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Dynamic wetting angle of a spreading droplet
Experimental Thermal and Fluid Science 29 (2005) 795–802 www.elsevier.com/locate/etfs
Dynamic wetting angle of a spreading droplet Sˇ. Sˇikalo a, C. Tropea
b,*
, E.N. Ganic´
a
a
b
Faculty of Mechanical Engineering, University of Sarajevo, Vilsonovo setaliste 9, Bosnia and Herzegovina Technische Universitat Darmstadt, Fachgebiet Stro¨mungslehre und Aerodynamik, Petersenstraße 30, 64287 Darmstadt, Germany
Abstract The dynamic contact angle is required as a boundary condition for modelling problems in capillary hydrodynamics, including certain stages of the droplet impact problem. The dynamic contact angle differs appreciably from the static advancing or receding values, even at low velocities. This paper presents experimental results of the dynamic contact angle of spreading droplets after impacting on horizontal and inclined surfaces. Droplets of an 85% (vol.) glycerin/water solution (D = 2.45 mm) and water (D = 2.7 mm) were used in the study. Two surfaces, wax (low wettable) and glass (high wettable) are used, to study the effect of surface wettability (static contact angle). The results include the dynamic contact angle of droplets together with the contact line velocity and the dimensionless spread factor as a function of time. 2005 Elsevier Inc. All rights reserved. Keywords: Drop impact; Dynamic contact angle
1. Introduction The wetting process is encountered in many coating flows, where a viscous liquid spreads over a dry substrate. Examples of industrial processes include dip coating of sheet metal, spin coating of surface layers, coating of inks on paper and gravity-driven drainage of paints. Achieving a spatially uniform coating requires careful control and understanding of the mechanisms that influence the spreading dynamics of liquid. In many of these different applications, the contact line dynamics is not completely understood. There is no complete and experimentally verified theory for the dynamic contact angle. Two different approaches to describe the wetting process are commonly used. The first approach is based on molecular-kinetic theory. The second approach is hydrodynamic in nat* Corresponding author. Tel.: +49 0 6151 162854; fax: +49 0 6151 164754. E-mail addresses: [email protected] (Sˇ. Sˇikalo), [email protected] (C. Tropea), [email protected] (E.N. Ganic´).
0894-1777/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2005.03.006
ure, in which the change of the contact angle during motion of the contact line is influenced by viscous stresses [1]. Shikhmuzaev [2] reviewed these different approaches to defining the contact angle and resolving the viscous singularity; however, neither are completely satisfactory for all applications. The typical procedure for comparing experiment and theory examines the dependence of the dynamic contact angle on the contact line speed. Here, the measured contact angle is referred to as the dynamic contact angle. Methods used to measure the dynamic contact angle (hD) are similar to those used to measure the static contact angle. In many experiments, the dynamic contact angle is measured directly through low-power optics. Therefore, some variations in reported values of contact angles can be attributed simply to different magnifications at which the measurements were made. The assignment of the tangent line, which defines the contact angle, is a factor that can lead to significant subjective error. Also the velocity of the contact line cannot be as well controlled during droplet spreading, as it can in capillary glass tubes. Thus, it must be calculated from spread
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Nomenclature Ca d D Ra t u v We x
capillary number spreading diameter, m droplet diameter before impact, m average surface roughness, lm time after impact, s impact velocity, m/s contact line velocity, m/s Weber number, We = qu2D/r the contact line distance from the point of impact, m
factor of the droplet and may involve some additional error. It is known from previous experiments that the dynamic contact angle hD depends on the spreading velocity v of the contact line (see for example the reviews [1,3–6]). Hoffman [7] carried out one of the most complete experimental studies of spreading viscous fluids in glass capillary tubes. Dynamic contact angles were calculated from the spherical meniscus by a micro photographic technique. One of the simplest approximate forms of the relationship between the dynamic contact angle and spreading velocity, typically in the dimensionless form of the capillary number Ca = lU/r with l and r denoting the dynamic viscosity and surface tension of the liquid is the Hoffman–Voinov–Tanner law for small capillary numbers, Ca < O(0.1) [8] h3D h3e ffi cT Ca
ð1Þ
where hD is the dynamic contact angle, he is the static equilibrium contact angle and cT a constant. HoffmanÕs experimental value for cT is about 72 rad3, although it does in fact depend weakly on the flow system size. For spontaneous spreading processes driven by molecular forces the dynamic contact angle hD is very close to the static value he and the interface near the contact line is wedge shaped. Under forced spreading, the dynamic contact angle can be very different from the static one and Ca is typically much larger than in the case of spontaneous spreading. Ngan and Dussan [9] showed that, in contrast to static wetting, the measurement of a dynamic contact angle near the contact line strongly depends on the macroscopic scale. Therefore, such measurements made in one geometry may not necessarily be used to predict the dynamics in another geometry. However, experimental results by Blake et al. [10], demonstrated that in curtain coating the dynamic contact angle varies not only with substrate speed but also with other (independent) parameters such as flow rate and curtain height. Summers et al. [11] made an attempt to explain this variation by accounting for the bending
Symbols h contact angle, rad l dynamic viscosity of the liquid, kg/m s q density of the liquid, kg/m3 r surface tension of the liquid, N/m Subscripts a advancing D dynamic r receding
of the meniscus close to the contact line by measuring an apparent contact angle at a certain resolution from the contact line. The apparent contact angle is taken to be the tangent to the free surface at a distance L (set to the experimental resolution of 20 lm) from the contact line. To explain the observed contact angle profile they developed a numerical model. There is a clear discrepancy of their value obtained from the numerical solution (at 20 lm from the contact line) in comparison to the experiments (at Ca = 0.27).
2. Experimental techniques and conditions To investigate the dynamic contact angle of impacting liquid droplets, a series of experiments were conducted with individual droplets impacting onto solid dry and smooth surfaces. The experimental method involved in these investigations has been described in detail in [12]. To observe the spreading droplet a high resolution CCD camera (Sensicam PCO, 1240 · 1024 pixels) equipped with a magnifying zoom lens is used. The magnification was manipulated so that the image could accommodate the maximum spread of the droplet. The spatial resolution (number of pixels per mm) was calibrated using a calibration scale. For example, at the scale of 213 px/mm one pixel resolution corresponds to 4.7 lm. The image processing was performed using Optimas image processing software. From the side view images, the spread factor and the dynamic angle are measured. In the present study experimental data about the dynamic contact angle of a glycerin (85%) droplet (D = 2.45 mm) spreading on a horizontal surface and a water droplet (D = 2.7 mm) spreading on 45 inclined smooth glass (Ra = 0.003 lm) and wax (Ra = 0.3 lm) are presented. The physical properties of the liquids and the wettability of the surfaces (advancing ha and receding hr static contact angles) are shown in Table 1. The values of static (equilibrium) contact angles are
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797
Table 1 Properties of the liquids and wettability of the surfaces Liquid
r (N/m)
l (mPa s)
q (kg/m3)
ha hr Glass
ha hr Wax
Water Glycerin
0.073 0.063
1.0 116
996 1220
10–6 17–13
105–95 97–90
found to depend on the recent history of the interaction. At the contact line of a stationary meniscus, the static contact angle he can assume any value between ha and hr ha 6 he 6 hr
ð2Þ
The Weber number (We = qDu2/r) based on droplet density, its diameter and its impact velocity ranges from 51 to 802 for the glycerin droplet and the Weber number for the water droplet is 90.
3. Results and discussion 3.1. Impact onto a horizontal surface The angle formed between the liquid–vapor interface and the liquid–solid interface at the solid–liquid–vapor three-phase contact line is conventionally defined as the contact angle. It should be noted that, at the molecular level, the three phases do not meet in a line but within a zone of small but finite dimensions in which the three interfacial regions merge. Therefore, the microscopic contact angle may be different from the contact angle measured by a conventional approach, such as the method presented here. However, discussion of such microscopic contact angles is beyond the scope of this work. The visualization method presented here also allows measurement of the evolution of the contact angle hD. Each exposure allows a double measurement of the wetting angle, on the left and the right of the droplet as Fig. 1a shows (notice that this is a half of the image). The value of dynamic contact angles may be appreciably larger (advancing) or smaller (receding) than its static value, as Fig. 1 illustrates. The contact angle is the angle between the tangent to the droplet profile and the tangent to the surface at the junction point between the air, fluid and solid, as Fig. 1 shows. The difficulty is in accurately estimating the tangent to the curved droplet profile because the curve ends at the point where one needs to measure it. This technique requires having the droplet in focus and having a clear baseline. Also the results are sensitive to the viewing angle. In this technique a tangent is aligned with the spreading-droplet profile at the point of contact with the solid surface. This is done directly using image processing software (Optimas 6.2), which allows the measurement
Fig. 1. Dynamic contact angles of: (a) glycerin droplet on horizontal smooth glass (ha = 17, hr = 13), exposure 1 (t1 = 1.7 ls) advancing motion and exposure 2 (t2 = 8.42 ls after impact) receding motion, (b) water droplet on an inclined (45) wax surface (ha = 105, hr = 95).
of line inclination. The results are somewhat subjective and depend on the experience of the operator, although certain training procedures can be used to improve both the accuracy and precision. Spatial filters can be applied to the image (noise reduction, edge enhancement, and sharpening) to increase the accuracy and repeatability of the measurement. With some training, this method of measurement of droplet contact angles can yield results to within approximately ±5. In reporting results, the dynamic contact angle (hD) is plotted together with the contact line velocity and the spread factor (d/D) as functions of time from impact, expressed in a dimensionless form as tu/D. Figs. 2–4 show the effect of the substrate surface (static contact angle) on the impact process of a 2.45 mm glycerin droplet with different droplet Weber numbers. Measurements of the contact angle at early times of the spreading process show a decrease of the contact angle with increasing spread factor. In the initial phase of spreading (droplet deformation phase) the contact angle decreases to a minimum value (about 130 for a glycerin). In the spreading phase (after ejection of the lamella, the contact angle increases to a local maximum (about 150). The contact angle of a glycerin droplet reaches its maximum at about the same instant as the spread factor (Figs. 2–4). Then the contact angle decreases rapidly from its maximum (larger than the equilibrium) to a minimum (smaller than the equilibrium) before returning to the equilibrium contact angle. The spread factor changes only slowly on the wax in reaching its final value. On the glass surface the contact angle of a glycerin droplet decreases monotonically from its local maximum value to its equilibrium value (static contact angle) and the spread factor increases slowly. The time to reach the equilibrium state is much longer than shown in the figures. In the later spreading phase, the spreading velocity is very small. For example, for
Sˇ. Sˇikalo et al. / Experimental Thermal and Fluid Science 29 (2005) 795–802
v 0.8
0 0.01
0.1
1
10
d/D, θ D (rad), v x 10-1 (m/s)
1000
(a)
tu/D 3.2
(b)
100
We = 802 Glycerin-Glass
2.4 d/D 1.6
θD v
0.8
static contact angle 0 0.01
d/D, θD (rad), v x 10-1 (m/s)
d/D
θD
0.1
1 tu/D
10
100
Fig. 2. Dynamic contact angle (hD), spread factor (d/D) and contact line velocity (v) of a glycerin droplet on (a) smooth wax and (b) smooth glass; D = 2.45 mm, We = 802.
spreading of the glycerin droplet on glass at We = 802, the spreading velocity is about 0.1 m/s for tu/D = 2.0 and for tu/D = 15.0 it is about 0.007 m/s (Fig. 2b). The dynamic contact angle differs appreciably from the static advancing (ha) or receding values (hr), even at velocities as low as 10 lm/s [1]. Fig. 5a and b compare the dynamic contact angle of a glycerin droplet on the glass and wax surfaces, which shows that the surface (static contact angle) does not affect the dynamic contact angle in the spreading phase up to the local maximum (maximum spread). No significant difference is seen in the early recoiling stage, because the inertia and viscous forces dominate the deformation. The effect of the static contact angle is observed only in the later recoiling stage (tu/D > 10). Surface tension then dominates, and the droplet slowly approaches the equilibrium state, where the contact angle reaches its static value (13–17 on glass or 90–97 on wax). The influence of the impact velocity (expressed here by the Weber number) on the dynamic contact angle of a glycerin droplet on glass is shown in Fig. 6. At lower impact velocities (We = 51) the dynamic contact angle reaches a minimum of about 138, at an earlier dimensionless than at larger impact Weber numbers (We = 802), where the minimum was about 125. The maximum of the dynamic contact angle for the impact Weber number of 51 occurs earlier at tu/D = 0.69 and
We = 93 Glycerin-Wax
2.4 d/D
θD
1.6
v 0.8
0 0.01
0.1
1
(b)
10
tu/D 3.2
d/D, θ D (rad), v x 10-1 (m/s)
(a)
2.4
1.6
3.2
We = 802 Glycerin-Wax
static contact angle
d/D, θ D (rad), v x 10-1 (m/s)
3.2
static contact angle
798
We = 93 Glycerin-Glass
2.4 d/D 1.6
θD v
0.8
static contact angle 0 0.01
0.1
1
10
tu/D
Fig. 3. Dynamic contact angle, spread factor and contact line velocity of a glycerin droplet on (a) smooth wax and (b) smooth glass; D = 2.45 mm, We = 93.
has the value of about 156, while for the Weber number of 802 the maximum of about 146 appears at tu/D = 1.5. In the later spreading phase, the dynamic contact angle decreases faster for a larger impact Weber number. 3.2. Impact onto an inclined surface When a droplet impacts onto an inclined surface, it spreads asymmetrically relative to the point of impact, as Fig. 7 illustrates [13]. The front edge (xfront) spreads below the point of impact faster than the back edge (xback) above the impact point. Note that both the values xfront and xback are defined as positive moving away from the point of impact. The dynamic contact angle of a water droplet low viscous liquid) impacting onto a horizontal surface is not easy to measure by the low power optics, used in this experimental study. The spreading lamella of a water droplet is very thin. Thus the radius of the lamella rim at the front, near the contact line is small and does not allow precise determination of the tangent to the lamellaÕs curved surface at the contact point. The analysis of a water droplet impacting onto an inclined surface offers the advantage of higher accuracy. The front and back sides of the droplet are thicker and it is easier locate the contact point and accurately estimate a tangent
Sˇ. Sˇikalo et al. / Experimental Thermal and Fluid Science 29 (2005) 795–802 180
We = 51 Glycerin-Wax
Glycerin-Glass 150
2.4 120
d/D
θ D (deg)
θD
1.6
static contact angle
d/D, θ D (rad), v x 10-1 (m/s)
3.2
v 0.8
0.01
(a)
0.1
1
90
We=51 We=93 We=802
60 30
0 10
tu/D
0 0.01
static contact angle 0.1
1
10
tu/D
3.2
d/D, θ D (rad), v x 10-1 (m/s)
799
Fig. 6. Comparison of dynamic contact angles of a glycerin droplet on smooth glass for We = 802, We = 93 and We = 51.
We = 51 Glycerin-Glass
2.4 d/D
θD
1.6
v 0.8
static contact angle 0 0.01
0.1
1
(b)
10
tu/D
Fig. 4. Dynamic contact angle (hD), spread factor (d/D) and contact line velocity (v) of a glycerin droplet on (a) smooth wax and (b) smooth glass; D = 2.45 mm, We = 51.
180
We = 802 Glycerin
150
θ D (deg)
120 90 60
Fig. 7. Side view of a liquid droplet on an inclined surface.
Wax
wax
Glass
θa θr static contact angle glass
30
(a)
0 0.01
0.1
1
10
100
1000
tu/D
180
We = 93 Glycerin
150
θ D (deg)
120 90
W ax Glass
wax
60 static contact angle
30
glass
0 0.01
(b)
0.1
1
10
tu/D
Fig. 5. Comparison of dynamic contact angles of a glycerin droplet on the smooth glass and smooth wax; (a) We = 802 and (b) We = 93.
to the curved droplet profile, as illustrated in Fig. 8 (see [14] for more details). Fig. 9a and b shows the measured contact angles and spread factors for a water droplet (We = 90) impacting onto a smooth glass and wax surfaces with 45 inclination. The decrease in the front contact angle corresponds to the increase in the front spread factor for both wax and glass surfaces. The contact angle of the dropletÕs rear contact line decreases to a minimum (about 46) on wax, which is reached considerably later (about tu/D = 4.7) than the time at which the rear contact line begins to slide (about tu/D = 0.85) on wax. After reaching the minimum, the back contact angle increases, approaching the static contact angle (95–105) as the droplet slows to rest. The contact angle on glass decreases, reaching its equilibrium state (6–10). The back contact angle decreases faster and may be lower than the static receding contact angle, as Fig. 9b shows. Fig. 10 shows the effect of the substrate surface on the dynamic contact angle of a water droplet. The difference between dynamic contact angles of a water droplet on
Sˇ. Sˇikalo et al. / Experimental Thermal and Fluid Science 29 (2005) 795–802
800
180
We=90, α =45°
150
θ Front, wax
90
θ Back, wax
60
θ Front, glass θ
static contact angle
θ D (deg)
120
Back, glass
30 0 0.01
Fig. 8. Definition of the contact angles between a water droplet and an inclined surface, where hFront and hBack are the dynamic contact angles in the front and back directions, respectively; hFront = 132, hBack = 46; a = 45, D = 2.7 mm, We = 90, time after impact is t = 8.08 ms.
1.6
We=90, α =45° Water-Wax x/D Front
x/DBack
θ Front
θ Back
vFront
vBack
static contact angle
2.4
-1
x/D, θ D (rad), v x 10 (m/s)
3.2
0.8
0
-0.8 0.01
0.1
(a)
1
10
0.1
1 tu/D
10
100
Fig. 10. Dynamic contact angles of a water droplet on the smooth wax and glass (a = 45, D = 2.72 mm, We = 90).
3.3. Dynamic contact angle as a function of capillary number The dynamic contact angles in terms of capillary number (Ca = vl/r) for the glycerin and water droplet are shown in Figs. 11 and 12. Comparisons with some existing correlations [15–17] are given. The data for glycerin in Fig. 11 show a rapid change in slope. There is a region where the dynamic contact angle falls as Ca (velocity of contact line) increases. The positive slope of hD with Ca strongly decreases at about Ca = 0.2 (v = 1.1 m/s). The slope increases at larger capillary numbers above about 10 (v about 5.5 m/s).
100
3.2
tu/D
Glycerin-Glass
1.6
We=90, α =45° Water-Glass
x/D
Front
x/D
Back
2.4
θ D , rad
2.4
-1
x/D, θ D (rad), v x 10 (m/s)
3.2
θ Front θ Back
v Front
We = 802 Jiang (1979) 0.8
static contact angle
Bracke (1989) Seebergh (1992)
v Back 0.8
(a) 0 0.01
(b)
We = 51 We = 93
1.6
0.1
1 tu/D
10
0 0.01
0.1
1 Ca
10
100
100
3.2 Glycerin-Wax
Fig. 9. Dynamic contact angle, spread factor and contact line velocity of a water droplet on inclined (a) smooth wax and (b) smooth glass; a = 45, D = 2.72 mm, We = 90.
θ D , rad
2.4
glass and wax is not observed in the first stage of spreading (tu/D = 0.1), where the inertial forces are strong. The dynamic contact angle decreases faster on glass than on wax when approaching the static contact angle. The back contact angle on glass decreases slightly below the static receding contact angle and then approaches the static value (6–10). Due to the strong elongation, the water droplet can rupture on the glass surface.
1.6
We = 51 We = 93 We = 802 Jiang (1979)
0.8
(b)
0 0.001
0.01
0.1
1
static contact angle
10
100
Ca
Fig. 11. Data of the dynamic contact angle hD versus Ca for glycerin droplet on horizontal: (a) smooth glass and (b) wax.
Sˇ. Sˇikalo et al. / Experimental Thermal and Fluid Science 29 (2005) 795–802 3.2
801
We = 90, α =45° Water Back-wax wax Front-wax Back-glass Front-glass Jiang (1979), glass Jiang (1979), wax
1.6
0.8
0 -0.05
static contant angle
θ D , rad
2.4
glass 0.05 Ca
0.15
Fig. 12. Data of the dynamic contact angles for water droplet on inclined smooth glass and wax, a = 45.
A faster increase of slope is observed at the smaller impact Weber number. This region belongs to the first spreading phase, in which an increase of the impact Weber number increases impact pressure and a thin film of air may be entrapped, which can affect the dynamic contact angle. At lower capillary numbers, Ca < 0.2, an increase of the impact Weber number decreases the slope of the data. This region corresponds to the later spreading phase. A decrease of the impact Weber number increases the thickness of the lamella and the radius of curvature of the interface at the contact line, which may also affect the dynamic contact angle. The irregular behavior of the data has been noted previously by Blake [6] for a tape submerged vertically into a pool of water (see [1, Figs. 3.17 and 3.20, pp. 82–84]). BlakeÕs results exhibit the changes in the slope of the data at much lower values of Ca. The data exhibit a negative slope at Ca above about 2.7 · 105 (v 0.002 m/s). The slope is again positive at Ca above about 0.007 (v 0.5 m/s). Differences between the experiments in material properties and geometry imply a difference in Ca and velocities at which the change in slope of the dynamic contact angle with the capillary number (contact line velocity) occurs. The data for the leading (front) and trailing (back) contact line also show differences, as Fig. 12 illustrates. However, for a water droplet at Ca > 0.075 (v > 5.5 m/s) the difference in the data for the dynamic contact angle of the trailing and the leading edge disappears. Also the influence of the static contact angle is obvious from a comparison of the data for wax and glass, as Fig. 12 shows. Data for dynamic contact angles of a water droplet on wax and glass are in agreement at capillary numbers above about 0.047 (v > 3.3 m/s). The empirical correlations do not agree well with experimental data as Figs. 11a and 12a illustrate. The correlation by Jiang [15] approaches 180 faster than the present experimental results do.
Fig. 13. The dynamic wetting failure at the initial phase of impact of a water droplet onto wax (side view, D = 2.7 mm, We = 90). The arrow indicates the dry area (black horizontal line).
In the initial phase of impact, the velocity of the contact line is large enough to entrap air between the liquid and solid, as Fig. 13 shows (the image has been enhanced). The liquid droplet contacts the surface with a ring-shape in the initial phase of the impact. The dry area, inside this ring-shaped liquid/solid contact area, decreases sharply with time and then the entrapped air forms an air bubble inside the droplet, which can be often observed for a droplet impact onto solid surfaces. The bubble stays on a low wettable surface while it leaves a high wettable surface. Fujimoto et al. [18] studied the time evolution of the circular dry area on the surface. They found that the initial dry area depends on the impact Weber number. At the instant of contact, the tangent to the droplet profile defines a wetting angle of 180. As the spreading velocity decreases, the air layer collapses and adhesion between liquid and the solid surface occurs. The dynamic contact angle decreases below 180. During adhesion the wetting angle seems to decrease to a minimum. Dynamic wetting failure occurs in a first phase of the contact. High liquid pressure near the contact line, at the initial phase of impact, may influence the contact line behavior and a decrease of the contact angle. After lamella ejection the pressure decreases (that can be numerically considered) and the contact angle increases. Blake et al. [19,10] have shown that the flow field can influence the onset of air entrainment and also the value of the dynamic contact angle. This implies that an empirical correlation can be applied only to particular classes of self-similar flows near the contact line [10]. Therefore, it is still necessary to conduct experimental and theoretical investigations to develop a more general model for the dynamic contact angle and its dependence on the parameters characterizing the flow and the material properties of the contacting media.
4. Conclusion There are two main conclusions from this study. First, the dynamic contact angle is not only a function of the contact line speed but also a function of the flow field in the vicinity of the moving contact line, as
802
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observed by Blake et al. [10]. As discussed above, the contact angle first decreases to a local minimum value and then increases to a local maximum value in the spreading phase, while the spreading factor increases monotonically and the contact line velocity decreases monotonically (see Fig. 2a). In the later recoiling phase the contact angle decreases to another minimum and again increases to reach the equilibrium state on the wax, while the spread factor decreases monotonically and thus the contact line (receding) velocity decreases until it becomes negative (to about 0.01 m/s). In the recoiling phase the dynamic contact angle decreases from the maximum to the final state on the glass surface, while the spread factor changes slowly and the contact line velocity is very small (<0.01 m/s). Therefore, a unique functional dependence of the dynamic contact angle on the contact line speed cannot be obtained to model the experimental results. Second, there is no effect of the static contact angle on the dynamic contact angle over a large range of experimental data for a glycerin droplet. This is observed in the spreading stage where the inertial and viscous forces dominate the deformation. In contrast, the experimental results for a water droplet on inclined glass and wax show that the static contact angle has an important influence on the dynamic contact angle (see Fig. 10). As noticed from the experimental data, a fixed contact angle (advancing or receding) as one of the boundary conditions commonly assumed in numerical calculation has obvious limitations. The dynamic contact angle, as a function of the flow variables, should be used as a boundary condition to obtain better quantitative agreement of numerical prediction with experimental data. The description of this boundary condition obviously deserves further theoretical and experimental studies.
References [1] D.T. Blake, J.K. Ruschak, Wetting: static and dynamic contact lines, in: F.S. Kistler, M.P. Schweizer (Eds.), Liquid Film Coating, Chapman & Hall, London, 1997, pp. 63–97.
[2] Y.D. Shikhmurzaev, Moving contact lines in liquid/liquid/solid systems, J. Fluid Mech. 334 (1997) 211–249. [3] V.E.B. Dussan, On the spreading of liquids on solid surfaces: static and dynamic contact lines, Ann. Rev. Fluid Mech. 11 (1979) 371–400. [4] P.G. de Gennes, X. Hua, P. Levinson, Dynamic of wetting: local contact angles, J. Fluid Mech. 212 (1990) 55–63. [5] S.F. Kistler, Hydrodynamics of wetting, in: Wettability, in: J. Berg (Ed.), Surfactant Science Series, vol. 49, Marcel Dekker, New York, NY, 1993, pp. 311–429. [6] T.D. Blake, Dynamic contact angles and wetting kinetics, in: Wettability, in: J.C. Berg (Ed.), Surfactant Science Series, vol. 49, Marcel Dekker, New York, 1993, pp. 251–309. [7] R.L. Hoffman, A study of the advancing interface. I. Interface shape in liquid–gas systems, J. Colloid Interface Sci. 50 (2) (1975) 228–241. [8] L.M. Hocking, On contact angles in evaporating liquids, Phys. Fluids. 7 (1995) 2950–2955. [9] C.G. Ngan, V.E.B. Dussan, On the dynamics of liquid spreading on solid surfaces, J. Fluid Mech. 209 (1989) 191–226. [10] T.D. Blake, M. Bracke, Y.D. Shikhmurzaev, Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle, Phys. Fluids. 11 (1999) 1995–2007. [11] J.L. Summers, P.H. Gaskell, M.D. Savage and M.C.T. Wilson, The kinematics and dynamics of free surface coating flows, Final Report on EPSRC Grant GL/L17191. School of Mech. Eng., Dep. of Physics and Astronomy, Univ. of Leeds, Leeds, LS2 9JT, 2000. Available from:. [12] Sˇ. Sˇikalo, M. Marengo, C. Tropea, E.N. Ganic´, Analysis of droplet impact on horizontal surfaces, Exp. Thermal Fluid Sci. 25 (2002) 503–510. [13] Sˇ. Sˇikalo, C. Tropea, M. Marengo, E.N. Ganic´, Oblique impact of droplets on walls and films, in: G.P. Celata, P. Di Marco, A. Mariani (Eds.), Proceedings of the 5th World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, 24–28 September, Thessaloniki, Greece, 2001, pp. 1553– 1558. [14] Sˇ. Sˇikalo, Analysis of droplet impact onto horizontal and inclined surfaces, Dissertation, TUD Darmstadt, 2002. [15] T. Jiang, S. Oh, J.C. Slattery, Correlation for dynamic contact angle, J. Colloid Interface Sci. 69 (1979) 74–77. [16] M. Bracke, F. De Voeght, P. Joos, The kinematics of wetting: the dynamic contact angle, Prog. Colloid Polym. Sci. 79 (1989) 142– 149. [17] J.E. Seebergh, J.C. Berg, Dynamic wetting in the low capillary number regime, Chem. Eng. Sci. 47 (1992) 4455–4464. [18] H. Fujimoto, H. Shiraishi, N. Hatta, Evolution of liquid/solid contact area of a drop impinging on a solid surface, Int. J. Heat Mass Transfer 43 (2000) 1673–1677. [19] T.D. Blake, A. Clarke, K.J. Ruschak, Hydrodynamic assist of dynamic wetting, AIChE. J. 40 (1994) 229–242.
Dynamic wetting angle of a spreading droplet Sˇ. Sˇikalo a, C. Tropea
b,*
, E.N. Ganic´
a
a
b
Faculty of Mechanical Engineering, University of Sarajevo, Vilsonovo setaliste 9, Bosnia and Herzegovina Technische Universitat Darmstadt, Fachgebiet Stro¨mungslehre und Aerodynamik, Petersenstraße 30, 64287 Darmstadt, Germany
Abstract The dynamic contact angle is required as a boundary condition for modelling problems in capillary hydrodynamics, including certain stages of the droplet impact problem. The dynamic contact angle differs appreciably from the static advancing or receding values, even at low velocities. This paper presents experimental results of the dynamic contact angle of spreading droplets after impacting on horizontal and inclined surfaces. Droplets of an 85% (vol.) glycerin/water solution (D = 2.45 mm) and water (D = 2.7 mm) were used in the study. Two surfaces, wax (low wettable) and glass (high wettable) are used, to study the effect of surface wettability (static contact angle). The results include the dynamic contact angle of droplets together with the contact line velocity and the dimensionless spread factor as a function of time. 2005 Elsevier Inc. All rights reserved. Keywords: Drop impact; Dynamic contact angle
1. Introduction The wetting process is encountered in many coating flows, where a viscous liquid spreads over a dry substrate. Examples of industrial processes include dip coating of sheet metal, spin coating of surface layers, coating of inks on paper and gravity-driven drainage of paints. Achieving a spatially uniform coating requires careful control and understanding of the mechanisms that influence the spreading dynamics of liquid. In many of these different applications, the contact line dynamics is not completely understood. There is no complete and experimentally verified theory for the dynamic contact angle. Two different approaches to describe the wetting process are commonly used. The first approach is based on molecular-kinetic theory. The second approach is hydrodynamic in nat* Corresponding author. Tel.: +49 0 6151 162854; fax: +49 0 6151 164754. E-mail addresses: [email protected] (Sˇ. Sˇikalo), [email protected] (C. Tropea), [email protected] (E.N. Ganic´).
0894-1777/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2005.03.006
ure, in which the change of the contact angle during motion of the contact line is influenced by viscous stresses [1]. Shikhmuzaev [2] reviewed these different approaches to defining the contact angle and resolving the viscous singularity; however, neither are completely satisfactory for all applications. The typical procedure for comparing experiment and theory examines the dependence of the dynamic contact angle on the contact line speed. Here, the measured contact angle is referred to as the dynamic contact angle. Methods used to measure the dynamic contact angle (hD) are similar to those used to measure the static contact angle. In many experiments, the dynamic contact angle is measured directly through low-power optics. Therefore, some variations in reported values of contact angles can be attributed simply to different magnifications at which the measurements were made. The assignment of the tangent line, which defines the contact angle, is a factor that can lead to significant subjective error. Also the velocity of the contact line cannot be as well controlled during droplet spreading, as it can in capillary glass tubes. Thus, it must be calculated from spread
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796
Nomenclature Ca d D Ra t u v We x
capillary number spreading diameter, m droplet diameter before impact, m average surface roughness, lm time after impact, s impact velocity, m/s contact line velocity, m/s Weber number, We = qu2D/r the contact line distance from the point of impact, m
factor of the droplet and may involve some additional error. It is known from previous experiments that the dynamic contact angle hD depends on the spreading velocity v of the contact line (see for example the reviews [1,3–6]). Hoffman [7] carried out one of the most complete experimental studies of spreading viscous fluids in glass capillary tubes. Dynamic contact angles were calculated from the spherical meniscus by a micro photographic technique. One of the simplest approximate forms of the relationship between the dynamic contact angle and spreading velocity, typically in the dimensionless form of the capillary number Ca = lU/r with l and r denoting the dynamic viscosity and surface tension of the liquid is the Hoffman–Voinov–Tanner law for small capillary numbers, Ca < O(0.1) [8] h3D h3e ffi cT Ca
ð1Þ
where hD is the dynamic contact angle, he is the static equilibrium contact angle and cT a constant. HoffmanÕs experimental value for cT is about 72 rad3, although it does in fact depend weakly on the flow system size. For spontaneous spreading processes driven by molecular forces the dynamic contact angle hD is very close to the static value he and the interface near the contact line is wedge shaped. Under forced spreading, the dynamic contact angle can be very different from the static one and Ca is typically much larger than in the case of spontaneous spreading. Ngan and Dussan [9] showed that, in contrast to static wetting, the measurement of a dynamic contact angle near the contact line strongly depends on the macroscopic scale. Therefore, such measurements made in one geometry may not necessarily be used to predict the dynamics in another geometry. However, experimental results by Blake et al. [10], demonstrated that in curtain coating the dynamic contact angle varies not only with substrate speed but also with other (independent) parameters such as flow rate and curtain height. Summers et al. [11] made an attempt to explain this variation by accounting for the bending
Symbols h contact angle, rad l dynamic viscosity of the liquid, kg/m s q density of the liquid, kg/m3 r surface tension of the liquid, N/m Subscripts a advancing D dynamic r receding
of the meniscus close to the contact line by measuring an apparent contact angle at a certain resolution from the contact line. The apparent contact angle is taken to be the tangent to the free surface at a distance L (set to the experimental resolution of 20 lm) from the contact line. To explain the observed contact angle profile they developed a numerical model. There is a clear discrepancy of their value obtained from the numerical solution (at 20 lm from the contact line) in comparison to the experiments (at Ca = 0.27).
2. Experimental techniques and conditions To investigate the dynamic contact angle of impacting liquid droplets, a series of experiments were conducted with individual droplets impacting onto solid dry and smooth surfaces. The experimental method involved in these investigations has been described in detail in [12]. To observe the spreading droplet a high resolution CCD camera (Sensicam PCO, 1240 · 1024 pixels) equipped with a magnifying zoom lens is used. The magnification was manipulated so that the image could accommodate the maximum spread of the droplet. The spatial resolution (number of pixels per mm) was calibrated using a calibration scale. For example, at the scale of 213 px/mm one pixel resolution corresponds to 4.7 lm. The image processing was performed using Optimas image processing software. From the side view images, the spread factor and the dynamic angle are measured. In the present study experimental data about the dynamic contact angle of a glycerin (85%) droplet (D = 2.45 mm) spreading on a horizontal surface and a water droplet (D = 2.7 mm) spreading on 45 inclined smooth glass (Ra = 0.003 lm) and wax (Ra = 0.3 lm) are presented. The physical properties of the liquids and the wettability of the surfaces (advancing ha and receding hr static contact angles) are shown in Table 1. The values of static (equilibrium) contact angles are
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797
Table 1 Properties of the liquids and wettability of the surfaces Liquid
r (N/m)
l (mPa s)
q (kg/m3)
ha hr Glass
ha hr Wax
Water Glycerin
0.073 0.063
1.0 116
996 1220
10–6 17–13
105–95 97–90
found to depend on the recent history of the interaction. At the contact line of a stationary meniscus, the static contact angle he can assume any value between ha and hr ha 6 he 6 hr
ð2Þ
The Weber number (We = qDu2/r) based on droplet density, its diameter and its impact velocity ranges from 51 to 802 for the glycerin droplet and the Weber number for the water droplet is 90.
3. Results and discussion 3.1. Impact onto a horizontal surface The angle formed between the liquid–vapor interface and the liquid–solid interface at the solid–liquid–vapor three-phase contact line is conventionally defined as the contact angle. It should be noted that, at the molecular level, the three phases do not meet in a line but within a zone of small but finite dimensions in which the three interfacial regions merge. Therefore, the microscopic contact angle may be different from the contact angle measured by a conventional approach, such as the method presented here. However, discussion of such microscopic contact angles is beyond the scope of this work. The visualization method presented here also allows measurement of the evolution of the contact angle hD. Each exposure allows a double measurement of the wetting angle, on the left and the right of the droplet as Fig. 1a shows (notice that this is a half of the image). The value of dynamic contact angles may be appreciably larger (advancing) or smaller (receding) than its static value, as Fig. 1 illustrates. The contact angle is the angle between the tangent to the droplet profile and the tangent to the surface at the junction point between the air, fluid and solid, as Fig. 1 shows. The difficulty is in accurately estimating the tangent to the curved droplet profile because the curve ends at the point where one needs to measure it. This technique requires having the droplet in focus and having a clear baseline. Also the results are sensitive to the viewing angle. In this technique a tangent is aligned with the spreading-droplet profile at the point of contact with the solid surface. This is done directly using image processing software (Optimas 6.2), which allows the measurement
Fig. 1. Dynamic contact angles of: (a) glycerin droplet on horizontal smooth glass (ha = 17, hr = 13), exposure 1 (t1 = 1.7 ls) advancing motion and exposure 2 (t2 = 8.42 ls after impact) receding motion, (b) water droplet on an inclined (45) wax surface (ha = 105, hr = 95).
of line inclination. The results are somewhat subjective and depend on the experience of the operator, although certain training procedures can be used to improve both the accuracy and precision. Spatial filters can be applied to the image (noise reduction, edge enhancement, and sharpening) to increase the accuracy and repeatability of the measurement. With some training, this method of measurement of droplet contact angles can yield results to within approximately ±5. In reporting results, the dynamic contact angle (hD) is plotted together with the contact line velocity and the spread factor (d/D) as functions of time from impact, expressed in a dimensionless form as tu/D. Figs. 2–4 show the effect of the substrate surface (static contact angle) on the impact process of a 2.45 mm glycerin droplet with different droplet Weber numbers. Measurements of the contact angle at early times of the spreading process show a decrease of the contact angle with increasing spread factor. In the initial phase of spreading (droplet deformation phase) the contact angle decreases to a minimum value (about 130 for a glycerin). In the spreading phase (after ejection of the lamella, the contact angle increases to a local maximum (about 150). The contact angle of a glycerin droplet reaches its maximum at about the same instant as the spread factor (Figs. 2–4). Then the contact angle decreases rapidly from its maximum (larger than the equilibrium) to a minimum (smaller than the equilibrium) before returning to the equilibrium contact angle. The spread factor changes only slowly on the wax in reaching its final value. On the glass surface the contact angle of a glycerin droplet decreases monotonically from its local maximum value to its equilibrium value (static contact angle) and the spread factor increases slowly. The time to reach the equilibrium state is much longer than shown in the figures. In the later spreading phase, the spreading velocity is very small. For example, for
Sˇ. Sˇikalo et al. / Experimental Thermal and Fluid Science 29 (2005) 795–802
v 0.8
0 0.01
0.1
1
10
d/D, θ D (rad), v x 10-1 (m/s)
1000
(a)
tu/D 3.2
(b)
100
We = 802 Glycerin-Glass
2.4 d/D 1.6
θD v
0.8
static contact angle 0 0.01
d/D, θD (rad), v x 10-1 (m/s)
d/D
θD
0.1
1 tu/D
10
100
Fig. 2. Dynamic contact angle (hD), spread factor (d/D) and contact line velocity (v) of a glycerin droplet on (a) smooth wax and (b) smooth glass; D = 2.45 mm, We = 802.
spreading of the glycerin droplet on glass at We = 802, the spreading velocity is about 0.1 m/s for tu/D = 2.0 and for tu/D = 15.0 it is about 0.007 m/s (Fig. 2b). The dynamic contact angle differs appreciably from the static advancing (ha) or receding values (hr), even at velocities as low as 10 lm/s [1]. Fig. 5a and b compare the dynamic contact angle of a glycerin droplet on the glass and wax surfaces, which shows that the surface (static contact angle) does not affect the dynamic contact angle in the spreading phase up to the local maximum (maximum spread). No significant difference is seen in the early recoiling stage, because the inertia and viscous forces dominate the deformation. The effect of the static contact angle is observed only in the later recoiling stage (tu/D > 10). Surface tension then dominates, and the droplet slowly approaches the equilibrium state, where the contact angle reaches its static value (13–17 on glass or 90–97 on wax). The influence of the impact velocity (expressed here by the Weber number) on the dynamic contact angle of a glycerin droplet on glass is shown in Fig. 6. At lower impact velocities (We = 51) the dynamic contact angle reaches a minimum of about 138, at an earlier dimensionless than at larger impact Weber numbers (We = 802), where the minimum was about 125. The maximum of the dynamic contact angle for the impact Weber number of 51 occurs earlier at tu/D = 0.69 and
We = 93 Glycerin-Wax
2.4 d/D
θD
1.6
v 0.8
0 0.01
0.1
1
(b)
10
tu/D 3.2
d/D, θ D (rad), v x 10-1 (m/s)
(a)
2.4
1.6
3.2
We = 802 Glycerin-Wax
static contact angle
d/D, θ D (rad), v x 10-1 (m/s)
3.2
static contact angle
798
We = 93 Glycerin-Glass
2.4 d/D 1.6
θD v
0.8
static contact angle 0 0.01
0.1
1
10
tu/D
Fig. 3. Dynamic contact angle, spread factor and contact line velocity of a glycerin droplet on (a) smooth wax and (b) smooth glass; D = 2.45 mm, We = 93.
has the value of about 156, while for the Weber number of 802 the maximum of about 146 appears at tu/D = 1.5. In the later spreading phase, the dynamic contact angle decreases faster for a larger impact Weber number. 3.2. Impact onto an inclined surface When a droplet impacts onto an inclined surface, it spreads asymmetrically relative to the point of impact, as Fig. 7 illustrates [13]. The front edge (xfront) spreads below the point of impact faster than the back edge (xback) above the impact point. Note that both the values xfront and xback are defined as positive moving away from the point of impact. The dynamic contact angle of a water droplet low viscous liquid) impacting onto a horizontal surface is not easy to measure by the low power optics, used in this experimental study. The spreading lamella of a water droplet is very thin. Thus the radius of the lamella rim at the front, near the contact line is small and does not allow precise determination of the tangent to the lamellaÕs curved surface at the contact point. The analysis of a water droplet impacting onto an inclined surface offers the advantage of higher accuracy. The front and back sides of the droplet are thicker and it is easier locate the contact point and accurately estimate a tangent
Sˇ. Sˇikalo et al. / Experimental Thermal and Fluid Science 29 (2005) 795–802 180
We = 51 Glycerin-Wax
Glycerin-Glass 150
2.4 120
d/D
θ D (deg)
θD
1.6
static contact angle
d/D, θ D (rad), v x 10-1 (m/s)
3.2
v 0.8
0.01
(a)
0.1
1
90
We=51 We=93 We=802
60 30
0 10
tu/D
0 0.01
static contact angle 0.1
1
10
tu/D
3.2
d/D, θ D (rad), v x 10-1 (m/s)
799
Fig. 6. Comparison of dynamic contact angles of a glycerin droplet on smooth glass for We = 802, We = 93 and We = 51.
We = 51 Glycerin-Glass
2.4 d/D
θD
1.6
v 0.8
static contact angle 0 0.01
0.1
1
(b)
10
tu/D
Fig. 4. Dynamic contact angle (hD), spread factor (d/D) and contact line velocity (v) of a glycerin droplet on (a) smooth wax and (b) smooth glass; D = 2.45 mm, We = 51.
180
We = 802 Glycerin
150
θ D (deg)
120 90 60
Fig. 7. Side view of a liquid droplet on an inclined surface.
Wax
wax
Glass
θa θr static contact angle glass
30
(a)
0 0.01
0.1
1
10
100
1000
tu/D
180
We = 93 Glycerin
150
θ D (deg)
120 90
W ax Glass
wax
60 static contact angle
30
glass
0 0.01
(b)
0.1
1
10
tu/D
Fig. 5. Comparison of dynamic contact angles of a glycerin droplet on the smooth glass and smooth wax; (a) We = 802 and (b) We = 93.
to the curved droplet profile, as illustrated in Fig. 8 (see [14] for more details). Fig. 9a and b shows the measured contact angles and spread factors for a water droplet (We = 90) impacting onto a smooth glass and wax surfaces with 45 inclination. The decrease in the front contact angle corresponds to the increase in the front spread factor for both wax and glass surfaces. The contact angle of the dropletÕs rear contact line decreases to a minimum (about 46) on wax, which is reached considerably later (about tu/D = 4.7) than the time at which the rear contact line begins to slide (about tu/D = 0.85) on wax. After reaching the minimum, the back contact angle increases, approaching the static contact angle (95–105) as the droplet slows to rest. The contact angle on glass decreases, reaching its equilibrium state (6–10). The back contact angle decreases faster and may be lower than the static receding contact angle, as Fig. 9b shows. Fig. 10 shows the effect of the substrate surface on the dynamic contact angle of a water droplet. The difference between dynamic contact angles of a water droplet on
Sˇ. Sˇikalo et al. / Experimental Thermal and Fluid Science 29 (2005) 795–802
800
180
We=90, α =45°
150
θ Front, wax
90
θ Back, wax
60
θ Front, glass θ
static contact angle
θ D (deg)
120
Back, glass
30 0 0.01
Fig. 8. Definition of the contact angles between a water droplet and an inclined surface, where hFront and hBack are the dynamic contact angles in the front and back directions, respectively; hFront = 132, hBack = 46; a = 45, D = 2.7 mm, We = 90, time after impact is t = 8.08 ms.
1.6
We=90, α =45° Water-Wax x/D Front
x/DBack
θ Front
θ Back
vFront
vBack
static contact angle
2.4
-1
x/D, θ D (rad), v x 10 (m/s)
3.2
0.8
0
-0.8 0.01
0.1
(a)
1
10
0.1
1 tu/D
10
100
Fig. 10. Dynamic contact angles of a water droplet on the smooth wax and glass (a = 45, D = 2.72 mm, We = 90).
3.3. Dynamic contact angle as a function of capillary number The dynamic contact angles in terms of capillary number (Ca = vl/r) for the glycerin and water droplet are shown in Figs. 11 and 12. Comparisons with some existing correlations [15–17] are given. The data for glycerin in Fig. 11 show a rapid change in slope. There is a region where the dynamic contact angle falls as Ca (velocity of contact line) increases. The positive slope of hD with Ca strongly decreases at about Ca = 0.2 (v = 1.1 m/s). The slope increases at larger capillary numbers above about 10 (v about 5.5 m/s).
100
3.2
tu/D
Glycerin-Glass
1.6
We=90, α =45° Water-Glass
x/D
Front
x/D
Back
2.4
θ D , rad
2.4
-1
x/D, θ D (rad), v x 10 (m/s)
3.2
θ Front θ Back
v Front
We = 802 Jiang (1979) 0.8
static contact angle
Bracke (1989) Seebergh (1992)
v Back 0.8
(a) 0 0.01
(b)
We = 51 We = 93
1.6
0.1
1 tu/D
10
0 0.01
0.1
1 Ca
10
100
100
3.2 Glycerin-Wax
Fig. 9. Dynamic contact angle, spread factor and contact line velocity of a water droplet on inclined (a) smooth wax and (b) smooth glass; a = 45, D = 2.72 mm, We = 90.
θ D , rad
2.4
glass and wax is not observed in the first stage of spreading (tu/D = 0.1), where the inertial forces are strong. The dynamic contact angle decreases faster on glass than on wax when approaching the static contact angle. The back contact angle on glass decreases slightly below the static receding contact angle and then approaches the static value (6–10). Due to the strong elongation, the water droplet can rupture on the glass surface.
1.6
We = 51 We = 93 We = 802 Jiang (1979)
0.8
(b)
0 0.001
0.01
0.1
1
static contact angle
10
100
Ca
Fig. 11. Data of the dynamic contact angle hD versus Ca for glycerin droplet on horizontal: (a) smooth glass and (b) wax.
Sˇ. Sˇikalo et al. / Experimental Thermal and Fluid Science 29 (2005) 795–802 3.2
801
We = 90, α =45° Water Back-wax wax Front-wax Back-glass Front-glass Jiang (1979), glass Jiang (1979), wax
1.6
0.8
0 -0.05
static contant angle
θ D , rad
2.4
glass 0.05 Ca
0.15
Fig. 12. Data of the dynamic contact angles for water droplet on inclined smooth glass and wax, a = 45.
A faster increase of slope is observed at the smaller impact Weber number. This region belongs to the first spreading phase, in which an increase of the impact Weber number increases impact pressure and a thin film of air may be entrapped, which can affect the dynamic contact angle. At lower capillary numbers, Ca < 0.2, an increase of the impact Weber number decreases the slope of the data. This region corresponds to the later spreading phase. A decrease of the impact Weber number increases the thickness of the lamella and the radius of curvature of the interface at the contact line, which may also affect the dynamic contact angle. The irregular behavior of the data has been noted previously by Blake [6] for a tape submerged vertically into a pool of water (see [1, Figs. 3.17 and 3.20, pp. 82–84]). BlakeÕs results exhibit the changes in the slope of the data at much lower values of Ca. The data exhibit a negative slope at Ca above about 2.7 · 105 (v 0.002 m/s). The slope is again positive at Ca above about 0.007 (v 0.5 m/s). Differences between the experiments in material properties and geometry imply a difference in Ca and velocities at which the change in slope of the dynamic contact angle with the capillary number (contact line velocity) occurs. The data for the leading (front) and trailing (back) contact line also show differences, as Fig. 12 illustrates. However, for a water droplet at Ca > 0.075 (v > 5.5 m/s) the difference in the data for the dynamic contact angle of the trailing and the leading edge disappears. Also the influence of the static contact angle is obvious from a comparison of the data for wax and glass, as Fig. 12 shows. Data for dynamic contact angles of a water droplet on wax and glass are in agreement at capillary numbers above about 0.047 (v > 3.3 m/s). The empirical correlations do not agree well with experimental data as Figs. 11a and 12a illustrate. The correlation by Jiang [15] approaches 180 faster than the present experimental results do.
Fig. 13. The dynamic wetting failure at the initial phase of impact of a water droplet onto wax (side view, D = 2.7 mm, We = 90). The arrow indicates the dry area (black horizontal line).
In the initial phase of impact, the velocity of the contact line is large enough to entrap air between the liquid and solid, as Fig. 13 shows (the image has been enhanced). The liquid droplet contacts the surface with a ring-shape in the initial phase of the impact. The dry area, inside this ring-shaped liquid/solid contact area, decreases sharply with time and then the entrapped air forms an air bubble inside the droplet, which can be often observed for a droplet impact onto solid surfaces. The bubble stays on a low wettable surface while it leaves a high wettable surface. Fujimoto et al. [18] studied the time evolution of the circular dry area on the surface. They found that the initial dry area depends on the impact Weber number. At the instant of contact, the tangent to the droplet profile defines a wetting angle of 180. As the spreading velocity decreases, the air layer collapses and adhesion between liquid and the solid surface occurs. The dynamic contact angle decreases below 180. During adhesion the wetting angle seems to decrease to a minimum. Dynamic wetting failure occurs in a first phase of the contact. High liquid pressure near the contact line, at the initial phase of impact, may influence the contact line behavior and a decrease of the contact angle. After lamella ejection the pressure decreases (that can be numerically considered) and the contact angle increases. Blake et al. [19,10] have shown that the flow field can influence the onset of air entrainment and also the value of the dynamic contact angle. This implies that an empirical correlation can be applied only to particular classes of self-similar flows near the contact line [10]. Therefore, it is still necessary to conduct experimental and theoretical investigations to develop a more general model for the dynamic contact angle and its dependence on the parameters characterizing the flow and the material properties of the contacting media.
4. Conclusion There are two main conclusions from this study. First, the dynamic contact angle is not only a function of the contact line speed but also a function of the flow field in the vicinity of the moving contact line, as
802
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observed by Blake et al. [10]. As discussed above, the contact angle first decreases to a local minimum value and then increases to a local maximum value in the spreading phase, while the spreading factor increases monotonically and the contact line velocity decreases monotonically (see Fig. 2a). In the later recoiling phase the contact angle decreases to another minimum and again increases to reach the equilibrium state on the wax, while the spread factor decreases monotonically and thus the contact line (receding) velocity decreases until it becomes negative (to about 0.01 m/s). In the recoiling phase the dynamic contact angle decreases from the maximum to the final state on the glass surface, while the spread factor changes slowly and the contact line velocity is very small (<0.01 m/s). Therefore, a unique functional dependence of the dynamic contact angle on the contact line speed cannot be obtained to model the experimental results. Second, there is no effect of the static contact angle on the dynamic contact angle over a large range of experimental data for a glycerin droplet. This is observed in the spreading stage where the inertial and viscous forces dominate the deformation. In contrast, the experimental results for a water droplet on inclined glass and wax show that the static contact angle has an important influence on the dynamic contact angle (see Fig. 10). As noticed from the experimental data, a fixed contact angle (advancing or receding) as one of the boundary conditions commonly assumed in numerical calculation has obvious limitations. The dynamic contact angle, as a function of the flow variables, should be used as a boundary condition to obtain better quantitative agreement of numerical prediction with experimental data. The description of this boundary condition obviously deserves further theoretical and experimental studies.
References [1] D.T. Blake, J.K. Ruschak, Wetting: static and dynamic contact lines, in: F.S. Kistler, M.P. Schweizer (Eds.), Liquid Film Coating, Chapman & Hall, London, 1997, pp. 63–97.
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