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Liquid drop spreading on solid surfaces at low impact speeds
Colloids and Surfaces A: Physicochemical and Engineering Aspects 163 (2000) 239 – 245 www.elsevier.nl/locate/colsurfa
Liquid drop spreading on solid surfaces at low impact speeds Yongan Gu, Dongqing Li * Department of Mechanical Engineering, Uni6ersity of Alberta, Edmonton, Alberta, T6G 2G8, Canada Received 5 March 1999; accepted 11 June 1999
Abstract In this paper, a previously developed drop spreading model is applied to studying the spreading kinetics of water drops with low impact speeds on an anodized aluminium surface and a glass plate. A set of measured spreading data is used to verify the model predictions for low-speed impact spreading cases, in which the impact speed was adjusted by varying the drop release height. It is found that, for relatively lower impact speeds up to U= 250 cm s − 1, the spreading model can simulate well, the spreading of the water drops on the flat metal surface and the glass plate. At higher impact speeds, the simple spreading model overestimates the spreading rates. Some possible reasons for the overestimation are discussed. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Drop spreading; Low-speed impact spreading; Moving contact line; Overall energy balance (OEB) method
1. Introduction The spreading process of a liquid drop on a solid surface is frequently encountered in many industrial applications, such as spray-cooling and metal forming in material processing operations, spray-forming, plasma coating, pesticide spraying and ink jet printing. A fundamental understanding of the spreading mechanism and the controlling factors is essential to these industrial processes. Generally, the liquid drop spreading on a horizontal solid surface can be classified into high-speed impact spreading and low-speed im* Corresponding author. Tel.: +1-780-4929734; fax: +1780-4922200. E-mail address: [email protected] (D. Li)
pact spreading. The spontaneous spreading occurs when the impact speed is equal to zero. In a high-speed impact spreading, the impact dynamics is important to the spreading kinetics of the liquid drops on the solid surfaces [1,2]. For a low-speed impact spreading, the spreading kinetics strongly depends on the inertial, viscous, gravitational, capillary forces, and the wettability of the spreading system. The liquid drop spreading phenomenon is an extremely complicated subject since it is related to both fluid mechanics and surface physics. In the past two decades, it has been proved to be unsatisfactory if this topic is studied either as a fluid dynamics problem or as a surface physics problem alone. In the classical fluid mechanics approach
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Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245
[3,4], the major difficulty arises in modelling the motion of the solid – liquid-fluid contact line. Usually, a slip boundary condition has to be imposed in the vicinity of the moving three-phase contact line. In addition, most fluid dynamics models either did not fully take account of the capillary effect [5] or did not have a proper model for determining the viscous drag force at the solid– liquid interface [6]. Hence, these models are applicable only to the impacting stage and the initial stage of drop spreading when the inertial effect is dominant. On the other hand, some researchers treated the spreading phenomenon as a pure surface physics problem [7,8]. However, the surface physics approach alone did not yield any predictive models mainly because the motion of the three-phase contact line is too complex to be modelled by this approach. Thus analytical solutions for the spreading kinetics are applicable only to the final stage of drop spreading when the inertial force is negligible in comparison with the capillary force and the viscous drag force. Recently, the authors developed a relatively simple theoretical spreading model that combines the fluid mechanics and the surface physics approaches together and contains minimum empiricism [9]. Based on the de Gennes’ model for the viscous drag force of a moving contact line, this new spreading model uses the overall energy balance (OEB) method to account for the effects of the inertial force at low impact speeds, viscous and gravitational forces, the interfacial tensions and the wettability of the solid – liquid-fluid system. It was shown that the spreading kinetics predicted by this model is in an excellent accordance with the experimental data of spontaneous spreading of silicone oil drops on a soda-lime glass plate [10]. The spontaneous spreading is an ideal spreading case. Most practical applications in industrial coating processes involve relatively low-speed impact spreading. Thus, a question that arises naturally is whether the previously developed spreading model is applicable to these low-speed impact spreading cases. Therefore, the purpose of the present paper is to compare the numerical predictions of the theoretical spreading model with the existing experimental data of water drop
spreading on an anodized aluminium surface and on a glass plate measured by Thoroddsen and Sakakibara [11] at different low impact speeds. In their measurements, they obtained the impact speeds in the range from U= 100 cm s − 1 to U= 500 cm s − 1 by adjusting the drop release height from H= 6 cm to H=124 cm. The comparison shows that the theoretical spreading model can also predict well the spreading processes with the lower impact speeds. But at the higher impact speeds, the spreading model always overestimates the inertial effect on the spreading process and thus the spreading rates. This can be attributed to some limitations of the simple theoretical spreading model, as discussed in a later section.
2. Overall energy balance (OEB) equation When a small liquid drop is deposited onto a smooth solid surface at low impact speeds, its spreading process will primarily depend on the inertial, viscous and gravitational forces, the interfacial tensions and the wettability of the spreading system. In the previous paper [9], an overall energy balance (OEB) method was employed to derive the theoretical spreading model. Madejski [12] was probably the first to use the OEB method in modeling the spreading motion of the liquid drop. Three components were considered in the first OEB equation: the kinetic energy, Ek(t); the surface energy, Ep(t); and the viscous dissipation work, Lf(t). Because gravity is always one of the driving forces for the spreading motion, the gravitational potential energy of the drop, Eg(t), was also included in the previous OEB equation for completeness. The magnitude of this potential merely depends on the size of liquid drops. Therefore, the spreading motion of the liquid drop is governed by the following OEB equation: d dL (t) [Ek(t)+ Ep(t)+ Eg(t)]+ f = 0, dt dt
(1)
where, dE(t)/dt stands for the change rate of each energy term and dLf(t)/dt represents the power of viscous dissipation. Unlike Madejski’s formulations for the three terms, Ek(t), Ep(t) and Lf(t),
Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245
each term in Eq. (1) was formulated differently in the previous spreading model [9]. In this paper, only some important assumptions used in deriving the theoretical spreading model will be reviewed as follows. Further discussion on some assumptions will be presented in the subsequent section. First, the liquid drop was assumed to be nonvolatile so that its volume during spreading remains constant. As will be seen later, this assumption will not cause an appreciable error since the total observation duration for water drops to spread on a solid surface is around a few milliseconds. Secondly, for mathematical simplicity, prior to the impact, the liquid drop was assumed to be a perfect sphere. During the spreading motion, the liquid drop was assumed to spread axisymmetrically on the horizontal, smooth and homogeneous solid surface, though evolution of the fingering pattern at the edge of an impacting drop was observed during its impact on a glass plate [11]. In addition, it is considered that no splashing occurs during the drop impacting on the surface, as confirmed experimentally for all the impact speeds tested [11]. Thirdly, without dealing with the impact dynamics, the impact kinetic energy of a liquid drop was assumed to be 100% transformed into its spreading kinetic energy after it impacts on a solid surface. Since the air drag force experienced by a falling water drop is much smaller than its gravity, its impact kinetic energy before it impacts on the solid substrate is equal to: Ei =1/2mwU 2 = mwgH where, g is the gravitational acceleration; mw = 4/ 3pR 3rw is the mass of the water drop; R is the radius of the spherical water drop and rw is the density of water. As will be shown later, this ideal energy transformation can cause greater error and eventually becomes unacceptable for higher impact speeds. Based on the above assumptions, the total surface energy term, Ep(t), can be calculated once the drop shape at any time is determined by numerically solving the classical Laplace equation of capillarity [9]. Furthermore, using a mass center approach [9], the total kinetic energy of the spreading drop, Ek(t), and its gravitational potential energy, Eg(t), during the drop spreading, can be determined accordingly. It should be empha-
241
sized that by means of the mass center approach, there is no need to obtain the complete information about the velocity field or to assume simple velocity distributions that should satisfy both the continuity equation and the proper boundary conditions [12,13]. In order to determine the viscous dissipation work, Lf(t), the slip boundary condition was applied in the vicinity of the moving three-phase contact line. In the theoretical spreading model, de Gennes’ model [13,14] was employed to calculate the dissipation work caused by the viscous drag force at the solid–liquid interface. Brochard–Wyart and de Gennes [13] used the fluid hydrodynamics approach to study the motion of a three-phase contact line. Their approach concentrates on the viscous losses inside the assumed flat liquid wedge of a small angle, where the so-called lubrication theory approximation is applicable and the molecular dissipation at the tip can be neglected [14]. They derived an analytical expression for the viscous drag force of per unit length of the three-phase contact line acting on the solid surface [13], which, for the water drop spreading on a solid surface studied in the present case, can be written as: 3mw dr(t) 1 (2) FV(t)= ln(o − d ), ud(t) dt where, mw is the viscosity of water; ud(t) is the dynamic contact angle; r(t) is the contact radius of the water drop on the solid surface and thus dr(t)/dt represents the spreading speed. In the above equation, od = Ld /L is a to-be-determined ratio of the microscopic scale cut-off length (Ld ), below which the continuum theory breaks down, to the macroscopic scale cut-off length (L), which is proportional to the horizontal characteristic length of the water drop. It is worthwhile emphasizing that the above expression for the viscous drag force is entirely based on the lubrication theory approximation [13,14] and that the viscous effect at a distance smaller than Ld from the contact line is completely ignored. Using the de Gennes’ model, the total viscous drag force of the circular contact line exerting on the solid substrate is equal to 2pr(t)Fv(t). Consequently, the dissipation work of the total viscous drag force can be calculated:
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Lf(t)=
&
Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245 r(t)
2pr(t)FV(t)dr(t)
0 1 =6pmw ln(o − d )
&
t
0
n
r(t) dr(t) ud(t) dt
2
dt
(3)
Eq. (3) involves the to-be-determined parameter, od. In the numerical simulations of the drop spreading process, od will be determined by finding the best fit of the theoretical spreading kinetics to one set of the measured spreading data. This parameter is anticipated to be approximately constant for a given solid – liquid-fluid spreading system. Integration of Eq. (1) with respect to time from t= 0 to any time t yields another format of the OEB equation: DEk(t)+DEp(t) + DEg(t) + DLf(t) = 0
(4)
Eq. (4) clearly indicates that there is always an overall energy balance among variations of the three energy components and the viscous dissipation work term at any moment. The detailed OEB equation was derived previously [9], which, for the present water drop spreading system, is given by: 1 m 2 w
!
dxm(t) dt
n 2
+
dym(t) dt
n
2
−U 2
"
+ gwv[Awv(t)− 4p R 2 −p r 2(t)cos ue] 1 +mg[ym(t)− R]+ 6pmw ln(o − d )
&
t
0
= 0,
n
r(t) dr(t) ud(t) dt
2
dt (5)
where, xm(t) and ym(t) are the coordinates of the mass centre point of the half cross-sectional area of the spreading water drop in a chosen cylindrical coordinate system; U is the impact speed; gwv and Awv are the surface tension and the interfacial surface area of the water – vapour interface; and ue is the so-called equilibrium contact angle and equal to the dynamic contact angle at the end of the spreading process. This contact angle represents the wettability of the solid – water–vapour system. Eq. (5) is a general OEB equation for modelling the spreading of a liquid drop on a horizontal solid surface. It can be solved numerically by using the finite difference method (FDM) and the iteration method simultaneously. The detailed numerical procedure for solving Eq. (5) and
determining the parameter od was described previously [9]. With the OEB equation, a thorough parametric study can be carried out to study the effects of the surface and physical properties of the solid–liquid-fluid system, the impact speed as well as the size of the liquid drop on a specific spreading process.
3. Results and discussion Recently, Thoroddsen and Sakakibara [11] studied the spreading phenomena of an impacting liquid drop on a flat metal surface and on a glass plate, respectively. Experimentally, they obtained accurate spreading data for a number of different drop release heights, H, and hence different impact speeds, U= 2gh. These experimental spreading data will be used to compare with the numerical predictions of the theoretical spreading model given by Eq. (5). The liquid tested was the distilled water. For imaging purpose, however, fluorescent dye was added to the water. The addition of the fluorescent dye slightly lowered the surface tension of the distilled water. In their experiments, weight measurements were used to determine the drop size, giving a size of around D:0.50 cm. No splashing was observed in these spreading experiments. Thoroddsen and Sakakibara’s experimental study was composed of two parts. First, the spreading rates were measured for a water drop impacting on a flat metal surface. A distilled water drop (D=2R= 0.55 cm) with very weak Fluorescein concentration (6 mg l − 1) was used, giving slightly lower surface tension (gwv =60 dyne cm − 1) measured with a ring tensiometer. The solid substrate was an anodized aluminium surface. In this part of the study, the detailed data of spreading rates and the vertical heights of water drops were measured for nine different drop release heights, H= 6, 10, 15, 22, 31, 40, 60, 80 and 124 cm (their Figs. 5 and 10). The corresponding impact speeds changed approximately from 100 to 500 cm s − 1. In the present study, only the spreading data for the following five drop release heights, H=6, 15, 31, 60 and 124 cm, are chosen in comparison with the theoretical
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spreading results for graphic consideration. In the second part of their experimental study, a glass plate was used as the solid substrate because it allows for observations through the substrate. A distilled water drop (D =2R = 0.52 cm) with very high Fluorescein concentration (1 g l − 1) was used in order to obtain the best imaging results. This lowered the surface tension of the distilled water to about gwv =50 dyne cm − 1. The detailed spreading rates were available only for one drop release height, H =50 cm (their Fig. 11), corresponding to the impact speed, U :313 cm s − 1. However, the true values of some other quantities such as the viscosity and density of water, mw and rw, the equilibrium contact angle, ue, required in Eq. (5) were not given in Ref. [11]. Here, in the numerical simulations, mw and rw are chosen as 0.01 P and 0.998 g cm − 3, respectively, i.e. the values of pure distilled water. The effects of the fluorescent dye on the viscosity and density of the water drop are neglected. The equilibrium contact angle of the water drops on the high-energy solid surfaces such as the aluminium surface and the glass plate is considered to be zero, ue =0°, that is, the perfect wetting of the water drops on these two solid substrates occurs eventually. Fig. 1 shows an overall comparison between the numerical spreading rates predicted by Eq. (5) 1 with ln(o − d ) = 9.20 and the measured spreading data of water drops on the aluminium surface for five different drop release heights, H = 6, 15, 31, 60 and 124 cm. The comparison for the drop height under the same testing conditions is plotted in Fig. 2. It can be seen from these two figures 1 that, by choosing ln(o − d ) to be 9.20, the numerical predictions give the overall best fit to the measured spreading data for all the drop release heights up to H =31 cm. Thus, the value of the parameter od =Ld /L used in the numerical simulations is about 1.0 × 10 − 4. This value corresponds to the microscopic scale cut-off length, Ld : 1 mm, if the maximum contact radius, r(t), i.e. the horizontal characteristic length of the water drops, is chosen as the macroscopic scale cut-off length and approximated as 1.0 cm, i.e. L :1.0 cm. This Ld value is close to Ld :2 mm obtained by the authors [9] for the experimental data of silicone oil drop spreading on a soda-lime glass plate in air
243
measured by Chen [10], and Ld : 5 mm published by Basu et al. [15] for glass–bitumen–water systems as well. On the one hand, physically, it is expected that the microscopic scale cut-off length, Ld, may depend on a specific spreading system, mainly reflecting the characteristics of the viscous drag force at the solid–liquid interface. On the other hand, it is unexpected to find out that the values of Ld for these three very different spreading systems [10,11,15] are so close. In fact, these three spreading systems differ from each other significantly. Their liquid viscosities are different by more than four orders of magnitude, the liquid drop sizes are different by one order of magnitude, and their impact speeds as well as the ranges of the spreading time are also rather different. Figs. 1 and 2 also indicate the strong effect of the impact speed on the spreading process. It is noted that, when H]60 cm, the theoretical spreading model always overestimates the spread-
Fig. 1. Spreading rates, r(t), versus time for a water drop (D= 0.55 cm, gwv =60 dyne cm − 1) impacting on an anodized aluminium surface for five different drop release heights, H (cm) =6 ( ); 15 (); 31 ( ); 60 (2); 124 (filled hexagon). The symbols represent the experimental spreading data measured by Thoroddsen and Sakakibara [11] whereas the solid lines represent the corresponding numerical spreading results 1 calculated by numerically solving Eq. (5) with ln(o − d ) =9.20 for the five drop release heights.
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4. Summary
Fig. 2. Heights of a water drop, h(t), versus time for five different drop release heights. The other captions are the same as those given in Fig. 1.
In conjunction with the previous paper [9], it has been shown that the previously developed theoretical spreading model is applicable not only to the spontaneous spreading processes but also to relatively low-speed impact spreading processes. It is also noted that this simple spreading model can not give excellent predictions for some relatively high-speed impact spreading cases. This may be remedied by incorporating the following improvements into the spreading model. First of all, the impact dynamics should be studied in order to understand the energy transformation process during actual impacting process. The ideal energy transformation always overestimates the spreading kinetic energy and thus leads to larger spreading rates. Secondly, the de Gennes’ model assumes that the dominant losses are due to hydrodynamic shear flows in the liquid wedge and that the molecular dissipation at the wetting line
ing rates. This overestimation may be caused by the following two major assumptions used in deriving the theoretical spreading model. First, in reality, a portion of the impact kinetic energy will be dissipated or transformed into other energy forms during impacting. The ideal energy transformation without any energy dissipation, i.e. the impact kinetic energy of a liquid drop is assumed to be 100% transformed into its spreading kinetic energy after impacting, will significantly overestimate the spreading kinetic energy and thus spreading rates. The higher the liquid drop is released from, the more serious error this assumption will cause. Second, at present, it is not quite clear that, during these relatively high-speed impact spreading processes, whether the drop shape at any moment is still completely governed by the classical Laplace equation of capillarity. Fig. 3 shows a comparison between the numerical spreading rates and the measured spreading data of a water drop on a glass plate for the drop release height, H=50 cm. In this case, as shown 1 in Fig. 3, by choosing ln(o − d ) : 9.20 in Eq. (5), the numerical simulation again gives the best fit to the measured spreading rates. Correspondingly, od =Ld /L: 1.0 ×10 − 4 or Ld :1 mm if L :1 cm.
Fig. 3. Spreading rates, r(t), versus time for a water drop (D= 0.52 cm, gwv =50 dyne cm − 1) impacting on a glass plate for one drop release height, H =50 cm. The open circle symbols represent the experimental spreading rates measured by Thoroddsen and Sakakibara [11] whereas the solid line represents the corresponding numerical spreading rates calcu1 lated by numerically solving Eq. (5) with ln(o − d ) =9.20 for the same drop release height.
Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245
is negligible. Therefore, this model should be modified properly or at best a completely new viscous drag force model of a moving contact line has to be developed for high-speed impact spreading processes. For relatively high-speed impact spreading cases, it is anticipated that the molecular dissipation at the moving contact line is important [13].
Acknowledgements The authors wish to acknowledge the financial supports of the Izaak Walton Killam Memorial Scholarship at the University of Alberta to Y. Gu and the research grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada to D. Li.
References [1] H. Liu, E.J. Lavernia, R.H. Rangel, Appl. Phys. 26 (1993) 1900.
.
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[2] J. Fukai, Y. Shiiba, T. Yamamoto, O. Miyatake, D. Poulikakos, C.M. Megaridis, Z. Zhao, Phys. Fluids 7 (1995) 236. [3] P. Ehrhard, S.H. Davis, J. Fluid Mech. 229 (1991) 365. [4] L.M. Hocking, J. Fluid Mech. 239 (1992) 671. [5] F.H. Harlow, J.P. Shannon, J. Appl. Phys. 38 (1967) 3855. [6] M. Pasandideh-Fard, Y.M. Qiao, S. Chandra, J. Mostaghimi, Phys. Fluids 8 (1996) 650. [7] F.T. Dodge, J. Colloid Interf. Sci. 121 (1988) 154. [8] J.C. Ambrose, M.G. Nicholas, A.M. Stoneham, Acta. Metal. Mater. 41 (1993) 2395. [9] Y. Gu, D. Li, A model for a liquid drop spreading on a solid surface, Colloids Surf. A 142 (1998) 243. [10] J.D. Chen, J. Colloid Interf. Sci. 122 (1988) 60. [11] S.T. Thoroddsen, J. Sakakibara, Phys. Fluids 10 (1998) 1359. [12] J. Madejski, Int. J. Heat Mass Transfer. 19 (1976) 1009. [13] F. Brochard-Wyart, P.G. de Gennes, Adv. Colloid Interf. Sci. 39 (1992) 1. [14] P.G. de Gennes, in: J. Meunier, D. Langevin, N. Boccara (Eds.), Physics of Amphiphilic Layers, vol. 21, Springer, New York, 1987, p. 64. [15] S. Basu, K. Nandakumar, J.H. Masliyah, J. Colloid Interf. Sci. 182 (1996) 82 Note that the value of od in this paper was corrected by Basu et al. to be 0.005 from personal communications (1997).
Liquid drop spreading on solid surfaces at low impact speeds Yongan Gu, Dongqing Li * Department of Mechanical Engineering, Uni6ersity of Alberta, Edmonton, Alberta, T6G 2G8, Canada Received 5 March 1999; accepted 11 June 1999
Abstract In this paper, a previously developed drop spreading model is applied to studying the spreading kinetics of water drops with low impact speeds on an anodized aluminium surface and a glass plate. A set of measured spreading data is used to verify the model predictions for low-speed impact spreading cases, in which the impact speed was adjusted by varying the drop release height. It is found that, for relatively lower impact speeds up to U= 250 cm s − 1, the spreading model can simulate well, the spreading of the water drops on the flat metal surface and the glass plate. At higher impact speeds, the simple spreading model overestimates the spreading rates. Some possible reasons for the overestimation are discussed. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Drop spreading; Low-speed impact spreading; Moving contact line; Overall energy balance (OEB) method
1. Introduction The spreading process of a liquid drop on a solid surface is frequently encountered in many industrial applications, such as spray-cooling and metal forming in material processing operations, spray-forming, plasma coating, pesticide spraying and ink jet printing. A fundamental understanding of the spreading mechanism and the controlling factors is essential to these industrial processes. Generally, the liquid drop spreading on a horizontal solid surface can be classified into high-speed impact spreading and low-speed im* Corresponding author. Tel.: +1-780-4929734; fax: +1780-4922200. E-mail address: [email protected] (D. Li)
pact spreading. The spontaneous spreading occurs when the impact speed is equal to zero. In a high-speed impact spreading, the impact dynamics is important to the spreading kinetics of the liquid drops on the solid surfaces [1,2]. For a low-speed impact spreading, the spreading kinetics strongly depends on the inertial, viscous, gravitational, capillary forces, and the wettability of the spreading system. The liquid drop spreading phenomenon is an extremely complicated subject since it is related to both fluid mechanics and surface physics. In the past two decades, it has been proved to be unsatisfactory if this topic is studied either as a fluid dynamics problem or as a surface physics problem alone. In the classical fluid mechanics approach
0927-7757/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 9 9 ) 0 0 2 9 5 - 2
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Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245
[3,4], the major difficulty arises in modelling the motion of the solid – liquid-fluid contact line. Usually, a slip boundary condition has to be imposed in the vicinity of the moving three-phase contact line. In addition, most fluid dynamics models either did not fully take account of the capillary effect [5] or did not have a proper model for determining the viscous drag force at the solid– liquid interface [6]. Hence, these models are applicable only to the impacting stage and the initial stage of drop spreading when the inertial effect is dominant. On the other hand, some researchers treated the spreading phenomenon as a pure surface physics problem [7,8]. However, the surface physics approach alone did not yield any predictive models mainly because the motion of the three-phase contact line is too complex to be modelled by this approach. Thus analytical solutions for the spreading kinetics are applicable only to the final stage of drop spreading when the inertial force is negligible in comparison with the capillary force and the viscous drag force. Recently, the authors developed a relatively simple theoretical spreading model that combines the fluid mechanics and the surface physics approaches together and contains minimum empiricism [9]. Based on the de Gennes’ model for the viscous drag force of a moving contact line, this new spreading model uses the overall energy balance (OEB) method to account for the effects of the inertial force at low impact speeds, viscous and gravitational forces, the interfacial tensions and the wettability of the solid – liquid-fluid system. It was shown that the spreading kinetics predicted by this model is in an excellent accordance with the experimental data of spontaneous spreading of silicone oil drops on a soda-lime glass plate [10]. The spontaneous spreading is an ideal spreading case. Most practical applications in industrial coating processes involve relatively low-speed impact spreading. Thus, a question that arises naturally is whether the previously developed spreading model is applicable to these low-speed impact spreading cases. Therefore, the purpose of the present paper is to compare the numerical predictions of the theoretical spreading model with the existing experimental data of water drop
spreading on an anodized aluminium surface and on a glass plate measured by Thoroddsen and Sakakibara [11] at different low impact speeds. In their measurements, they obtained the impact speeds in the range from U= 100 cm s − 1 to U= 500 cm s − 1 by adjusting the drop release height from H= 6 cm to H=124 cm. The comparison shows that the theoretical spreading model can also predict well the spreading processes with the lower impact speeds. But at the higher impact speeds, the spreading model always overestimates the inertial effect on the spreading process and thus the spreading rates. This can be attributed to some limitations of the simple theoretical spreading model, as discussed in a later section.
2. Overall energy balance (OEB) equation When a small liquid drop is deposited onto a smooth solid surface at low impact speeds, its spreading process will primarily depend on the inertial, viscous and gravitational forces, the interfacial tensions and the wettability of the spreading system. In the previous paper [9], an overall energy balance (OEB) method was employed to derive the theoretical spreading model. Madejski [12] was probably the first to use the OEB method in modeling the spreading motion of the liquid drop. Three components were considered in the first OEB equation: the kinetic energy, Ek(t); the surface energy, Ep(t); and the viscous dissipation work, Lf(t). Because gravity is always one of the driving forces for the spreading motion, the gravitational potential energy of the drop, Eg(t), was also included in the previous OEB equation for completeness. The magnitude of this potential merely depends on the size of liquid drops. Therefore, the spreading motion of the liquid drop is governed by the following OEB equation: d dL (t) [Ek(t)+ Ep(t)+ Eg(t)]+ f = 0, dt dt
(1)
where, dE(t)/dt stands for the change rate of each energy term and dLf(t)/dt represents the power of viscous dissipation. Unlike Madejski’s formulations for the three terms, Ek(t), Ep(t) and Lf(t),
Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245
each term in Eq. (1) was formulated differently in the previous spreading model [9]. In this paper, only some important assumptions used in deriving the theoretical spreading model will be reviewed as follows. Further discussion on some assumptions will be presented in the subsequent section. First, the liquid drop was assumed to be nonvolatile so that its volume during spreading remains constant. As will be seen later, this assumption will not cause an appreciable error since the total observation duration for water drops to spread on a solid surface is around a few milliseconds. Secondly, for mathematical simplicity, prior to the impact, the liquid drop was assumed to be a perfect sphere. During the spreading motion, the liquid drop was assumed to spread axisymmetrically on the horizontal, smooth and homogeneous solid surface, though evolution of the fingering pattern at the edge of an impacting drop was observed during its impact on a glass plate [11]. In addition, it is considered that no splashing occurs during the drop impacting on the surface, as confirmed experimentally for all the impact speeds tested [11]. Thirdly, without dealing with the impact dynamics, the impact kinetic energy of a liquid drop was assumed to be 100% transformed into its spreading kinetic energy after it impacts on a solid surface. Since the air drag force experienced by a falling water drop is much smaller than its gravity, its impact kinetic energy before it impacts on the solid substrate is equal to: Ei =1/2mwU 2 = mwgH where, g is the gravitational acceleration; mw = 4/ 3pR 3rw is the mass of the water drop; R is the radius of the spherical water drop and rw is the density of water. As will be shown later, this ideal energy transformation can cause greater error and eventually becomes unacceptable for higher impact speeds. Based on the above assumptions, the total surface energy term, Ep(t), can be calculated once the drop shape at any time is determined by numerically solving the classical Laplace equation of capillarity [9]. Furthermore, using a mass center approach [9], the total kinetic energy of the spreading drop, Ek(t), and its gravitational potential energy, Eg(t), during the drop spreading, can be determined accordingly. It should be empha-
241
sized that by means of the mass center approach, there is no need to obtain the complete information about the velocity field or to assume simple velocity distributions that should satisfy both the continuity equation and the proper boundary conditions [12,13]. In order to determine the viscous dissipation work, Lf(t), the slip boundary condition was applied in the vicinity of the moving three-phase contact line. In the theoretical spreading model, de Gennes’ model [13,14] was employed to calculate the dissipation work caused by the viscous drag force at the solid–liquid interface. Brochard–Wyart and de Gennes [13] used the fluid hydrodynamics approach to study the motion of a three-phase contact line. Their approach concentrates on the viscous losses inside the assumed flat liquid wedge of a small angle, where the so-called lubrication theory approximation is applicable and the molecular dissipation at the tip can be neglected [14]. They derived an analytical expression for the viscous drag force of per unit length of the three-phase contact line acting on the solid surface [13], which, for the water drop spreading on a solid surface studied in the present case, can be written as: 3mw dr(t) 1 (2) FV(t)= ln(o − d ), ud(t) dt where, mw is the viscosity of water; ud(t) is the dynamic contact angle; r(t) is the contact radius of the water drop on the solid surface and thus dr(t)/dt represents the spreading speed. In the above equation, od = Ld /L is a to-be-determined ratio of the microscopic scale cut-off length (Ld ), below which the continuum theory breaks down, to the macroscopic scale cut-off length (L), which is proportional to the horizontal characteristic length of the water drop. It is worthwhile emphasizing that the above expression for the viscous drag force is entirely based on the lubrication theory approximation [13,14] and that the viscous effect at a distance smaller than Ld from the contact line is completely ignored. Using the de Gennes’ model, the total viscous drag force of the circular contact line exerting on the solid substrate is equal to 2pr(t)Fv(t). Consequently, the dissipation work of the total viscous drag force can be calculated:
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Lf(t)=
&
Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245 r(t)
2pr(t)FV(t)dr(t)
0 1 =6pmw ln(o − d )
&
t
0
n
r(t) dr(t) ud(t) dt
2
dt
(3)
Eq. (3) involves the to-be-determined parameter, od. In the numerical simulations of the drop spreading process, od will be determined by finding the best fit of the theoretical spreading kinetics to one set of the measured spreading data. This parameter is anticipated to be approximately constant for a given solid – liquid-fluid spreading system. Integration of Eq. (1) with respect to time from t= 0 to any time t yields another format of the OEB equation: DEk(t)+DEp(t) + DEg(t) + DLf(t) = 0
(4)
Eq. (4) clearly indicates that there is always an overall energy balance among variations of the three energy components and the viscous dissipation work term at any moment. The detailed OEB equation was derived previously [9], which, for the present water drop spreading system, is given by: 1 m 2 w
!
dxm(t) dt
n 2
+
dym(t) dt
n
2
−U 2
"
+ gwv[Awv(t)− 4p R 2 −p r 2(t)cos ue] 1 +mg[ym(t)− R]+ 6pmw ln(o − d )
&
t
0
= 0,
n
r(t) dr(t) ud(t) dt
2
dt (5)
where, xm(t) and ym(t) are the coordinates of the mass centre point of the half cross-sectional area of the spreading water drop in a chosen cylindrical coordinate system; U is the impact speed; gwv and Awv are the surface tension and the interfacial surface area of the water – vapour interface; and ue is the so-called equilibrium contact angle and equal to the dynamic contact angle at the end of the spreading process. This contact angle represents the wettability of the solid – water–vapour system. Eq. (5) is a general OEB equation for modelling the spreading of a liquid drop on a horizontal solid surface. It can be solved numerically by using the finite difference method (FDM) and the iteration method simultaneously. The detailed numerical procedure for solving Eq. (5) and
determining the parameter od was described previously [9]. With the OEB equation, a thorough parametric study can be carried out to study the effects of the surface and physical properties of the solid–liquid-fluid system, the impact speed as well as the size of the liquid drop on a specific spreading process.
3. Results and discussion Recently, Thoroddsen and Sakakibara [11] studied the spreading phenomena of an impacting liquid drop on a flat metal surface and on a glass plate, respectively. Experimentally, they obtained accurate spreading data for a number of different drop release heights, H, and hence different impact speeds, U= 2gh. These experimental spreading data will be used to compare with the numerical predictions of the theoretical spreading model given by Eq. (5). The liquid tested was the distilled water. For imaging purpose, however, fluorescent dye was added to the water. The addition of the fluorescent dye slightly lowered the surface tension of the distilled water. In their experiments, weight measurements were used to determine the drop size, giving a size of around D:0.50 cm. No splashing was observed in these spreading experiments. Thoroddsen and Sakakibara’s experimental study was composed of two parts. First, the spreading rates were measured for a water drop impacting on a flat metal surface. A distilled water drop (D=2R= 0.55 cm) with very weak Fluorescein concentration (6 mg l − 1) was used, giving slightly lower surface tension (gwv =60 dyne cm − 1) measured with a ring tensiometer. The solid substrate was an anodized aluminium surface. In this part of the study, the detailed data of spreading rates and the vertical heights of water drops were measured for nine different drop release heights, H= 6, 10, 15, 22, 31, 40, 60, 80 and 124 cm (their Figs. 5 and 10). The corresponding impact speeds changed approximately from 100 to 500 cm s − 1. In the present study, only the spreading data for the following five drop release heights, H=6, 15, 31, 60 and 124 cm, are chosen in comparison with the theoretical
Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245
spreading results for graphic consideration. In the second part of their experimental study, a glass plate was used as the solid substrate because it allows for observations through the substrate. A distilled water drop (D =2R = 0.52 cm) with very high Fluorescein concentration (1 g l − 1) was used in order to obtain the best imaging results. This lowered the surface tension of the distilled water to about gwv =50 dyne cm − 1. The detailed spreading rates were available only for one drop release height, H =50 cm (their Fig. 11), corresponding to the impact speed, U :313 cm s − 1. However, the true values of some other quantities such as the viscosity and density of water, mw and rw, the equilibrium contact angle, ue, required in Eq. (5) were not given in Ref. [11]. Here, in the numerical simulations, mw and rw are chosen as 0.01 P and 0.998 g cm − 3, respectively, i.e. the values of pure distilled water. The effects of the fluorescent dye on the viscosity and density of the water drop are neglected. The equilibrium contact angle of the water drops on the high-energy solid surfaces such as the aluminium surface and the glass plate is considered to be zero, ue =0°, that is, the perfect wetting of the water drops on these two solid substrates occurs eventually. Fig. 1 shows an overall comparison between the numerical spreading rates predicted by Eq. (5) 1 with ln(o − d ) = 9.20 and the measured spreading data of water drops on the aluminium surface for five different drop release heights, H = 6, 15, 31, 60 and 124 cm. The comparison for the drop height under the same testing conditions is plotted in Fig. 2. It can be seen from these two figures 1 that, by choosing ln(o − d ) to be 9.20, the numerical predictions give the overall best fit to the measured spreading data for all the drop release heights up to H =31 cm. Thus, the value of the parameter od =Ld /L used in the numerical simulations is about 1.0 × 10 − 4. This value corresponds to the microscopic scale cut-off length, Ld : 1 mm, if the maximum contact radius, r(t), i.e. the horizontal characteristic length of the water drops, is chosen as the macroscopic scale cut-off length and approximated as 1.0 cm, i.e. L :1.0 cm. This Ld value is close to Ld :2 mm obtained by the authors [9] for the experimental data of silicone oil drop spreading on a soda-lime glass plate in air
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measured by Chen [10], and Ld : 5 mm published by Basu et al. [15] for glass–bitumen–water systems as well. On the one hand, physically, it is expected that the microscopic scale cut-off length, Ld, may depend on a specific spreading system, mainly reflecting the characteristics of the viscous drag force at the solid–liquid interface. On the other hand, it is unexpected to find out that the values of Ld for these three very different spreading systems [10,11,15] are so close. In fact, these three spreading systems differ from each other significantly. Their liquid viscosities are different by more than four orders of magnitude, the liquid drop sizes are different by one order of magnitude, and their impact speeds as well as the ranges of the spreading time are also rather different. Figs. 1 and 2 also indicate the strong effect of the impact speed on the spreading process. It is noted that, when H]60 cm, the theoretical spreading model always overestimates the spread-
Fig. 1. Spreading rates, r(t), versus time for a water drop (D= 0.55 cm, gwv =60 dyne cm − 1) impacting on an anodized aluminium surface for five different drop release heights, H (cm) =6 ( ); 15 (); 31 ( ); 60 (2); 124 (filled hexagon). The symbols represent the experimental spreading data measured by Thoroddsen and Sakakibara [11] whereas the solid lines represent the corresponding numerical spreading results 1 calculated by numerically solving Eq. (5) with ln(o − d ) =9.20 for the five drop release heights.
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4. Summary
Fig. 2. Heights of a water drop, h(t), versus time for five different drop release heights. The other captions are the same as those given in Fig. 1.
In conjunction with the previous paper [9], it has been shown that the previously developed theoretical spreading model is applicable not only to the spontaneous spreading processes but also to relatively low-speed impact spreading processes. It is also noted that this simple spreading model can not give excellent predictions for some relatively high-speed impact spreading cases. This may be remedied by incorporating the following improvements into the spreading model. First of all, the impact dynamics should be studied in order to understand the energy transformation process during actual impacting process. The ideal energy transformation always overestimates the spreading kinetic energy and thus leads to larger spreading rates. Secondly, the de Gennes’ model assumes that the dominant losses are due to hydrodynamic shear flows in the liquid wedge and that the molecular dissipation at the wetting line
ing rates. This overestimation may be caused by the following two major assumptions used in deriving the theoretical spreading model. First, in reality, a portion of the impact kinetic energy will be dissipated or transformed into other energy forms during impacting. The ideal energy transformation without any energy dissipation, i.e. the impact kinetic energy of a liquid drop is assumed to be 100% transformed into its spreading kinetic energy after impacting, will significantly overestimate the spreading kinetic energy and thus spreading rates. The higher the liquid drop is released from, the more serious error this assumption will cause. Second, at present, it is not quite clear that, during these relatively high-speed impact spreading processes, whether the drop shape at any moment is still completely governed by the classical Laplace equation of capillarity. Fig. 3 shows a comparison between the numerical spreading rates and the measured spreading data of a water drop on a glass plate for the drop release height, H=50 cm. In this case, as shown 1 in Fig. 3, by choosing ln(o − d ) : 9.20 in Eq. (5), the numerical simulation again gives the best fit to the measured spreading rates. Correspondingly, od =Ld /L: 1.0 ×10 − 4 or Ld :1 mm if L :1 cm.
Fig. 3. Spreading rates, r(t), versus time for a water drop (D= 0.52 cm, gwv =50 dyne cm − 1) impacting on a glass plate for one drop release height, H =50 cm. The open circle symbols represent the experimental spreading rates measured by Thoroddsen and Sakakibara [11] whereas the solid line represents the corresponding numerical spreading rates calcu1 lated by numerically solving Eq. (5) with ln(o − d ) =9.20 for the same drop release height.
Y. Gu, D. Li / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 239–245
is negligible. Therefore, this model should be modified properly or at best a completely new viscous drag force model of a moving contact line has to be developed for high-speed impact spreading processes. For relatively high-speed impact spreading cases, it is anticipated that the molecular dissipation at the moving contact line is important [13].
Acknowledgements The authors wish to acknowledge the financial supports of the Izaak Walton Killam Memorial Scholarship at the University of Alberta to Y. Gu and the research grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada to D. Li.
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