unknown, which are required to solve ߚ . From the experimental results, two features can be observed. First, it is shown that ߚ is the biggest for droplet impacting on the Glass surface (with the smallest ߠ ), while the smallest on the PDMS401 surface (with the largest ߠ ). The relation of ߚǡீ௦௦ ߚǡ௬ ߚǡௌ ߚǡ் ߚǡெௌଵଵ ߚǡெௌସଵ is always true when ܹ݁ ൏ ͳͲͲ, which indicates wettability dominates in this regime. Second, when ܹ݁ ͳͲͲ, ߚ for the six different surfaces collapses to the same and the wettability effect dies out. So, these two experimentally observed features can generally serve as the two criteria to evaluate different theoretical models. Some of these models cannot predict the two features very well due to inappropriate selection of
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Page 14 of 22
14 contact angle19,21-22 or calculation of spontaneous viscous dissipation28 for our examined surfaces with certain CAH. Moreover, deviations shown by two specific models22,32 indicate that neither spherical-cap nor ring-like shape can be assumed for the full range of investigated ܹ݁. Figure 7 compares the newly proposed rim-lamella model with the experimental data for the six examined surfaces. It is shown that this new model can generally well predict the maximum spread factor for a full range of Weber numbers fromͳͲିଵ to ͳͲଷ . The above two criteria can be well satisfied. Especially for ܹ݁ ͳ, the predictions for all the six surfaces are more accurate than that for ܹ݁ ൏ ͳ as validated by the experimental data.
ࣄ
ࣄ൏
Fig. 7 Comparison of the prediction of ߚ from the newly proposed model with the experimental data for Glass, Acrylic, Silicon, Teflon, PDMS101 and PDMS401 surfaces.
Chandra
Pasandideh-
Mao
Park
Ukiwe
Gao
Lee
Yonemoto
Huang
Current
et al.
Fard et al.
et al.
et al.
et al.
et al.
et al.
et al.
et al.
model
Glass
5.7%
25.8%
28.1%
47.6%
20.5%
26.3%
14.7%
10.3%
9.1%
5.9%
Acrylic
16.6%
14.2%
11.5%
30.3%
8.5%
26.8%
11.8%
8.2%
7.8%
7.1%
Silicon
22.6%
12.3%
6.9%
27.8%
5.7%
25.5%
8.9%
10.2%
13.5%
4.3%
Teflon
26.9%
12.5%
7.8%
25.8%
8.6%
24.3%
12.3%
8.9%
18.7%
6.9%
PDMS101
25.0%
10.1%
5.7%
27.0%
13.0%
30.7%
14.5%
11.4%
12.7%
3.0%
PDMS401
14.5%
8.9%
27.6%
50.2%
12.5%
30.7%
13.7%
10.2%
12.3%
4.4%
Substrate
Table 3 Least-square error (ߝ) of different models for the six surfaces. To better quantify the improvement in the proposed model, an L2 error is defined as:
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Langmuir
15
ߝൌ
ฮఉǡ ିఉǡೣ ฮ ฮఉǡೣ ฮ
,
(25)
and ߚǡௗ are the experimentally measured and theoretically predicted maximum spread factor. ԡߚ ԡ where ߚǡ௫
is the L2 norm49 of the maximum spread factor ߚ . Table 3 summarizes the least-square error for the 9 previous models and the current new model. It is shown that the current model gives the best overall prediction for the whole range of Weber number (ܹ݁ ൌ ͳͲିଵ ̱ͳͲଷ ) among other available models. The least-square errors for all the six examined surfaces are below 10%. To further investigate the capability of the proposed model, the relative error at each experimental ܹ݁ condition has also been investigated (see Section S4 of the Supporting Information). It is shown that within the full ܹ݁ region ͳͲିଵ ൏ ܹ݁ ൏ ͳͲଷ , relative errors for more hydrophilic surfaces (Glass, Acrylic, Silicon) are smaller than 10%. This indicates that the proposed rim-lamella model can have a stable, accurate prediction of ߚ for hydrophilic surfaces with relatively small ߠ . For hydrophobic surfaces (Teflon, PDMS101, PDMS401 in current study), although the relative error can be as high as 25% when ܹ݁ ൏ ͳ, it generally decreases to much smaller values below 5% for high Weber numbers (ܹ݁ ͷͲ). This also explains why PDMS101 and PDMS401 even have smaller overall Least-square error albeit the existence of large local errors for small Weber numbers. Weakness of the rim-lamella model Before extending the rim-lamella model to investigate the energy dissipation and effects of wettability, we would like to make brief comments on the weakness of the proposed model. First, a “transition point” can be observed at ܹ݁ ൌ ͳǤͶ on each prediction curve in Fig. 7, which connects two curve segments. The gradients of the two curve segments at the “transition point” are different. Such gradient discontinuity is more obvious for hydrophobic surfaces (Teflon, PDMS101, PDMS401) than in the other three surfaces. Moreover, with the existence of the gradient discontinuity, the variation of ߚ decreases abruptly, leading to inaccurate predictions for ܹ݁ ൏ ͳ on Teflon, PDMS101 and PDMS401 surfaces. Physically, the “transition point” marks the change of a convex central lamella (ߢ ͳ) to a concave central lamella (ߢ ൏ ͳ). From the experimental data, it is shown that the rim-convex central lamella shape cannot well represent the shape of droplet at maximum spread when ܹ݁ ൏ ͳ for hydrophobic surfaces. The second weakness lies in the neglect of the viscoelastic effects50-52 for PDMS401 surface. The viscoelastic substrate is so soft that wetting ridge can be formed during the advancing process of TPCL53, which can yield extra energy dissipation mechanism. Analysis of energy dissipation and effects of wettability (ࣂࢇ ) In Fig. 8, the proportion of the viscous dissipated energy (ܹ௩௦ Ȁሺܧǡ ܧ௦ǡ ሻ), the spontaneous dissipation (ܹ௦ Ȁሺܧǡ ܧ௦ǡ ሻ), the friction dissipation at the TPCL (ܹ Ȁሺܧǡ ܧ௦ǡ ሻ), and the surface energy (ܧ௦ǡଵ Ȁሺܧǡ ܧ௦ǡ ሻ) at the maximum spread are plotted as a function of ܹ݁ for the five surfaces excluding PDMS401 (due to the unknown viscoelastic dissipation), based on the new model. Generally, for all the five surfaces at small ܹ݁, a significant amount of the initial energy is transferred to the surface energy at the maximum spread. With the increase of ܹ݁, the proportion of ܧ௦ǡଵ decreases. On the other hand, the energy dissipated by viscous effects is tiny at small ܹ݁. However, it can increase to a very high value with the increase of ܹ݁. The variation trend of the proportion of viscous dissipation and surface energy are consistent with the simulation results by Lee et al..18 The proportion of the
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Page 16 of 22
16 friction dissipation ܹ decreases with the increase of ܹ݁ and generally stays below 10%. Friction dissipation at contact line (ܹ ) is more dominant for PDMS101 than the other four surfaces. At low ܹ݁, it is shown that the spontaneous dissipation related to “interface relaxation” should account for the major part of energy dissipation.
ܹ௩௦ ܹ௦ ܹ ܧ௦ǡଵ
Glass
Acrylic
Silicon
Teflon PDMS101
Fig. 8 Variation of the proportion of different energy terms at the maximum spread with ܹ݁ (ܹ݁ ͳǤͶ) for droplet impacting on Glass, Acrylic, Silicon, Teflon and PDMS101 surfaces. With this new model, the influence of contact angle on the maximum spread for droplet impacting onto smooth, rigid surfaces can be further scrutinized. As shown in Eq. (22), both the advancing contact angle (ߠ ) and the dynamic contact angle at the maximum spread (ߠఉ ) are involved in determining the maximum spread factor (ߚ ). In the derivation of the proposed model, ߠ is introduced by the contact-line dissipation, while ߠఉ is involved by considering the droplet shape at the maximum spread. In the current study, ߠఉҧ is used as an approximate of ߠఉ as aforementioned. As can be seen from Table 2, ߠఉ is almost the same as ߠ for Acrylic, Silicon, Teflon and PDMS101 surfaces. However, for Glass and PDMS401, ߠఉ can be different from ߠ with a difference of ͳͲι. A detailed study of such deviation shall be carried out in the future to determine whether the difference is caused by either the impacting inertia effects, the viscoelastic behavior of PDMS401, or by the error of the measurement (currently, ߠఉ is measured from high-speed video with a resolution of 22ߤ݉Ȁ݈݁ݔ݅, while ߠ is measured with a resolution of 4ߤ݉Ȁ)݈݁ݔ݅. Considering the small difference, for the current investigation of the effect of wettability on ߚ for rigid surfaces, we would like to assume that ߠఉ ൌ ߠ and thus, only ߠ is involved in the model. It has also been validated that such a
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17 small difference between ߠఉ and ߠ does not result in obvious deviation in the prediction of ߚ (see S5 of Supporting Information). Therefore, the variation of ߚ with ߠ is plotted in Fig. 9 for 6 different ܹ݁ conditions varying between ͳͲ and ͳͲଷ . It is shown that at the same ܹ݁, ߚ generally decreases with the increase of ߠ . This makes sense as a high ߠ means more hydrophobic, which tends to prevent the droplet from spreading. Moreover, from Fig. 9, the influence of ߠ on ߚ is shown to be weaker as ܹ݁ increases. At high ܹ݁, the inertial effect dominates over the wettability in determining ߚ . This finding is also consistent with the observation of data collapse in Fig. 7 for all the six different surfaces. The effect of the advancing contact angle (ߠ ) on the spreading process has also been recently elaborated by Jiang et al. in the study of spreading force during spontaneous spreading of a droplet.41
Fig. 9 Effect of advancing contact angle (ߠ ) on the maximum spread factor for droplet impacting onto rigid, smooth surface with different Weber numbers: ܹ݁ ൌ ͳǡ ͳͲǡ ͷͲǡ ʹͲͲǡ ͷͲͲǡ ͳͲͲͲ.
Conclusions and outlook In the present work, we conducted both experimental and theoretical investigations of the maximum spread factor (ߚ ) of an impacting drop onto solid surfaces with different wettabilities and certain contact angle hysteresis (CAH). The experiment was done with deionized water on six surfaces (Glass, Acrylic, Silicon, Teflon, PDMS101, PDMS401) with different ߠ . The experimental impact Weber numbers (ܹ݁) falled between ͳͲିଵ and ͳͲଷ . The quasi-static advancing (ߠ ) and receding (ߠ ) contact angles were measured with the sessile drop technique. The dynamic contact angle at maximum spread (ߠఉ ) was measured through the high-speed videos. A detailed derivation of the rim (with variable ߠఉ )-lamella (with variable ߢ) model was provided in this work. Compared to previous models, the new one features two major modifications in approximating actual droplet shape at maximum spread. First, the influence of the dynamic contact angle at maximum spread (ߠఉ ) is incorporated in the model to account for the curvature of the rim, which is determined by the surface wettability. The other modification
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18 focuses on the depict of the central lamella at its maximum spread. It is shown that different droplet deformation modes exist at different ܹ݁. This indicates that none of the cylinder, spherical-cap, or hollow ring-like shape can depict the droplet shape for a wide range of ܹ݁, i.e., ͳͲିଵ ൏ ܹ݁ ൏ ͳͲଷ . In the new model, a variable lamella/rim thickness ratio ߢ is suggested, where ߢ is found to well scale as ߢ̱ି ܹ݁ܣ (ܣǡ ܤ Ͳ). Such a scaling relation enables us to describe the droplet shape for all the six surfaces at different ܹ݁ in a unifying manner. A comprehensive evaluation of the new theoretical model has been presented. Comparison with the experimental data shows that the new model can well recover the two features of the experimental observations: (1) ߚǡீ௦௦ ߚǡ௬ ߚǡௌ ߚǡ் ߚǡெௌଵଵ ߚǡெௌସଵ is true for ܹ݁ ൏ ͳͲͲ, and (2) ߚ of the six surfaces collapses when ܹ݁ ͳͲͲ. The L2-error analysis demonstrates the improvement of this new model in predicting ߚ for a wide range of ܹ݁. The L2-error of the predictions by the new model are smaller than 8% for all the six examined surfaces. Further investigation of the variation of the local relative error indicates that the new model is most accurate for droplet impact with ܹ݁ ൌ ͳͲିଵ ̱ͳͲଷ on hydrophilic surfaces and for droplet impact with ܹ݁ ൌ ͳͲ ̱ͳͲଷ on hydrophobic surfaces. Based on the new model, the energy transfer and dissipation were studied for drop impact process till its maximum spread. The variation trend of the portion of viscous dissipation and surface energy are generally consistent with the simulation results by Lee et al..18 It is shown that the surface energy occupies a large portion especially when ܹ݁ ൏ ͳͲଶ . This indicates the high necessity of accurately approximating droplet shape as it is related to the calculation of the surface energy at maximum spread. For low Weber numbers, interface relaxation (spontaneous dissipation) is found to account for the major energy dissipation. However, for a high Weber number ܹ݁ ͳͲଶ , most portion of the energy is taken away by the viscous dissipation. The approximation of the droplet shape and the inclusion of the friction dissipation are not that important any more. Finally, the proposed model is used to study the effect of ߠ on the maximum spread factor ߚ . Generally, it is found that a higher ߠ leads to a smaller ߚ as the surface gets more hydrophobic. The influence of ߠ weakens as ܹ݁ becomes higher, when the inertia starts to dominate in determining ߚ . In the end, we would like to point out that our study suggests that there exists anomaly in the new model when predicting ߚ for extremely small ܹ݁ (especially for droplet impacting onto hydrophobic surfaces). Currently, we hypothesize that this anomaly is from the approximation of the droplet shape. The current rim-lamella model cannot approximate the shape of a spherical cap very well. Further theoretical efforts should be devoted to better incorporating the shape of a spherical cap into current model. Moreover, further experimental investigations of the real 3D droplet structure at the maximum spread are highly desired to improve the current model. Supporting Information. A note on the measurement of wettability of PDMS substrates; Measurement of dynamic contact angle at maximum spread; Comparison of the selected energetic models for the prediction of ߚ ; Investigation of the local relative error for the rim-lamella model; Comparison of the predictions using different contact angles in the model.
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19 REFERENCES (1) Yarin, A. L. Drop impact dynamics: splashing, spreading, receding, bouncing…. Annu. Rev. Fluid Mech. 2006, 38, 159-192. (2) Josserand, C.; Thoroddsen, S. T. Drop impact on a solid surface. Annual Review of Fluid Mechanics 2016, 48, 365-391. (3) Bordbar, A.; Taassob, A.; Khojasteh, D.; Marengo, M.; & Kamali, R. Maximum Spreading and Rebound of a Droplet Impacting onto a Spherical Surface at low Weber numbers. Langmuir 2018, 34(17), 5149-5158. (4) Vaikuntanathan, V.; Sivakumar, D. Maximum spreading of liquid drops impacting on groove-textured surfaces: effect of surface texture. Langmuir 2016, 32(10), 2399-2409. (5) Guo, C.; Zhao, D.; Sun, Y.; Wang, M.; Liu, Y. Droplet Impact on Anisotropic Superhydrophobic Surfaces. Langmuir 2018, 34(11), 3533-3540. (6) Zhao, P.; Hargrave, G. K.; Versteeg, H. K.; Garner, C. P.; Reid, B. A.; Long, E. J.; Zhao, H. The dynamics of droplet impact on a heated porous surface. Chemical Engineering Science 2018, 190, 232-247. (7) Ahmed, A.; Fleck, B. A.; Waghmare, P. R. Maximum spreading of a ferrofluid droplet under the effect of magnetic field. Physics of Fluids 2018, 30(7), 077102. (8) Fedorchenko, A. I.; Wang, A. B. The formation and dynamics of a blob on free and wall sheets induced by a drop impact on surfaces. Physics of Fluids 2004, 16(11), 3911-3920. (9) Eggers, J.; Fontelos, M. A.; Josserand, C.; Zaleski, S. Drop dynamics after impact on a solid wall: theory and simulations. Physics of Fluids 2010, 22(6), 062101. (10) Roisman, I. V.; Rioboo, R.; Tropea, C. Normal impact of a liquid drop on a dry surface: model for spreading and receding. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 2002, 458, 1411-1430. (11) Roisman, I. V. Inertia dominated drop collisions. II. An analytical solution of the Navier–Stokes equations for a spreading viscous film. Physics of Fluids 2009, 21(5), 052104. (12) Clanet, C.; Béguin, C.; Richard, D.; Quéré, D. Maximal deformation of an impacting drop. Journal of Fluid Mechanics 2004, 517, 199-208. (13) Laan, N.; de Bruin, K. G.; Bartolo, D.; Josserand, C.; Bonn, D. Maximum diameter of impacting liquid droplets. Physical Review Applied 2014, 2(4), 044018. (14) Lee, J. B.; Laan, N.; de Bruin, K. G.; Skantzaris, G.; Shahidzadeh, N.; Derome, D.; Carmeliet, J.; Bonn, D. Universal rescaling of drop impact on smooth and rough surfaces. Journal of Fluid Mechanics 2016, 786. (15) Madejski, J. Solidification of droplets on a cold surface. International Journal of Heat and Mass Transfer 1976, 19(9), 1009-1013. (16) Scheller, B. L.; Bousfield, D. W. Newtonian drop impact with a solid surface. AIChE Journal 1995, 41(6), 1357-1367. (17) Dong, H.; Carr, W. W.; Bucknall, D. G.; Morris, J. F. Temporally̺resolved inkjet drop impaction on surfaces. AIChE Journal 2007, 53(10), 2606-2617.
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21 (37) Tadmor, R. Line energy and the relation between advancing, receding, and young contact angles. Langmuir 2004, 20(18), 7659-7664. (38) Butt, H. J.; Gao, N.; Papadopoulos, P.; Steffen, W.; Kappl, M.; Berger, R. Energy dissipation of moving drops on superhydrophobic and superoleophobic surfaces. Langmuir 2016, 33(1), 107-116. (39) Jiang, X.; Li, H. Z. Liquid drops hurdling barriers of various geometries. Advanced Materials Interfaces 2017, 4(17), 1700516. (40) Wang, Y.; Andrews, J. E.; Hu, L.; Das, S. Drop spreading on a superhydrophobic surface: pinned contact line and bending liquid surface. Physical Chemistry Chemical Physics 2017, 19(22), 14442-14452. (41) Jiang, Y.; Sun, Y.; Drelich, J. W.; Choi, C. H. Spontaneous Spreading of a Droplet: The Role of Solid Continuity and Advancing Contact Angle. Langmuir 2018, 34(17), 4945-4951. (42) Guo, C.; Zhao, D.; Sun, Y.; Wang, M.; Liu, Y. Droplet Impact on Anisotropic Superhydrophobic Surfaces. Langmuir 2018, 34(11), 3533-3540. (43) Liu, M.; Chen, X. P. Numerical study on the stick-slip motion of contact line moving on heterogeneous surfaces. Physics of Fluids 2017, 29(8), 082102. (44) Kim, H. Y.; Lee, H. J.; Kang, B. H. Sliding of liquid drops down an inclined solid surface. Journal of Colloid and Interface Science 2002, 247(2), 372-380. (45) Liu, T. L.; Chen, Z.; Kim, C. J. A dynamic Cassie–Baxter model. Soft Matter 2015, 11(8), 1589-1596. (46) Wildeman, S.; Visser, C. W.; Sun, C.; Lohse, D. On the spreading of impacting drops. Journal of fluid mechanics 2016, 805, 636-655. (47) Roisman, I. V.; Berberović, E.; Tropea, C. Inertia dominated drop collisions. I. On the universal flow in the lamella. Physics of Fluids 2009, 21(5), 052103. (48) Lagubeau, G.; Fontelos, M. A.; Josserand, C.; Maurel, A.; Pagneux, V.; Petitjeans, P. Spreading dynamics of drop impacts. Journal of Fluid Mechanics 2012, 713, 50-60. (49) Horn, R. A.; Johnson, C. R. Norms for vectors and matrices. Matrix analysis 1990, 313-386. (50) Carré, A.; Gastel, J. C.; Shanahan, M. E. Viscoelastic effects in the spreading of liquids. Nature 1996, 379(6564), 432. (51) Extrand, C. W.; Kumagai, Y. Contact angles and hysteresis on soft surfaces. Journal of colloid and interface science 1996, 184(1), 191-200. (52) Sokuler, M.; Auernhammer, G. K.; Roth, M.; Liu, C.; Bonacurrso, E.; Butt, H. J. The softer the better: fast condensation on soft surfaces. Langmuir 2009, 26(3), 1544-1547. (53) Kajiya, T.; Daerr, A.; Narita, T.; Royon, L.; Lequeux, F.; Limat, L. Advancing liquid contact line on viscoelastic gel substrates: stick-slip vs. continuous motions. Soft Matter 2013, 9(2), 454-461.
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Maximum spread factor ܍܋܉ܚܝܛܖܗ܋ܑܔܑ܁ܖܗǡ ࣂࢼ ൎ ૢι
Rim-lamella model
ࣄ൏
ࣄൌ
ࣄ
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