ARTICLE pubs.acs.org/Langmuir
Dynamic Wetting and Spreading Characteristics of a Liquid Droplet Impinging on Hydrophobic Textured Surfaces Jae Bong Lee and Seong Hyuk Lee* School of Mechanical Engineering, Chung-Ang University, 221 Heuksuk-dong, Dongjak-gu, Seoul 156-756, Korea ABSTRACT: We report on the wetting dynamics of a 4.3 μL deionized (DI) water droplet impinging on microtextured aluminum (Al 6061) surfaces, including microhole arrays (hole diameter 125 μm and hole depth 125 μm) fabricated using a conventional microcomputer numerically controlled (μCNC) milling machine. This study examines the influence of the texture area fraction φs and drop impact velocity on the spreading characteristics from the measurement of the apparent equilibrium contact angle, dynamic contact angle, and maximum spreading diameter. We found that for textured surfaces the measured apparent contact angle (CA) takes on values of up to 125.83°, compared to a CA of approximately 80.59° for a nontextured bare surface, and that the spreading factor decreases with the increased texture area fraction because of increased hydrophobicity, partial penetration of the liquid, and viscous dissipation. In particular, on the basis of the model of Ukiwe and Kwok (Ukiwe, C.; Kwok, D. Y.Langmuir 2005, 21, 666), we suggest a modified equation for predicting the maximum spreading factor by considering various texturing effects and wetting states. Compared with predictions by using earlier published models, the present model shows better agreement with experimental measurements of the maximum spreading factor.
I. INTRODUCTION Since the pioneering work of Worthington (1876), who studied the impact of oil and mercury droplets at low pressure and found the splashing pattern and jet formation during the impact, many researchers have studied the behavior of a droplet impinging on solid surfaces.110 Droplet impact and spreading are crucial in a variety of technical applications, including thin film coating, pesticide application, spray painting, surface cooling, deposition of solder bumps on printed circuit boards, and inkjet printing.35 Many researchers have conducted extensive studies on the spreading behavior of impinging droplets.68 When a droplet impinges on a solid surface, the dynamic behavior of the droplet after impingement depends substantially on fluid properties, including surface tension and viscosity, kinematics related to the droplet size and velocity, and surface characteristics that can be altered by surface roughness or texturing. Those factors can affect the corresponding wettability9,10 and different outcomes after impact such as deposition, splashing, and rebound.11 Among these, droplet spreading phenomena are of great interest in many industrial applications, including inkjet printing, electrical cooling, spray painting, and others. At low impact velocity of a droplet, a droplet is far from equilibrium when first deposited and the dynamic contact angle changes over time. Some research has been conducted regarding the spreading behavior of a droplet after impact.1214 In examining the spreading characteristics of a liquid droplet on a surface, an accurate estimation of the maximum spreading diameter would be of key importance in understanding the detailed r 2011 American Chemical Society
physics behind both the dynamic behavior of the liquid droplet and the thermal energy transport occurring at the surface. The maximum spreading factor (Dm*) is defined as the ratio of the maximum spreading diameter (Dm) to droplet diameter before impact (d0); this quantity is closely associated with the energy balance among droplet kinetic energy, viscous dissipation, and surface energy during impingement. Pasandideh-Fard et al.12 suggested a prediction model for the maximum spreading factor by considering capillary effects, assuming a cylindrically shaped liquid film and taking into account the advancing contact angle during film spreading. Since then, this model has been modified by Ukiwe and Kwok13 to consider the lateral area of a liquid film on the walls and to use the dynamic contact angle measured at maximum spreading. More recently, Vadilo et al.14 proposed a prediction model using the mean value of the apparent dynamic contact angle during the spreading phase (plateau contact angle, θd) to reflect capillary effects. However, these models were developed for smooth, flat surfaces. As was noted previously, the behavior of a droplet impinging on a solid surface can be controlled by three representative factors: the fluid properties (surface tension and viscosity), kinematics (droplet size and velocity), and surface characteristics (roughness and controllable surfaces). Much research has been reported on the variation of the apparent contact angle (θCA) of a gently deposited droplet in order to describe the wetting Received: December 4, 2010 Revised: March 8, 2011 Published: May 03, 2011 6565
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Langmuir characteristics of microstructured surfaces.1517 The change in the apparent contact angle depends on the wetting state (e.g., the Cassie or Wenzel state).18,19 In particular, some researchers described both a metastable regime representing a transition from the Cassie state to the Wenzel state and a mixed state involving the Cassie and Wenzel states.20 In particular, active surface characterizations such as texturing have recently emerged as one of the feasible methods for controlling droplet behavior. Some researchers have examined droplet behavior on textured surfaces with superhydrophobic characteristics2125 and measured the dynamic contact angle and spreading diameter in the rebound regime for water-repellent surfaces. Xu21 showed that droplet behavior in the splashing regime depends on roughness and surface texturing. Malouin et al.24 examined directed rebounding characteristics of droplets impinging on microscale surface roughness gradients and reported that rebounding on textured surfaces was substantially affected by the uniformity of the surface roughness. They also suggested that the use of nonuniform textured surfaces would be practical in controlling the trajectory of droplets after impact. Kannan and Sivakumar25 compared microgroove surfaces with smooth surfaces and showed that the presence of grooves reduced the maximum spreading diameter and delayed the receding phase. Zheng et al.26 numerically studied the detailed mechanisms of stability, metastability, and instability of the wetting models and their transitions on superhydrophobic surfaces made with periodic micropillars. They investigated the effects of hydraulic pressure on the stability and the transition and proposed the concept of the mixed wetting mode that was associated with the multimetastable states between the CassieBaxter and Wenzel wetting modes. Lee et al.27 fabricated self-replication with hydrophobic PDMS using a CNC machine to examine the wetting transition behavior of water droplets. They demonstrated the existence of a wetting transition from the Cassie state to the Wenzel state, which is dependent on the spacing-todiameter ratio, and observed the partial penetration of the liquid meniscus moving downward in a groove formed by four pillar posts during the transition. Deng et al.28 studied the nonwetting of liquid droplets impinging on superhydrophobic textured surfaces and proposed design guidelines for nonwetting surfaces under droplet impingement. In particular, they introduced a new notion of wetting pressure, using the concept of effective water hammer pressure, to define wetting states for impinging droplets clearly. Recently, Li et al.29 investigated the dynamic behavior of the water droplet impact on a textured hydrophobic/superhydrophobic surface and demonstrated that the difference in contact time depends on the solid fraction, meaning the ratio of the actual area in contact with the liquid with respect to its projected area on the textured surface. In addition, they examined the effect on the contact time of the liquid film arising at the tops of pillars. Even though many investigators as noted previously have discussed the behavior of droplets impinging on textured or bare surfaces, most of this research has employed lithographic or molding methods to create hydrophobic surfaces with embossed patterns using silicon wafers or elastic materials such as PDMS. Because most industrial applications tend to employ metallic materials (e.g., conventional steel), embossed patterns are not suitable for real industry because of fabrication problems and issues with surface reliability. To resolve this problem, in the present study we used an engraved pattern including microhole arrays fabricated using conventional CNC methods; this pattern
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Figure 1. Top-view SEM of the bare aluminum surface and textured surfaces: (a) φs = 0.28, (b) φs =0.43, and (c) φs =0.56.
Figure 2. Schematic of the experimental setup.
changes the wetting characteristics and dynamic behavior of a water droplet impinging on a textured surface. To characterize the dynamic behavior of an impinging droplet, we measured the spreading diameter, apparent dynamic contact angle, and maximum spreading factor during droplet impingement. To investigate the effects of texture on the spreading behavior, we fabricated several textured surfaces with different area fractions and classified the resulting wetting states under different impingement conditions. In particular, we suggest a new refinement of a theoretical model to predict the maximum spreading factor by considering geometrical effects due to texturing and compare this with earlier published models and experimental data.
II. EXPERIMENTAL DETAILS Fabrication Methods of the Textured Surfaces. For our target surface specimens, we used aluminum (Al 6061) surfaces polished with alumina paste. The surface roughness average (Ra) of the polished surfaces was approximately 0.037 ( 0.003 μm, as measured by a profilometer (SJ-201, Mitutoyo). The textured surface consisted of microholes in a square-patterned array drilled using a micro CNC machine (EGX-350, Roland). With holes of 125 μm diameter (d) and a spindle speed of 10 000 rpm, the hole depth (hd) was set to 125 μm. Figure 1 shows an SEM image of the resulting microtextured aluminum surfaces. The edge-to-edge spacing between textured holes (s) was fixed at 22.6, 44.9, and 85.2 μm for different patterns. In addition to a bare surface (φs = 0.0), we created textured surfaces with texture area fractions of 0.28, 0.43, and 0.56; the tolerance achieved was (2 μm, indicating that the reliability of CNC machining is very high. Here, the texture area fraction (φs), which can be defined as (πd2/4)/(s þ d)2, represents the ratio of the area of a textured hole to the solid surface area. The surface morphology of the textured aluminum was examined using scanning electron microscopy (SEM, S-3400N, Hitachi). Measurements. As shown in Figure 2, which gives a schematic of the experimental apparatus for measurement, we used a pendant method to generate and dispense a water droplet using a custom injection system consisting of a flat-tipped metal hub needle (gage 33, Hamilton) and a syringe pump (LSP01-1A, LongerPump). The injection system slowly pushes liquid through the needle, from which a droplet eventually 6566
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Table 1. Physical Properties of Water Droplets and Experimental Conditions Used in the Present Study under Controlled Conditions (Temperature 25 °C, Relative Humidity 50%, and Atmospheric Pressure) H (mm)
u (m/s)
d0 (mm)
viscosity (mPas)
density (kg/m3)
surface tension (mN/m)
12
0.40
2.018
1.002
998.2
72.8
4.43
804.14
25 115
0.64 1.45
2.018 2.018
1.002 1.002
998.2 998.2
72.8 72.8
11.33 58.18
1286.62 2915.00
detaches owing to its weight. We used deionized (DI) water droplets, with a volume of 4.3 μL each as measured by a microbalance (AC 121 S, Sartorius) under controlled environmental conditions (temperature 25 °C, relative humidity 50%, and atmospheric pressure). To visualize an impinging droplet, we used a Telecentric lens (TEC-M55, Computar) with a 2 converter and a high-speed camera (Hotshot 1280, NAC); this setup gave a field of view (FOV) of 3.3 mm 9.9 mm, with a resolution of approximately 7 μm per pixel and a distortion angle of 0.6%. All droplet spreading images were captured in front view at 2000 images/s with a 5 μs exposure time. The digital images were analyzed using a free open-source Java image-processing program (ImageJ).30,31 In particular, we measured the sphericity of the droplets to confirm the spherical shape of a droplet prior to impact. Top-view images were obtained using a Telecentric lens (TML-HP, Edmond) and a CMOS camera (XLi, Meyer) with a resolution of 6 μm per pixel and a distortion angle of 0.3%. We found that the droplet sphericity was approximately 0.996, indicating a nearly spherical shape, and concluded that the pendant method used in the present study would be suitable for consistent droplet generation. There are several important dimensionless parameters associated with the impact dynamics of a droplet; these include the Weber number (We), Reynolds number (Re), and Ohnesorge number (Oh), which can be expressed in terms of the droplet impact velocity (u), initial droplet diameter (d0), and liquid properties pffiffiffiffiffiffi We Fu2 d0 Fud0 , Re ¼ , and Oh ¼ We ¼ ð1Þ σ μ Re where, F, σ, and μ are the density, surface tension, and viscosity of the fluid. Using a center-to-center distance and a time interval between two captured images of droplets prior to impact, the instantaneous impact velocity of a droplet can be estimated. The physical properties of DI water and the experimental conditions are summarized in Table 1.
III. RESULTS AND DISCUSSION Hydrophobic Characteristics of the Textured Surface. This study used the measured apparent contact angle to represent surface wetting under the gentle deposition of a droplet. After a drop was gently deposited on a surface, an equilibrium state was observed following a certain time. Figure 3 compares the measured apparent contact angles on different surfaces with the estimations obtained by using the well-known CassieBaxter equation18
cos θCA ¼ ð1 φs Þðcos θeq þ 1Þ 1
ð2Þ
where θeq denotes an apparent contact angle (CA) measured in the equilibrium state for a bare surface. The apparent CA of a nontextured (bare) aluminum surface (φs = 0.0) is 80.59 ( 1.90°, showing that a bare surface is intrinsically hydrophilic. On the contrary, the apparent CA of a textured surface increases to 125.83 ( 0.49°, indicating such a hydrophobic nature occurring at a composite interface among solid, liquid, and air that is induced by textured holes. Measurements of apparent CAs were
We
Re
Oh
0.0026
Figure 3. Comparison of the measured apparent contact angle on a target surface with the theoretical prediction from the CassieBaxter equation. Measured data: θCA = 80.59 ( 1.90° (φs = 0.0, bare), θCA = 100.05 ( 0.22° (φs = 0.28), θCA = 118.53 ( 1.02° (φs =0.43), and θCA = 125.83 ( 0.49° (φs = 0.56).
conducted 10 times, ensuring consistent results, and experimental uncertainties were shown in Figure 3 for different φs values. The measured apparent CAs are in good agreement with the values estimated by the CassieBaxter equation, showing that the increase in the apparent CA can be seen as the number density of textured holes increases. In particular, similar to the embossed patterns with microscale post arrays, the engraved patterns seem to contribute to make an intrinsically hydrophilic solid surface more hydrophobic. This tendency may be interpreted as the formation of air pockets under a droplet. Spreading Characteristics of Impinging Droplets on Textured Surfaces. For the impingement of a droplet, the spreading behavior is closely associated with the dynamic contact angle that is formed at the interface among three phases. This composite interface can be destroyed by dynamic effects such as squeezing, impact, vibration, or evaporation.29 As discussed previously, there may be a partial penetration of liquid into the holes and a formation of air pockets during impact, consequently resulting in a variation in the dynamic contact angles. Figure 4 displays the transient behavior of a water droplet impinging on a nontextured (bare) surface and on a textured surface with respect to the droplet impact velocity. Just before impact, the droplet clearly has a spherical shape; it deforms and spreads rapidly in the radial direction upon impact owing to a rapid increase in pressure at the point of impact.32 The droplet reaches its maximum spread and is momentarily at rest because of the depletion of kinetic energy. In this process, the droplet undergoes viscous dissipation and moves back toward the center after the time of maximum spread. Over time, the droplet finally 6567
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Figure 4. High-speed camera images of impacts on bare and textured aluminum surfaces at different impact velocities.
Figure 5. Schematic representations of different phases after impingement.
reaches an equilibrium state in which the appearance of the liquid droplet’s meniscus shape depends on the texture area fraction. It should be noted that, as the initial Weber number increases, the differences in liquid meniscus shapes are more apparent. In addition, the spreading diameter increases with the impact velocity; for a given impact velocity, slight changes in the spreading diameter depending on the texture area fraction can be seen. As noted previously, the present study focuses only on the deposition regime occurring at impact velocities lower than approximately 1.45 m/s. According to Rioboo et al.,33 who observed evolving droplet behavior in the deposition regime, there are four representative phases—the kinematic phase, spreading phase, relaxation phase, and final equilibrium phase—as illustrated in Figure 5. Following the impact on the textured surface, the dimensionless contact diameter increases as approximately t*1/2 in
the kinematic phase, where t* = t(u/d0) denotes the nondimensional time. Among the various phases, the spreading phase was the primary target of the present study. For a given liquid, the contact angle characterizes the energy level of the solid surface, and determines the surface energies of the solidliquidvapor system as well as the wettability. Thus, we measured the dynamic contact angle to investigate changes in surface energy resulting from surface texturing. From these results, we obtain a useful understanding of dynamic changes in wettability as well as design parameters for such surfaces that could facilitate active control of liquid motion. Moreover, the nondimensional spreading factor D*, defined as the spreading diameter normalized by the droplet size before impact, is estimated from the experimental measurements so as to examine the spreading characteristics. Figure 6 represents the dimensionless spreading factor (D*) with respect to time for different texture area fractions. We can see that the spreading factor increases with the impact velocity of a droplet. In an early stage of impingement, as seen in Figure 5, the droplet resembles a truncated sphere and the spreading factor increases rapidly regardless of the texture area fraction; this indicates that initial droplet spreading is dominated by inertial effects, which typically appear in the kinematic phase. As time increases, a liquid lamella is ejected from the base of the droplet and forms a thin liquid film propagating in the radial direction. After the maximum spreading state is reached, we can clearly observe the effects of texturing on the spreading factor, in particular, that the spreading factor decreases with increased texture area fraction. This phenomenon occurs because, when the Weber number is small, surface effects such as interfacial and viscous forces are crucial to droplet spreading.34 In other words, changes in surface energy due to texturing affect the evolution of droplet spreading on textured surfaces. Figure 7 presents measured values of the dimensionless maximum spreading factor (Dm*) with respect to the texture area fraction at different impact velocities. In particular, it can be seen that the maximum spreading factor slightly decreases with 6568
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Figure 7. Effect of texture area fraction on the maximum spreading factor.
Figure 8. Schematics of (a) three different wetting states and (b) the geometry of textured holes.
hammer pressure PEWH, modeled as PD ¼
Figure 6. Transient evolution of the spreading factor on nontextured (bare) and textured surfaces. The filled and open symbols correspond to bare and textured surfaces, respectively: (a) u = 0.40 m/s, (b) u = 0.64 m/s, and (c) u = 1.45 m/s.
increased texture area fraction. For this decreasing tendency, it is conjectured that there are three possible reasons. The first is partial liquid penetration through the textured holes. Deng et al.28 reported that, when a droplet impinges on a textured surface, three different wetting states may be seen: a total wetting state (Wenzel state), a partial wetting state, and a total nonwetting state (Cassie state). The state observed depends on the dynamic pressure PD, capillary pressure PC, and effective water
1 2 θa Fu , PC ¼ 2γLV cos , and PEWH 0:2FCu 2 d ð3Þ
where C is the speed of sound in water, θa indicates the advancing angle, and γLV is the surface tension at the liquidair interface. Using eq 3, each wetting state can be determined according to the impinging conditions. As shown in Figure 8, three different wetting states can be classified for our texturing surface geometry, according to the approach of Deng et al.28 Depending on the dynamic pressure, capillary pressure, and effective water hammer pressure, as listed in Table 2, we determined the wetting states under our experimental conditions. As a result, it was found that when the impact velocity is lower than 0.64 m/s, impinging droplets are in the partial wetting state, whereas at u = 1.45 m/s the total wetting state exists. Under our experimental conditions, there is no Cassie state that indicates a total nonwetting state. Because of the optical limitation used to measure the exact amount of the liquid volume penetrating through the holes in space, we calculated a liquid penetration ratio defined as the 6569
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Table 2. Corresponding Wetting States under Our Experimental Conditions and Estimates of Various Pressures for Different Textured Surfaces and Impact Velocities φs u (m/s)
φs
PC (Pa)
u (m/s)
0.28
0.43
0.56
118.34
0.28
651.35
0.40
partial wetting
partial wetting
partial wetting
189.34
0.43
717.12
0.64
partial wetting
partial wetting
partial wetting
429.01
0.56
823.64
1.45
total wetting
total wetting
total wetting
PD (Pa)
PEWH (kPa)
0.40
79.06
0.64
202.38
1.45
1038.84
Figure 9. Measured dynamic contact angle with time on bare and textured surfaces: (a) u = 0.40 m/s, (b) u = 0.64 m/s, and (a) u = 1.45 m/s.
amount of liquid volume occupied inside holes (=(π/4)Dm2hdφs) to the initial volume of a liquid droplet under the assumption that all of the textured holes underneath a liquid droplet at maximum spread are fully occupied by the liquid. Even though this assumption is impracticable, it would be helpful for us to understand the effect of the liquid penetration on the spreading factor by considering this extreme case. The maximum liquid penetration ratio attained is approximately 0.32, implying that liquid penetration may be an important factor affecting the spreading characteristics. The second mechanism describing the effect of texturing on the spreading factor is viscous dissipation inside the holes. Using the lattice Boltzmann model, Hyvaluoma and Timonen35 simulated impact states and energy dissipation in bouncing and nonbouncing droplets on superhydrophobic surfaces made up of an array of micrometer-scale posts. They reported that when the liquid retreats from the surface texture, significant energy dissipation occurs. From our measurement and calculation using the energy conservation equation, the viscous energy dissipation is estimated by using the expressions of Chandra and Avedisian36 and Pasandideh-Fard et al.,12 and its amount is approximately 11.4 to 24.4% of the summation of kinetic energy and surface energy of a droplet before impact. Moreover, the viscous dissipation energy is varied depending on the texture area fraction and the impact droplet velocity. This indicates that viscous dissipation cannot be ignored when predicting spreading factors during impact. Finally, we suggest dynamic changes in the surface energy of the liquid film due to the surface texture as an explanation of the decreasing tendency of the spreading factor. The contact angle is one of the important factors that determine surface wettability. Dynamic changes in surface energy due to the surface texture can be directly analyzed from in situ measurements of the contact angle during the impact process. Figure 9 shows the measured apparent dynamic contact angle (DCA: θDCA) for different textured surfaces in the spreading and receding stages. The drastic decrease in DCAs is found, implying that the decreasing rate of DCAs is independent of the surface properties and is mainly dependent on the kinetic energy. The DCAs of the droplets on the textured surfaces are slightly higher than those of the bare surface in the spreading stage at approximately t < 3 ms as shown in Figure 6, indicating that textured surfaces become more hydrophobic under the dynamic condition of liquid lamella propagation in the radial direction. In fact, the DCAs are closely associated with the surface tension and liquid pressure due to inertia,37 and the fraction of solid underneath the droplet can alter the dynamic contact angle and the surface energy of the liquid film on the textured surface.38 Surface roughness can also promote the formation of air pockets under a droplet, which further reinforces the increase in advancing angle during the impact 6570
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Table 3. Summary of Representative Theoretical Models for the Maximum Spreading Factor
Dm
correlations sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We þ 12 pffiffiffiffiffi ¼ 3ð1 cos θa Þ þ 4ðWe= ReÞ
ðWe þ 12ÞDm
Pasandideh-Fard et al.12
We 3ð1 cos θY Þ þ 4pffiffiffiffiffi Re We ¼ 8 þ Dm 3 3ð1 cos θd Þ þ 4pffiffiffiffiffi Re We ¼ 8 þ Dm 3 4pffiffiffiffiffi 3ψ Re
ðWe þ 12ÞDm ¼ 8 þ Dm ðWe þ 12ÞDm
references
Ukiwe and Kwok13
3
Vadillo et al.14 present study
where ψ = (cos θm(1 φs) þ φs)(1 ωs) þ ωsrs cos θm 1
Figure 10. Comparison between experimental and computed maximum spreading factors using the present model, UK model, and Vadillo model.
process. This change plays an important role in decreasing the spreading factor. Prediction of the Maximum Spreading Factor. The accurate prediction of the maximum spreading factor is important for providing researchers with a fundamental understanding of the spreading characteristics affected by surface characterization. Many earlier researchers1214 have proposed theoretical models, as listed in Table 3, to predict the maximum spreading factor accurately; these models generally rely on the energy conservation principle, analytical methods from mass and momentum equations, and experimental data. However, most extant theoretical models do not take into account changes in surface energies induced by geometrical modifications such as surface texturing. In particular, the choice of the appropriate contact angle at maximum spread is still an open point, and the measurement of the dynamic contact angle is crucial in determining the maximum spreading factor. A determination of the maximum spreading factor can produce conflicting results depending on the contact angle used, such as the advancing contact angle, the contact angle at the maximum spreading, and the plateau contact angle.14 As in previous discussions by several authors,12,36 it seems that the use of the advancing contact angle measured at maximum spread is most reasonable in describing the wettability of a liquid in the maximum spreading state.
In this study, we suggest a theoretical model incorporating a geometrical modification due to surface texturing and considering different wetting states depending on impact conditions. Beginning with the energy conservation principle, both prior to impact and at maximum spread, we have KE1 þ SE1 ¼ KE2 þ SE2 þ W
ð4Þ
The total energy at the instant of impact is the sum of the kinetic energy (KE1) and surface energy (SE1), which can be written as 1 2 π 3 Fu d0 KE1 ¼ ð5Þ 2 6 SE1 ¼ πd0 2 γLV
ð6Þ
After impact, the kinetic energy of the impinging droplet is stored in its deformation and the liquid lamella moves parallel to the surface during spreading. At the maximum spreading state, a liquid film of cylindrical shape moving on the wall is assumed;12,13 at the moment of maximum spread, the kinetic energy (KE2) is negligible. Thus, two important energies should be modeled. First, for the viscous dissipation 6571
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energy (W), we used the original expression of Chandra and Avedisian36 Z td Z W ¼ Φ dΩ dt ΦΩtd ð7Þ 0
Ω
where Φ is the mean value of the dissipation energy per unit time and volume. Ω and td are the volume of a drop and the time of dissipation, respectively. We used the assumption by PasandidehFard et al.12 to define Φ, Ω, and td. Here, the dissipation time td is expressed as a function of the characteristic time of impact tc ( d0/u) as follows: td = 8/3tc. Next, by considering the texturing effects and wetting state, we propose a new model of surface energy at the maximum spread (SE2) as follows ! 2π d0 3 π 2 SE2 ¼ þ Dm ð1 þ ð1 ωs Þφs Þ γLV 3 Dm 4 π þ Dm 2 ðð1 ωs Þð1 φs Þ þ ωs rs Þ ðγSL γSV Þ ð8Þ 4 where γSV and γSL denote the surface tensions at the solidair interface and the solidliquid interface, respectively. The Wenzel roughness, rs, is defined as 1 þ (πdhd)/(s þ d)2), and ωs can be defined as (Dw/Dm)2, indicating the area fraction of the partial wetting area to the liquid film area at a maximal extension. These two parameters are introduced in this study to describe the effects of texturing and wetting on the maximum deformation of the liquid film. Moreover, Young’s equation is used to eliminate surface tensions γSV and γSL, which are difficult to measure directly, as follows γSV γSL ¼ γLV cos θY
literature, it has been reported that both the Vadillo model and UK model showed good agreement with experimental data for nontextured (bare) surfaces. However, both models show some discrepancies relative to experimental data for droplet impingement on textured surfaces because they cannot describe changes in wetting and surface morphology that are dependent on impact conditions. In particular, there are no terms that can describe different wetting states due to the presence of textured holes. It should be noted that the present study examines only low maximum spreading factors (below 3.0) that arise when the corresponding Weber number is lower than 58.2. In this regime, hydrodynamic forces play the most important role. In this regime, texturing effects and changes in interfacial energies among solid, liquid, and air should be considered. By contrast, the present model predicts the maximum spreading factor better than other models and reflects changes in the interfacial energy due to the surface texture as well as changes in the wetting state according to different impact conditions. At We = 4.4, we can see that the predicted values for the nonwetting state (ωs = 0) are in better agreement with experimental data than those for the total wetting state (ωs = 1). At a high Weber number of 58.2, the Wenzel state exists regardless of the texture area fraction, as shown in Table 2. In Figure 10b, it can be seen more clearly that a total wetting state exists; this is because, in the present model, the maximum spreading factor predicted in the total wetting state is in better agreement with the experimental data than that in the nonwetting state. Moreover, even at higher Weber numbers, the present model shows better predictability for the maximum spreading factor than do the other models.
ð9Þ
where θY indicates the Young’s angle and the surface tension γLV can be easily measured. For the contact angle in eq 9, we used the measured contact angle θm at the instant of maximum extension of the liquid film. By substituting eqs 59 into eq 4, a final form for the maximum spreading factor is derived as follows We p ffiffiffiffiffi 4 3Ψ ð10Þ ðWe þ 12ÞDm ¼ 8 þ D3 m Re where Ψ, a geometrical modification due to surface texture considering the wetting state and texture area fraction (φs), can be expressed as Ψ ¼ ðcos θm ð1 φs Þ þ φs Þð1 ωs Þ þ ωs rs cos θm 1 ð11Þ The present portion of the study determines the wetting states corresponding to the classifications of Deng et al.28 as mentioned earlier. In fact, for metal surfaces, it is very difficult to directly measure the amount of liquid area wetted on textured surfaces because of the optical opacity of the surface and the very short timescale of the experiments. Therefore, this study deals with two limiting cases—a total nonwetting state (representing ωs = 0) and a total wetting state (Wenzel state showing ωs = 1)—even though there exists a partial wetting state during impact, as shown in Table 2. Figure 10 represents the comparison of predictions obtained using previously published theoretical models and the present model against measured maximum spreading factors. The respective reported mean measurement errors for the Vadillo model, the UK model, the present model for the Cassie state (ωs = 0), and the present model for the Wenzel state (ωs = 1) are 7.81 ( 5.06, 9.46 ( 4.39, 5.44 ( 4.32, and 4.72 ( 3.00%. In the
IV. CONCLUSIONS We conducted extensive experiments on impinging droplets on textured surfaces to investigate dynamic wetting and spreading characteristics during impact and presented both a comparative analysis of earlier published models and a new model developed in this study for the prediction of the maximum spreading factor. The following conclusions are drawn: first, when the metal surface was textured with a square-pattered array using a micro CNC machine, the apparent contact angle increased to 125.8°, representing hydrophobic characteristics due to the presence of a composite interface among solid, liquid, and air induced by textured holes. The measured apparent CAs show good agreement with the values estimated by the CassieBaxter equation. Second, we investigated the dynamic wetting and spreading characteristics of a water droplet impinging on textured surfaces. From the results, it was observed that the spreading factor decreases with increasing texture area fraction. To explain this decreasing tendency, we suggested three reasons: the partial penetration of liquid through the textured holes, viscous dissipation effects, and dynamic changes in surface energy resulting from the texturing. Finally, we proposed a new prediction model for the maximum spreading factor that considers changes in interfacial energy due to surface texturing and variation in the wetting state depending on impact conditions. From the results, we concluded that the present model was in better agreement with experimental data than the earlier published models. ’ AUTHOR INFORMATION Corresponding Author
*Tel: þ82 2 820 5254. Fax: þ82 2 823 9780. E-mail: shlee89@ cau.ac.kr. 6572
dx.doi.org/10.1021/la104829x |Langmuir 2011, 27, 6565–6573
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