Article pubs.acs.org/Langmuir
Modeling the Maximum Spreading of Liquid Droplets Impacting Wetting and Nonwetting Surfaces Jae Bong Lee,† Dominique Derome,‡ Robert Guyer,§,∥ and Jan Carmeliet*,†,‡ †
Chair of Building Physics, ETH Zurich, Stefano-Franscini-Platz 5, CH-8093 Zürich, Switzerland Laboratory for Multiscale Studies in Building Physics, Swiss Federal Laboratories for Materials Science and Technology, Empa, Ü berlandstrasse 129, CH-8600 Dübendorf, Switzerland § Solid Earth Geophysics Group, Los Alamos National Laboratory, MS D446, Los Alamos, New Mexico 87545, United States ∥ Department of Physics, University of Nevada, Reno, Nevada 89557, United States ‡
S Supporting Information *
ABSTRACT: Droplet impact has been imaged on different rigid, smooth, and rough substrates for three liquids with different viscosity and surface tension, with special attention to the lower impact velocity range. Of all studied parameters, only surface tension and viscosity, thus the liquid properties, clearly play a role in terms of the attained maximum spreading ratio of the impacting droplet. Surface roughness and type of surface (steel, aluminum, and parafilm) slightly affect the dynamic wettability and maximum spreading at low impact velocity. The dynamic contact angle at maximum spreading has been identified to properly characterize this dynamic spreading process, especially at low impact velocity where dynamic wetting plays an important role. The dynamic contact angle is found to be generally higher than the equilibrium contact angle, showing that statically wetting surfaces can become less wetting or even nonwetting under dynamic droplet impact. An improved energy balance model for maximum spreading ratio is proposed based on a correct analytical modeling of the time at maximum spreading, which determines the viscous dissipation. Experiments show that the time at maximum spreading decreases with impact velocity depending on the surface tension of the liquid, and a scaling with maximum spreading diameter and surface tension is proposed. A second improvement is based on the use of the dynamic contact angle at maximum spreading, instead of quasi-static contact angles, to describe the dynamic wetting process at low impact velocity. This improved model showed good agreement compared to experiments for the maximum spreading ratio versus impact velocity for different liquids, and a better prediction compared to other models in literature. In particular, scaling according to We1/2 is found invalid for low velocities, since the curves bend over to higher maximum spreading ratios due to the dynamic wetting process.
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INTRODUCTION
where Dmax is the maximum spreading diameter, and D0 is the initial drop diameter prior to impact. The maximum spreading ratio is directly related with the performance of spray systems in industrial processes,13 or water transport from a rain droplet to soil or porous building in natural phenomena.1 Controlling or predicting maximum spreading therefore is essential for many problems involving the deposition of an impacting drop. A large number of parameters, such as drop size, impact velocity, liquid properties (density, viscosity and surface tension), surface roughness, and wettability, play a role in the maximal spreading achieved by a droplet. Spreading is driven by the kinetic energy (given as Ek = ρD03Vi2) and countered by the capillary energy (as Eγ = γLVD02) and viscous energy (as Eμ = μViD02).41 Most of the existing models are formulated based on two dimensionless parameters: Weber number (We = ρVi2D0/γLV), the ratio between the kinetic and capillary energy, Ek/Eγ, and Reynolds
The impact of a liquid droplet on a surface is an everyday and ubiquitous phenomenon with important applications in natural, agricultural, and industry processes such as raindrop erosion,1,2 pesticide spraying,3,4 thermal spraying,5 and inkjet printing.6 In these applications, the prediction or the control of the contact area of the liquid droplet with the surface during deposition is critical.7 Thus, a fundamental understanding of droplet spreading is crucial for determining the relative effects of controllable factors such as fluid properties, impact conditions, wettability of surface, and roughness. Such fundamental understanding of spreading and maximum spreading of a liquid droplet is still incomplete and correct modeling of the phenomenon remains a challenge, despite the large number of existing models using empirical,8,9 semi-empirical,5,10−30 analytical or numerical approaches,22,31−39 as detailed in a recent comprehensive review40 of liquid droplet impact modeling. A common interest in drop impact dynamics is the analysis and prediction of the maximum spreading ratio βmax = Dmax/D0, © XXXX American Chemical Society
Received: December 14, 2015 Revised: January 4, 2016
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DOI: 10.1021/acs.langmuir.5b04557 Langmuir XXXX, XXX, XXX−XXX
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Langmuir number (Re = ρViD0/μ), the ratio between the kinetic and viscous energy, Ek/Eμ, where ρ is the liquid droplet density, γLV is the surface tension, μ is the viscosity, and Vi is the impact velocity. Maximum spreading can be controlled by the liquid properties, namely, viscosity and surface tension. For instance, surfactant additives are used to enhance the deposition efficiency for pesticide applications.42 The effect of viscosity on spreading has been studied in many experiments using water−glycerol mixtures.8,16,26,41,43−45 Laan et al.41 studied the spreading of water−glycerol mixtures on various surfaces mimicking the impact of blood droplets for bloodstain analysis. In several studies, the impact of silicone oil,29,45,46 isopropanol,44 ethanol,45,47 and surfactant42,48 droplets was studied in order to understand the effect of surface tension. Zhang et al.48 studied the maximum spreading of surfactant solutions with different concentrations on a glass surface, showing that the maximum spreading can be increased by the addition of surfactants at low impact velocity. A large number of studies deal with the prediction of maximum spreading on a solid surface.40 In theoretical approaches, maximum spreading is commonly estimated based on the energy balance, in terms of kinetic energy and surface energy prior to impact and surface energy along with viscous dissipation at its maximum spreading.34 Chandra and Avedisian25 formulated the energy balance equation based on a simplified cylindrical disk shape for maximum spreading on hot flat surfaces. Pasandideh-Fard et al.22 proposed a prediction model for maximum spreading on cool surfaces, taking into account the advancing contact angle and the viscous boundary layer. Mao et al.30 and Ukiwe and Kwok20 proposed a modification to the Pasandideh-Fard model taking into account the surface energy of the lateral sides of a cylinder disk. Mao et al.30 proposed an empirical equation based on a large amount of literature data and evaluated the prediction in a large domain of We and Re numbers. Although good agreement was found between experiments in a wide range of impact parameters,49 this energy balance approach shows two problems: (i) the viscous dissipation energy is integrated over time to the point of maximum spreading, and, thus, the time to maximum spreading tmax has to be known; (ii) the surface energy of the liquid−vapor and liquid−solid interfaces is estimated using Young’s equation introducing a contact angle, which during dynamic spreading is not directly defined. Pasandideh-Fard et al.22 derived the time at maximum spreading as tmax = (8/3)· D0/Vi based on simple geometric assumptions. Vadillo et al.16 showed a disagreement between the measured time at maximum spreading and the time predicted by the model proposed by Pasandideh-Fard et al.22 Different contact angles have been proposed to be used for estimating the surface energy in the energy balance approach such as equilibrium contact angle,30,50 contact angle at maximum spreading,22 advancing contact angle,20 averaged advancing contact angle,16 and nonwetting contact angle (θ = 180°).51 In this paper, we investigate the role of parameters, such as surface tension, viscosity, wettability, and surface roughness, on the maximal spreading ratio βmax experimentally and propose a modified energy balance model for the maximum spreading ratio. Three different liquids with different surface tension and viscosity, three different surfaces with different wettability, and three surface roughnesses are used to achieve a comprehensive and complete set of data from low to high impact velocity in order to analyze the quality of the prediction model for the
maximum spreading diameter. More specifically, the aim of this study is to extend the approach of Lee et al.52 by including the scaling of the time at maximum spreading with the maximum spreading diameter, as observed in our experimental data.
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DROP IMPACT EXPERIMENTS Material and Methods. We investigate the influence of the liquid−gas surface tension and the liquid viscosity on the spreading of a droplet using three different liquids: pure ethanol (absolute ≥99.8%, SIGMA-Aldrich, referred to as ethanol), deionized water (referred to as water), and a 55% glycerol− water mixture (glycerol ≥99%, SIGMA-AlDRICH and deionized water, referred to as glycerol). The properties of the liquids at 25 °C are given in Table 1. The density, surface Table 1. Properties of Liquids at 25°C
ethanol water glycerol
density ρ (kg/m3)
surface tension γLV (N/m)
viscosity μ (Pa·s)
789 998 1 158
2.3 × 10−2 7.3 × 10−2 6.8 × 10−2
1.2 × 10−3 1.0 × 10−3 1.0 × 10−2
tension and viscosity of the glycerol−water mixture are provided by the manufacture or determined using empirical formula.53 Taking water as the reference liquid, ethanol shows a surface tension three times smaller than water, while the viscosity is quite similar. Glycerol shows a larger viscosity 10 times than water, while the surface tension is quite similar to water. Three substrates are considered: steel, aluminum and a paraffin/thermoplastic film (trademark Parafilm). For aluminum, the original smooth surface is roughened by sand paper, attaining three roughness values, referred to as Al sm (smooth), Aluminum (reference) and Al ro (rough). For the five surfaces, the roughness is measured by a contact profilometer (Surftest211, Mitutoyo). Each specimen is traced in 10 arbitrary locations near the place of droplet impact over a 0.8 mm sampling length. The results are given in Table 2. The arithmetic average roughness (average height of irregularities) Ra ranges between 0.41 to 0.69 μm for the rough surfaces and is less than 0.05 μm for smooth aluminum. The wettability of the substrate is characterized by the equilibrium (θeq), advancing (θadv) and receding (θrec) contact angles. The contact angle is measured using the sessile drop method. A 3.0 μL droplet is very gently deposited on the surface and imaged with a pixel resolution of 7.38 μm. The contact angle is obtained by taking the tangent to the piecewise polynomial fit of the segmented droplet using MATLAB (MathWorks Inc.). θadv and θrec are measured by quasi-statically expanding or contracting the deposited sessile droplet, respectively. The increasing and decreasing volume rate of the sessile drop is controlled by a syringe pump and is equal to 30 μL/min. All contact angle measurements are repeated 10 times for each sample to guarantee reproducibility. The results are reported in Table 3. Ethanol shows in most cases a zero contact angle, indicating that ethanol is totally wetting the surface in quasi-static conditions. Ethanol shows on smooth aluminum and parafilm an equilibrium contact angle higher than zero, i.e., 11° and 22°. Glycerol shows a wetting behavior on steel and aluminum (from smooth to rough) with an equilibrium contact angle between 52° and 60°, while it is nonwetting on parafilm. Water shows the highest equilibrium contact angles and is wetting on steel and aluminum, while it is B
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Langmuir Table 2. Mean and Standard Deviation of Surface Roughness Ra for Five Substrates Ra (μm)
Al sm
aluminum
Al ro
steel
parafilm
0.05 ± 0.00
0.41 ± 0.06
0.69 ± 0.08
0.42 ± 0.05
0.45 ± 0.06
Al sm: smooth; Aluminum: reference; Al ro: rough.
Table 3. Wettability Characterized by the Contact Angle for Ethanol, Water, and Glycerol on Steel, Smooth, Reference, and Rough Aluminum and Parafilm: Equilibrium θeq, Advancing θadv, Receding θrec, Dynamic at Maximum Spreading θD(tmax) surface
Al sm
aluminum
Al ro
steel
parafilm
∼0 ∼0 ∼0 41.2 ± 4.8
∼0 ∼0 ∼0 43.9 ± 2.5
21.3 ± 1.9 22.5 ± 1.4 ∼0 63.0 ± 5.9
92.9 ± 10.2 99.4 ± 2.8 ∼0 111.1 ± 6.9
60.9 ± 1.3 61.5 ± 3.3 6.8 ± 1.0 102.9 ± 3.2
109.6 ± 2.6 115.0 ± 6.3 86.1 ± 5.5 107.6 ± 5.6
53.5 ± 1.7 51.8 ± 1.5 ∼0 110.7 ± 5.6
52.4 ± 3.7 48.5 ± 4.3 ∼0 121.2 ± 6.9
94.4 ± 3.4 105.4 ± 4.5 71.3 ± 2.0 116.2 ± 5.9
Ethanol θeq (deg) θadv (deg) θrec (deg) θD(tmax) (deg)
11.2 ± 1.6 ∼0 ∼0 42.2 ± 3.8
∼0 ∼0 ∼0 51.4 ± 5.8
θeq (deg) θadv (deg) θrec (deg) θD(tmax) (deg)
62.8 73.2 16.2 93.1
± ± ± ±
5.6 1.4 4.1 8.8
87.6 ± 7.1 94.0 ± 2.8 ∼0 115.5 ± 5.8
θeq (deg) θadv (deg) θrec (deg) θD(tmax) (deg)
60.1 ± 1.8 65.2 ± 4.2 ∼0 104.2 ± 9.8
59.8 ± 2.6 59.4 ± 1.4 ∼0 107.6 ± 2.9
Water
Glycerol
developed code tracking the droplet contour (Figure 1a). Results include the initial droplet diameter D0, the droplet
nonwetting on rough aluminum and parafilm. In general, we found that the advancing contact angle is quite similar to the equilibrium contact angle (differences less than 16%). The receding contact angle is much lower than the equilibrium contact angle and, in several cases, cannot be measured, since the droplet remained pinned. In the following, we will compare the equilibrium contact angle to the dynamic contact angle during droplet spreading. For the impact measurements, a droplet is generated at the flattened tip of a needle by pushing a syringe pump. The droplet has an initial diameter D0, reproducible with a relative error of 1%. When the droplet is released, it accelerates by gravity reaching an impact velocity Vi. As the release height varies from 3 to 820 mm, the impact velocity ranges between 0.2 and 3.7 m/s (see Table 4) with a relative error of 2%. The
Figure 1. Image analysis for a spreading droplet: (a) segmented outlines of a spreading droplet in color lines; (b) contact line profiles at the surface at three times to a pixel height Py = 10 and schematic representation of the area angle relationship.
Table 4. Droplet Impact Conditions D0 (m) ethanol water glycerol
1.8 × 10−3, 2.0 × 10−3a 2.0 × 10−3 1.8 × 10−3
Vi (m/s)
Weber number
Reynolds number
0.2−2.4
2.5−315
260−2900
0.2−3.2 0.2−3.7
1.0−290 1.1−414
350−6300 40−750
diameter D(t), and normalized spreading ratio β(t)= D(t)/D0 versus time. The dynamic contact angle θD is determined using the goniometric mask method54 obtained from the contact line profile (Figure 2b). The goniometric mask method is the emulation of a goniometer for a binary image, and is preferred to edge fitting with polynomial curves for determining the dynamic contact angle. The dynamic contact angle represents a macroscopic apparent contact angle, which is measured over a height of 100 μm, i.e., 14 pixels, from the contact point. At this height, the contact angle is shown to reach a constant value and is not influenced anymore by blurring at the contact line. The goniometric mask method is validated by comparison with the polynomial fitting method, and a maximal difference of 5% between the two methods is observed. Finally, the maximum droplet diameter Dmax, the maximum spreading ratio βmax = Dmax/D0, and the time at maximum spreading tmax are determined. Each measurement is repeated at least three times. Experimental Results. We first mention that the results presented below are also illustrated by Movie S1 at droplet
a
Ethanol droplets of D0 = 1.8 mm for steel and parafilm, and of D0 = 2.0 mm for Al sm, Aluminum, and Al ro.
possible deformation of an impacting droplet due to drag force was measured by determining the difference between horizontal and vertical diameter. We found this difference to be less than 6% for all measurements, showing that the drag force does not have a substantial influence here. The droplet impact and spreading on the surface is recorded by a high-speed camera (model NX7-S2 from IDT) with a frame rate of 10 000 fps and a pixel resolution of 7.38 μm (12x zoom lens, Navitar). The droplet is illuminated by a LED lamp (model LLS2 from SCHOTT) with 5 μs exposure time per frame. Images are analyzed in MATLAB (MathWorks Inc.) using an in-house C
DOI: 10.1021/acs.langmuir.5b04557 Langmuir XXXX, XXX, XXX−XXX
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ethanol forms a flat liquid layer with a dynamic contact angle lower than 60°, showing a dynamic wetting behavior. In Figure 3a−d, the results for two liquids, water and glycerol, at an impact velocity of Vi = 2.1 m/s are represented.
Figure 2. Spreading of droplet at an impact velocity of Vi = 0.5 m/s on three substrates: steel (blue), aluminum (red), and parafilm (black) for ethanol and water droplets. Vertical lines indicate time at βmax. (a) spreading diameter ratio β; (b) dynamic contact angle θD; snapshots of (c) ethanol droplet and (d) water droplet impacting on aluminum. Movies for three liquids on aluminum are found in the Supporting Information (Movie S1).
Figure 3. Spreading of droplet at an impact velocity Vi = 2.1 m/s on three substrates: steel (blue), aluminum (red), and parafilm (black) for water and glycerol droplets. Vertical lines indicate time at βmax. (a) spreading diameter ratio β; (b) dynamic contact angle θD; snapshots of (c) water droplet and (d) glycerol droplet impacting on aluminum. Movies for three liquids on aluminum are found in the Supporting Information (Movie S2).
impact velocity Vi ∼ 0.5 m/s and in Movie S2 at droplet impact speed Vi ∼ 2.1 m/s for impact of the three liquids on aluminum (see the Supporting Information). We observe in Figure 2a that, at impact velocity of Vi = 0.5 m/s, the ethanol droplet spreads more than the water droplet on all three surfaces. Ethanol reaches its maximum diameter later in time than water, indicated by the vertical lines in Figure 2a. Until maximum spreading, the droplet diameter evolution for water and ethanol is rather similar for the all three surfaces. The dynamic contact angle at maximum spreading for ethanol is around 60°, showing a dynamic wetting behavior (Figure 2b). The dynamic contact angle of water is higher than the one of ethanol and reaches a value of around 105° at maximum spreading, showing a dynamic nonwetting behavior for all surfaces during spreading. Figure 2c,d show images of the droplets at different times for ethanol and water, respectively. At maximum spreading (tmax = 2.7 ms), the water droplet shows a pancake form with a contact angle higher than 90°, showing a dynamic nonwetting behavior. At the same time, the ethanol droplet shows a spreading lamella at the droplet contact line. At maximum spreading (7.8 ms),
We observe in Figure 3a that glycerol spreads less than water, reaching its maximum diameter earlier in time, which is attributed to its higher viscosity, causing the kinetic energy to be dissipated faster. Figure 3b shows that the dynamic contact angle during spreading and at maximum spreading is quite similar for water and glycerol ranging from 107 to 123°, thus showing a dynamic nonwetting behavior. Figure 3c shows that the water droplet displays a spreading lamella at short impact time. Visual observation from above indicates that the lamella evolves to a fingering shape at longer time. In Figure 3d, glycerol also initially shows a lamella, but later a pancake form appears. Figure 4a shows that the maximum spreading ratio βmax for three substrates with the same roughness (steel, aluminum, parafilm), given a specific liquid, is almost identical for all the surfaces. Ethanol shows the highest maximal spreading, glycerol D
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Figure 4. Maximum spreading ratio βmax versus impact velocity Vi: (a) surfaces with same roughness (Ra ∼ 0.41); (b) aluminum with different roughness. (c) Dynamic contact angle at maximum spreading θD(tmax) as a function of the impact velocity Vi on steel substrate.
Figure 5. Time at maximum spreading tmax versus impact velocity: (a) measured data; (b) rescaled data.
range and depends mainly on the nature of the fluid. We remark that, in supplementary experiments, we found that the dynamic contact angle for the other substrates follows the same observations. The average values of the dynamic contact angle at maximum spreading are reported in Table 3. Ethanol shows a dynamic contact angle between 41 and 63° and is dynamically wetting all substrates. Water and glycerol show a contact angle ranging from 93 to 121° and are dynamically nonwetting. Comparing the dynamic and equilibrium contact angles (see Table 3), we observe that the dynamic contact angle is higher than the equilibrium contact angle, except for water on parafilm, where the two contact angles are quite similar. It is observed that the dynamic contact angle depends less on the nature and roughness of the surface than the equilibrium contact angle does. This observation is attributed to the presence of a thin air layer between the drop and the surface
the lowest spreading, while water shows a transition behavior between both. At low impact velocity, the data level off to a constant spreading ratio, which we will denote further by βVi=0. The data for ethanol and glycerol at high impact velocity tend toward an asymptotic behavior with constant slope. Figure 4b shows the maximum spreading ratio for the aluminum surface with increasing roughness. The maximum spreading ratio data almost coincide, showing that roughness has a minimal influence on maximum spreading. We may conclude that the nature, not of the substrate, but of the liquid plays the most important role in the maximum spreading of a droplet on a solid surface. Figure 4c shows, as an example, the dynamic contact angle at maximum spreading θD(tmax) as a function of impact velocity for the different liquids on the steel substrate. The dynamic contact angle θD(tmax) is quite constant over the impact velocity E
DOI: 10.1021/acs.langmuir.5b04557 Langmuir XXXX, XXX, XXX−XXX
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to surface. The transverse momentum diffusion constant equals Dμ = μ/ρ. The energy balance equation between state 1 (before impact) and 2 (at maximum spreading) is KE1 + SE1 = SE2 + W. Different models have been proposed,20,22,25,30 which differ in the choice of terms included in this energy balance, as well as the geometry of the droplet during spreading and the specific descriptions of the terms SE2 and W. Models in Literature. At low impact velocity, we can neglect the dissipation energy W and the balance equation becomes KE1 = SE2. Assuming the spherical droplet being deformed into a pancake-like droplet and SE2 = (π/4)Dmax2·γLV, it can be shown that βmax scales with We1/2. This scaling is regularly assumed in literature.41,60,61 On the other hand, Clanet et al.18 developed a model based on momentum and mass conservation, and found a scaling of βmax with We1/4. Our measuring results of the maximum spreading ratio versus We number in Figure 7 indicate that the data do not scale with We1/4 nor with We1/2. While a We1/4 scaling could be consistent for water, it is not for ethanol and glycerol.
during spreading for all cases considered. Due to the presence of this air layer, the dynamic contact angle depends primarily on the surface tension of the liquid, and, to a lesser extent, on the nature and roughness of the substrate. On the contrary, for the equilibrium contact, where the liquid directly contacts the surface, the nature and roughness of the surface plays a more important role. Figure 5a shows that the time at maximum spreading decreases with impact velocity. Ethanol shows a higher spreading time than water and glycerol do, which is attributed to its lower surface tension. The time at maximum spreading is similar for water and glycerol, as their values of surface tension are similar. This observation suggests that the time at maximum spreading scales with the surface tension. Figure 5b shows that, after scaling the time at maximum spreading with the surface tension, tmax·γ/γwater, all data collapse onto a single curve. Finally, in Figure 6, we compare our data with measurements from literature, and a good agreement is found. In the next
Figure 6. Comparison of experimental data of this paper for water on aluminum surface with other experiments of water on different substrates from the literature5,51,55−58. Figure 7. Maximum spreading ratio as a function of Weber number, and scaling with We1/2 (dashed line) and We1/4 (dotted line).
section, we will use the experimental data to model the maximum spreading ratio. Since our data also contain detailed information on the time at maximum spreading and dynamic contact angle at maximum spreading, we will mainly use our own data to compare the different models.
The scaling of βmax ∼ Weα implies that, at zero impact velocity, the maximum spreading ratio equals zero, which is physically not possible (βmax ≥ 1). Our results in Figures 4a and b show that, at low impact velocity, the data does not tend to zero, but levels off to a constant maximum spreading ratio, which we denote by βVi=0. To account for the dynamic wetting behavior correctly, both the surface tension energy before impact SE1 and at maximum spreading SE2 have to be included in the energy balance. Assuming the spherical droplet is deformed into a pancake-like droplet with height h and using Young’s equation γSV = γSL + γLVcos θ, the surface tension energy at maximum spreading equals SE2 = π/4D2maxγLV(1 − cos θ) + π/4DmaxhγLV.20,30 For determining the dissipation energy W, the time from impact to maximum spread tmax is derived assuming that the spherical drop spreads into a cylindrical disk22 to be 8 D0 tmax = 3 Vi (2)
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MODELING The energy balance of a droplet impacting on a solid surface takes into account the kinetic energy, capillary or surface tension energy and energy loss due to viscous dissipation. The kinetic energy before impact is KE1 = (π/12)·D03ρVi2 and the surface tension energy before impact equals SE1 = πD02γLV. The surface tension energy at maximum spreading is SE2 = SLV·γLV + SSL·(γSL − γSV), where SLV is the droplet surface contacting the vapor phase, while SSL is the droplet surface contacting the solid surface. The loss of energy due to viscous dissipation during spreading is22 W=
t max
∫0 ∫Ω τ dΩdt ≈ τ Ωtmax
(1)
where τ is the mean value of the viscous dissipation energy per unit time and volume59 and is τ ≈ μ(Vi/δ)2. Ω in eq 1 is the volume where viscous dissipation occurs and Ω = (π/4)Dmax2·δ, where δ is the boundary layer thickness. The boundary layer thickness is the depth to which momentum can diffuse on the time scale of the drop motion, or δ = (cDμ·tmax)1/2, where c is a constant equal to 2 for one-dimensional diffusion perpendicular
The boundary layer thickness then equals δ = (cDμ·tmax)1/2 = 2D0/(Re)1/2, where c = 3/2.22,49,58 Note that the coefficient c in this model does not equal the theoretical value 2 as reported before. Solving the energy balance for the maximum spreading ratio, we get F
DOI: 10.1021/acs.langmuir.5b04557 Langmuir XXXX, XXX, XXX−XXX
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Figure 8. Comparison between experiments and models for maximum spreading ratio as a function of impact velocity: (a) ethanol, (b) water, and (c) glycerol on aluminum surface. 2 = βmax
12γLV + ρVi2D0 3Γ + 2ρVi2D0Δ
−
8γLV
1 2 3Γ + 2ρVi D0Δ βmax
overpredict the maximum spreading ratio, while the Visser and Roisman models underpredict the spreading ratio. The goodness of model prediction for the Visser and Roisman models are lower than the one of the Mao and Pasandideh-Fard models. In the next section, we propose an improved model to predict the maximum spreading ratio, with special attention to the lower impact velocity range. Improved Model. The Mao model is based on the assumption that the time at maximum spreading is properly described by eq 2, indicating that tmax is only dependent on the impact velocity given the initial diameter and not on surface tension. In Figure 5a,b our data showed that tmax is also dependent on the surface tension and can be rescaled by the ratio of liquid surface tension. Figure 9a shows that eq 2 for predicting tmax is not in good agreement with the experimental data. It is more reasonable to assume that the time at maximum spreading scales with Dmax/Vi instead of with D0/Vi. We therefore analyze how our measured data of Dmax in function of the impact velocity Vi to predict the time at maximum spreading. Figure 9b confirms that the equation
·
(3)
with Δ = δ/D0 the dimensionless boundary layer height and Γ = γLV(1 − cos θ). This equation is referred to as the Mao model in this paper (Mao et al.30). In the Mao model, the equilibrium contact angle is used for determining Γ. In Figure 8a−c, we compare the predictions of the Mao model with our experimental data. Also included are predictions of other models such as the models of Pasandideh-Fard et al.,22 Roisman49 and Visser et al.51 The goodness of model prediction is determined as g=1−
1 n
i
∑
(yi − fi )2
n
yi2
(4)
where yi is the measured maximum spreading ratio, f i is predicted maximum spreading ratio by the model, and n is the number of data. The goodness of model predictions is summarized in Table 5. We observe that most models give a good prediction in the higher impact velocity range. At lower impact velocity, the Mao and Pasandideh-Fard models
tmax = b
Table 5. Goodness of Model Prediction for Maximum Spreading Ratio (Aluminum Surface)
g
PasandidehFard et al. (1996)
Mao et al. (1997)
Visser et al. (2012)
Roisman (2009)
improved model
ethanol water glycerol
0.9898 0.9491 0.9767
0.9253 0.9329 0.9348
0.8906 0.9813 0.9572
0.8973 0.5103 0.9209
0.9927 0.9771 0.9930
Dmax Vi
(5)
is in much better agreement with our measurements. The parameter b equals the ratio of surface tension of liquid versus surface tension of the reference fluid, in this case water, counting for the scaling with surface tension. Assuming tmax = b· Dmax/Vi, the boundary layer thickness equals δ = (cDμ·tmax)1/2 = (bc/Re)1/2(D0Dmax)1/2, where the constant c = 2 for onedimensional problems. Solving the energy balance for the maximum spreading ratio, we get G
DOI: 10.1021/acs.langmuir.5b04557 Langmuir XXXX, XXX, XXX−XXX
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Figure 9. Time at maximum spreading tmax versus impact velocity; (a) comparison of scaled measured data with eq 2; (b) comparison of measured data with eq 5. 2 ρVi2D0 + 12γLV = 3Γβmax + 8γLV
1 5/2 1 + 3 b/c ρVi2D0βmax βmax Re (6)
been identified to properly characterize this dynamic wetting process, especially important at low impact velocity. Experiments show that the time at maximum spreading decreases with impact velocity depending on the surface tension of the liquid and a new scaling of the time at maximum spreading with maximum spreading diameter and surface tension is proposed. Based on this finding, an improved energy balance model for maximum spreading ratio is proposed using an improved analytical model of the time at maximum spreading, which is an input to the model for predicting the viscous energy dissipation at maximum spreading. A second improvement is based on the use of the dynamic contact angle at maximum spreading, instead of quasi-static contact angles, to describe the dynamic wetting process at low impact velocity. This new model shows good agreement compared to experiments for the maximum spreading ratio versus impact velocity for different liquids and a better prediction compared to other models in the literature. The dynamic contact angle is found to be generally higher than the equilibrium contact angle, showing that statically wetting surfaces can become less wetting or even nonwetting under dynamic droplet impact. Surface roughness and type of surface (steel, aluminum, and parafilm) slightly affect the dynamic wettability and maximum spreading at low impact velocity, while the type of liquid is found to play a major role. Scaling according to We1/2 is found to be invalid for low velocities, since the curves bend over to higher maximum spreading ratios due to a dynamic wetting process.
This model is referred to as the improved model. This model cannot be solved directly for βmax and is solved numerically. A second improvement is that we do not use a quasi-static contact angle, but the dynamic contact angle at maximum spreading θD(tmax) as determined in our experiments and documented in Table 3. The comparison of the results of the improved model with the measurement data in Figure 8 shows that improved model predicts the maximum spreading ratio quite well, especially in the low impact velocity region. When comparing the goodness of model prediction for the different models in Table 5, we observe that the proposed model improves the predictions of the maximum spreading ratio for all liquids. This observation suggests that the appropriate contact angle to be used is the dynamic contact angle during spreading, indicating that the influence of dynamic wetting on maximum spreading is properly described by the dynamic contact angle at maximum spreading. The asymptotic solution for maximum spreading ratio at high impact velocity is normally based on the assumption that the surface tension energies SE1 and SE2 can be neglected resulting in KE1 = W. When the volume of the droplet at maximum spreading is assumed to form a cylinder with diameter Dmax and height h and we further assume that the boundary layer height δ equals the height of the cylinder h and that the time at maximum spreading is given by tmax = Dmax/Vi, it is found that the maximum spreading ratio βmax scales with Re1/5.18,25 We also determine that the asymptote of the improved model (eq 6) at large impact velocity (Vi → ∞) is 5/2 βmax
=
1/5
c /9b Re → βmax ∼ Re
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b04557. Video explaining spreading of liquid droplets (ethanol, water, and glycerol) on aluminum surface at low (Vi = 0.5 m/s, S1) impact velocities (AVI) Video explaining spreading of liquid droplets (ethanol, water, and glycerol) on aluminum surface at high (Vi = 2.1 m/s, S2) impact velocities (AVI)
(7)
Equation 7 shows that the maximum spreading ratio in our improved model scales at high impact velocity with Re1/5 as is also commonly documented in the literature.
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CONCLUSION Droplet impact has been imaged on rigid substrates of different roughness for three liquids with different viscosity and surface tension, with special attention to the lower impact velocity range. The maximum spreading ratio as a function of impact velocity clearly shows the main role of surface tension and viscosity. The dynamic contact angle at maximum spreading has
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. H
DOI: 10.1021/acs.langmuir.5b04557 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir Notes
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The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors acknowledge the support of Swiss National Science Foundation Grant No. 200021_135510.
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DOI: 10.1021/acs.langmuir.5b04557 Langmuir XXXX, XXX, XXX−XXX
