Spreading of a Water Drop Triggered by the Surface Tension Gradient

Sep 28, 2006 - Anoop Chengara, Alex D. Nikolov, and Darsh T. Wasan*. Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Il...
0 downloads 0 Views 167KB Size
Download PDF
Ind. Eng. Chem. Res. 2007, 46, 2987-2995

2987

Spreading of a Water Drop Triggered by the Surface Tension Gradient Created by the Localized Addition of a Surfactant Anoop Chengara, Alex D. Nikolov, and Darsh T. Wasan* Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

The spreading of aqueous solutions of trisiloxane surfactants on solid surfaces has been studied extensively. Trisiloxane surfactants are used in pesticide delivery as adjuvants to promote spreading on leaves and provide a larger area for solute transfer. The spreading of a dew-drop on a leaf when a spray of pesticide is delivered is simulated by studying the spreading of a water drop on a hydrophobic surface when a small drop of aqueous trisiloxane surfactant is brought in contact with it. This study reveals many new features that differ from the spreading of an aqueous trisiloxane drop on a solid surface; the spreading of the substrate drop is characterized by an inertial rather than a viscous response, the imposed surface tension gradient dies out rapidly, and the spreading velocity is consistent with a balance of kinetic energy imparted to the substrate drop and the decrease in its surface energy. 1. Introduction The presence of a surfactant generally enhances the spreading of a liquid on a solid surface. The spreading behavior of trisiloxanes has been studied extensively, and different explanations have been provided and reviewed for the high rate and degree of spreading on hydrophobic surfaces.1-4 The use of surfactants as adjuvants in pesticide formulation is common;5-6 the surfactant creates a well-wetted leaf surface and increases the interfacial area available for the pesticide transfer. Recently, a study7 compared various trisiloxane surfactants’ ability to spread over different leaf surfaces. Our study was motivated by the desire to model the essential elements of pesticide delivery to crops after dew formation. When the pesticide spray strikes the dew drops, it transfers the pesticide to the drop and the presence of the surfactant in the pesticide causes the drop to spread on the leaf surface. Thus, the spreading drop now mediates the pesticide uptake to the leaf. Similar principles have also been exploited in medical applications.8,9 We seek to understand the spreading phenomena driven by the localized surfactant application. In our study, this situation is simulated by a water droplet on a polystyrene surface (a hydrophobic surface like the leaf on which water beads) and a small drop of surfactant that is brought into contact with the air-liquid interface of the water drop. The localization of the contact between the surfactant-laden drop and the water causes a surface tension gradient to arise, resulting in the spreading of the water drop. Most studies in the literature10-13 on spreading driven by an externally imposed surface tension gradient rely on applying a spatial temperature gradient along the length of the solid surface; however, the surface tension gradients studied were small in magnitude and constant throughout the experiment. Theoretical studies14-16 have mostly focused on the instability of the contact line. We will discuss the spreading of a liquid drop under the action of a large surface tension gradient created by the localized application of an aqueous trisiloxane surfactant on a substrate water drop’s air interface. This experiment simulates a sudden (impulselike) reduction in the surface tension over a portion of the drop’s surface and creates a large surface tension gradient. * To whom correspondence should be addressed. E-mail: wasan@ iit.edu. Tel: (312) 567-3001. Fax: (312) 567-3003.

The resulting motion of the liquid drop exhibits new and interesting characteristics that are reported here. The localized surfactant quickly spreads on the surface of the liquid drop, causing a reduction in the surface tension gradient from its large initial value. This reduction is further accentuated by the dissolution of the surfactant into the liquid drop. We show that an inviscid flow model, rather than the commonly adopted viscous flow model, better explains the observed high velocity and short time scale of spreading. A variation of this experiment has been conducted17-21 to study the deformation of a thin flat liquid film when a drop of surfactant is brought in contact with its free surface. However, in these studies, the liquid film was infinite, unlike the finite drop studied here. Thus, their studies were concerned with the propagation of a shock front in the liquid film and with the spreading of the surfactant drop on the liquid surface. Further, the theoretical model for the dynamics of the liquid film was based on the viscous flow approximation, which we show is not valid for our case. The importance of the spreading dynamics to the mode of pesticide delivery is also discussed. 2. Materials and Methods The surfactant used in the following experiment was Silwet L-77 (a trisiloxane surfactant with an 8 ethoxy chain length) supplied by Witco (Tarrytown, NY). The solid substrates on which spreading experiments were conducted were sterile polystyrene Petri dishes (either 10 or 15 cm in diameter) procured from Fisher Scientific. Each Petri dish was cleaned with deionized water and dried at room temperature before experimental use and was discarded after each run. The surfactant was dissolved in deionized water obtained from a Millipore filter (18 MΩ cm resistivity). Deionized water of the same resistivity was used as the liquid substrate on which the surfactant was dosed. In some experiments, a solution of deionized water and glycerol (from Fisher Scientific) in a specific volume ratio was used as the liquid substrate. This was done to study the effect of substrate viscosity on the spreading. The added advantage of this choice is that the surface tension of pure glycerol is 68 dyn/cm, so mixing with water does not appreciably change the surface tension. 2.1. Spreading Rate. The schematic of the experimental setup used to study the spreading rate of a liquid drop under the action

10.1021/ie060695y CCC: $37.00 © 2007 American Chemical Society Published on Web 09/28/2006

2988

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007

Figure 2. Schematic showing the relationship between wavelength and optical fringes. Figure 1. Schematic of experimental setup.

of imposed surface tension gradient is in Figure 1. A liquid drop of a given volume is first deposited on the polystyrene Petri dish. The volume of the deposited liquid drop (henceforth referred to as the “substrate drop”) is measured with a graduated 2 mL glass pipet used to dispense the drop on the solid and is correct to within 10% error. A 5 µL volume of trisiloxane surfactant is dosed onto the surface of the substrate drop in the form of a 0.1 wt % aqueous solution. The volume of the dosed surfactant solution is carefully controlled using a 5 µL volume syringe (Hamilton) that permits accurate dispensing of small liquid volumes. In some experimental runs, a larger syringe (25 µL) was used in conjunction with a linear stepper motor (model 860A Motorizer from Newport, CA) to accurately dispense the surfactant solution. Care was taken to minimize the kinetic impact of the surfactant drop on the surface of the substrate dropsin some experiments, the surfactant drop flowed over a glass bead that was placed in the center of the substrate drop and extended above its free surface. This arrangement virtually eliminated the impact of the surfactant drop with the substrate drop. The spreading rates measured in these experiments were almost identical with those measured in experiments conducted without the glass beads, confirming that the effect of the impact of the surfactant drop on the motion of the substrate drop is negligible. A magnified image of the spreading substrate drop was projected onto a screen using an overhead projector. The image on the screen was recorded using a digital video camera (Sony TRV-310) and played back to capture successive images separated by 1/30 s in real time. The instantaneous diameter of the substrate drop was measured using image analysis software (Image Pro v.3.0 from Media Cybernetics, MD), with the pixel/ length calibration done using a known length. Two perpendicular diameters were measured and averaged to get the instantaneous diameter; the two values differed by less than (10% in magnitude. Three runs were conducted for each experimental condition to check for reproducibility and the scatter in the spreading rate is shown in the relevant plots. The main source of error arises from the short time scale of the spreading process; the initial acceleration of the substrate drop soon after contact with the surfactant drop influences the calculation of the spreading rate, particularly when the spreading process is very short, as with large substrate drops. 2.2. Surface Tension Gradient. The method used to estimate the initial surface tension gradient is based on the measurement

of the wavelength and velocity of a wave created on the surface of the liquid surface by the addition of the surfactant. The experimental setup is similar to that shown in Figure 1 for the measurement of the spreading rate except that, in this case, the Petri dish (diameter 15 cm) was filled with water to a depth of 1 cm, creating a flat interface that extended to the boundary of the Petri dish. A glass bead was placed in the center of the Petri dish with a small portion of it extending out of the water. A 5 µL volume of 0.1 wt % Silwet L77 is dosed onto the glass bead, over which it flows and comes into contact with the flat water surface. The contact between the surfactant solution and water created capillary-gravity waves on the water surface, whose wavelength is equal to twice the distance between the successive dark fringes (Figure 2). The videotape of the wave motion was played back at time intervals of 1/30 s, and in each frame the distance between was measured by Image Pro. Three such frames, corresponding to a total time lapse of 0.1 s since the deposition of surfactant, were analyzed before the boundary effects of the Petri dish interfered with the wave front. The wave velocity was measured by the distance traveled by each fringe in a time interval of 1/30 s, denoted by successive images captured from the videotape. The wave velocity of all fringes was the same within experimental error and was constant in the total time frame of measurement. The wavelength and velocity of the wave traveling on the liquid surface is related to the surface tension by22

Vp )

x2πgλ + 2πσ λF

(1)

where F is the density of the substrate fluid. The first term denotes the gravitational contribution while the second term denotes the capillary one. Having measured the wave velocity (Vp) and wavelength (λ), we estimate the local surface tension (σ) from a rearrangement of eq 1. When applied to a particular time “snapshot” of the process, eq 1 gives a plot of the surface tension as a function of the radial distance from the center, the latter being marked by the mean distance of each of the dark fringes from the center of symmetry. Since most of the dark fringes (or wave fronts) were visible after 0.1 s elapsed after the deposition of the surfactant, this particular snapshot of the process was used to plot the local surface tension as a function of the radial distance. The surface tension gradient is then estimated from the slope of the curve of the surface tension vs the radial distance.

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 2989

3. Theoretical Model 3.1. Surfactant Transport. This section describes a simple approach to estimate the time over which the concentration of the surfactant in the dosed drop changes appreciably. In this treatment, the surfactant species in the dosed drop is assumed to be bounded by a control volume whose boundary is determined by the flow rate set up in the substrate liquid. The control volume is an artificial construction because we do not expect the dosed drop to retain its distinct identity from the substrate drop since the predominant chemical species in both is water. We also assume that the concentration of surfactant in the control volume is radially uniform and changes only with time as it is transferred to the substrate drop. Despite these limitations, the model provides a useful estimate of the time scale over which the surfactant concentration reduces significantly as a result of the convective flow in the substrate drop. This time scale is relevant because a reduction in the surface concentration of the surfactant reduces the gradient of the surface tension, since the entire surface of the substrate drop will become depleted of surfactant. The main idea is to determine if the surface tension gradient continues to exist for the entire duration of the spreading phenomenon. Consider the mass balance of the surfactant species in the control volume. The time rate of change of the surfactant concentration (C) in a dosed drop of volume (Vs) is given by

d (V C) ) -ws(t) As(t) (C - Cb) dt s

(2)

where ws(t) is the molar average velocity of the surfactant species, As(t) is the contact area between the surfactant control volume and the substrate liquid drop, and Cb is the concentration of the surfactant in the bulk of the substrate drop. Since the volume of the substrate drop is orders of magnitude larger than that of the dosed surfactant drop, there is no appreciable change in the bulk concentration of the surfactant in the substrate drop even after the entire transfer has occurred (i.e. Cb ) 0). The molar velocity, ws(t), can be approximated by the velocity at which the air-liquid interface of the substrate drop approaches the solid surface due to its radial spreading. This, in turn, is equal to the rate of the change in height (Hd(t)) of the substrate drop of volume Vd:

-ws(t) )

( )

2Vd d Vd )Ud dt πR 2(t) πRd3(t) d

(3)

Rs(t) ) Rso + tUd

(4)

Substituting eqs 3 and 4 into eq 2 and integrating with respect to time, t, gives

(

)

(Rso + t′Ud)2

∫0 (R t

so

+ t′Ud)3

∆P )

2σ + Fwgh b

(6)

where σ is the uniform surface tension of the substrate drop and b is the radius of curvature of the drop at the apex. The contact of the surfactant with the substrate drop creates a sudden decrease in the tension over a portion of the surface of the substrate drop. The resulting radial nonuniformity of the surface tension creates a radial nonuniformity of the capillary pressure and alters the pressure distribution in the interior of the substrate drop (eq 6). Since the surface tension of the substrate drop is lowest at the apex (because the surfactant drop makes contact with the substrate drop), the capillary pressure in this portion cannot balance the atmospheric pressure. The resulting imbalance in the pressure acts as an impulse of the energy transferred to the substrate drop, causing it to spread. An energy conservation condition written for the substrate drop before and after the addition of the surfactant yields a simple expression for the velocity of the drop. The surface energy of the substrate drop before the surfactant was added (Ein) is given by the product of the surface tension (σw) and the area of the air-liquid interface (Aw). If As is the area of contact of the surfactant drop and the substrate drop which has surface tension equal to that of the surfactant solution at cmc (the dosed surfactant solution is at a concentration greater than 10× cmc), then the remaining area has a surface tension equal to that of water (Aw - As). The dosage of the surfactant then creates an excess surface energy (E) of the initial state (prior to surfactant addition) relative to that after addition, given by

∆E ) Awσw - [(Aw - As)σw + Asσs] ) As(σw - σs) (7)

Here the substrate drop has been approximated to spread like a flat disk of radius Rd(t) and the definition of the spreading velocity (Ud) was used in the last equality. Experimental observations show that the assumption of a constant value of Ud is valid. The area of contact for mass transfer between the control volume and the substrate drop is equal to the area of the circular disk surfactant control volume of radius Rs(t). The radius of the surfactant disk is simply related to the initial radius, Rso, and the horizontal velocity at which the control volume is stretched (which is equal to Ud):

2VdUd C )ln Vs C(t ) 0)

In eq 5, all the quantities on the right-hand side are experimentally known, permitting the calculation of the surfactant concentration profile as a function of time. This is an advantage of this approach as compared to the possibility of using an unknown mass transfer coefficient in eq 2. As remarked earlier, the main utility of eq 5 is to provide an estimate of the time scale for the substantial decrease in the surfactant concentration (C). 3.2. Inertial Force Dominated Spreading. In this section, we model the spreading of the substrate drop under the assumption that the inertial forces dominate the viscous forces. Prior to the contact of the surfactant drop with the substrate drop, the excess pressure in the substrate drop over that of the atmospheric pressure (P) at a height, h, below the apex is given by the Laplace equation for a sessile droplet23

dt′

(5)

This excess energy is transformed into the kinetic energy (Ek) of movement of the drop of mass (FVd/2) moving with velocity Ud. Note that the effective mass moving is half the drop’s mass because the two halves of the drop move away from each other at the radial velocity Ud. The energy balance on the substrate drop gives

1 As(σw - σs) ) FVdUd2 4

(8)

In eq 8, the energy loss due to the viscous dissipation has been ignored. This neglect of the viscous term is prompted by the experimental results on the spreading velocity of the substrate drops of varying viscosities (described later). Another reason to ignore the viscous dissipation is the short time scale of spreading; the viscous dissipation is expected to be large when the film thickness (or drop height) is very small and the steady state flow is set up. The extremely short spreading time ensures

2990

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007

Figure 3. Increase in area of substrate water drop as a function of initial drop size.

that the final film thickness (or drop height) is of the order of 0.1 cm in the extreme case and that a steady-state viscous flow pattern has not yet been established. Equation 8 is accurate only in providing the order of magnitude of the expected velocity. It ignores the energy needed to accelerate the substrate drop from rest to velocity Ud. Furthermore, it assumes that the surface energy of the substrate drop can be split into two parts, one of which has a saturation (minimum) surface tension and the other that has a surface tension of pure water. This is not strictly correct since the surfactant is transferred to the bulk and radially along the surface. However, since the energy balance is written for the situation just after contact and since the spreading velocity, Ud, was experimentally observed to be a constant, this assumption is not too restrictive. 4. Results and Discussion 4.1. Final Diameter of Substrate Drop. In this section, we examine the final radius (or area) achieved when the spreading of the substrate drop ceased. Spreading experiments were conducted for substrate water drops of different volumes in the range 0.5-5 cm3 while maintaining a constant surfactant concentration of 0.1 wt % in the 5 µL dosed aqueous drop. The initial radius of the substrate drops varied from 0.75 to 2 cm. In all cases, the height of the drop was constant and was achieved by choosing sufficiently high drop volumes. The constancy of the height of the substrate drop was checked by plotting the volume of the drop as a function of the experimentally measured surface area. A constant value of the slope equal to 0.3 cm was observed, indicating that the substrate drop adopts a flattened disklike shape on the solid surface. Maintaining a constant drop height is equivalent to maintaining a constant capillary pressure in the substrate drop before the surfactant is added. This is a consequence of the balance of forcessthe Laplace equation governing the drop’s shape dictates that the constant hydrostatic pressure (determined by the constant height of the drop) must be balanced by the (constant) capillary pressure. In this manner, we maintained a constant resisting force, viz. capillary pressure, while varying the length over which the surface tension changes, thereby varying only the driving force, viz. the surface tension gradient. Figure 3 shows a well-defined maximum in the percentage change in the surface area of the substrate drop as a function of the initial drop radius. This indicates that the effect of the surface tension gradient on

spreading is most pronounced when there is a sufficient length over which it can act. It is also interesting to examine the deviation of the final diameter of the substrate drop from the diameter that is expected under equilibrium conditions. To determine the diameter that will be achieved under equilibrium conditions, we performed independent experiments with different concentrations of Silwet L77 in aqueous solutions. The solutions chosen for the experiments were Cb ) 2.5 × 10-3 wt % and Cb ) 7.5 × 10-4 wt %; these concentrations were chosen because drops of the aqueous solutions at these conditions spread only partially on the polystyrene surface, permitting the calculation of the contact angle. A drop of a known volume of a specific Silwet L-77 concentration was placed on a polystyrene dish, and the radius of the drop was measured. The surface tension of the aqueous solution was measured by the Wilhelmy plate method using a Pt plate and a type K-8 tensiometer obtained from Kru¨ss GmbH, Hamburg, Germany. Using 4 drops of varying volumes (in the range 0.1-1 cm3), the equilibrium radius was measured for each volume. The drop volume (Vd), surface tension (σ), and radius of the contact of drop (Rc) for a particular concentration are known. These three quantities are sufficient to solve the Laplace equation (eq 6). The calculation procedure23 involves the iteration on the apex radius (b) such that the calculated volume and radius of contact match the experimentally determined quantities. The contact angle is determined from the calculation as that angle at which the calculated radius of contact and volume subtended between the apex and the contact line match the experimental quantities; it is accurate to within 5°. For the two concentrations of interest, the surface tensions and contact angle are reported in the box in Figure 4, while the contact angle for water on polystyrene is 90°. The contact angle sharply changes for a small change in the surface tension in the lower range of the surface tension. We calculated the concentration of the surfactant in the substrate drop after equilibrium was reached from a mass balance on the surfactant species. For example, the equilibrium concentration of the surfactant in the substrate drop of volume 0.5 cm3 after the dosage of a 5 µL drop of 0.1 wt % is Cb ) 0.1 × 0.005/0.5 ) 1 × 10-3 wt %. The surface tension corresponding to this effective concentration is read from the surface tension isotherm reported in the literature.24 Note that the surface tensions of the substrate drops at equilibrium are very close to the test cases in the box, allowing

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 2991

Figure 4. Deviation of final diameter of substrate drop from equilibrium value as a function of initial drop diameter.

us to predict the equilibrium contact angles. For substrate drops with initial diameters greater than 2.5 cm, the contact angle is virtually 90° and only the surface tension needs to be read from the isotherm for the equilibrium surfactant concentrations determined from the mass balance. Knowing the surface tension and contact angle expected when the equilibrium concentration of the surfactant in the substrate drop is reached, we can now use the Laplace equation, eq 6, to calculate the contact diameter at equilibrium. This is called the “Laplace diameter” in the ordinate axis of Figure 4, which plots the difference of the actual final contact diameter obtained from the spreading experiments and the Laplace diameter as a function of the initial size of the substrate drop. In all the cases, the actual diameter exceeds the Laplace diameter, indicating that an extra force (over and above the equilibrium capillary pressure) drives the spreading of the substrate drop. The fact that the diameter does not retract to its equilibrium value is a consequence of the hysteresis of the contact on the solid surface. Figure 4 reveals that the excess spreading (as denoted by the difference between the final diameter and equilibrium diameter) passes through a maximum as the initial size of the substrate drop increases. This is again indicative of an excess force (i.e. a force over and above the capillary forces resulting from a reduction in the surface tension upon the addition of the surfactant) that manifests itself most strongly when the size of the substrate drop is intermediate. It is tempting to explain the maximum in total expansion (and in the excess expansion over the equilibrium configuration) in the following simplistic manner: when the substrate drop is small, the dosed surfactant can cover almost the entire surface immediately, thereby providing only a small gradient of surface concentration (and hence the gradient of the surface tension), leading to a small expansion. When the radius of the substrate drop becomes larger, the dosed surfactant cannot spread over the entire surface immediately, leading to a larger surface tension gradient and more spreading. When the substrate drop is very large, much of the surface will remain uncovered by the dosed surfactant, resulting in a large fraction of the surface area with a uniform tension corresponding to that of water, again leading to a low surface tension gradient and a low degree of expansion. This explanation is oversimplified, as we show in section 4.3. In the next section, we compare the observed spreading rate of the substrate with the prediction from a simplified lubrication

Figure 5. Diameter of spreading substrate water drop as a function of time.

approximation to show the failure of the often-used viscous flow model in describing these experiments. 4.2. Comparison of Experimental Velocity and Viscous Flow Model. A plot of the diameter of the substrate drop of volume as a function of time is shown in Figure 5. The initial radial spreading velocity, defined as half the slope of the curve, is seen to be a constant and is of the order of 1.5 cm/s (the half contribution is due to the fact that the diameter, rather than the radius, of the drop is plotted). Although the spreading velocity decreases toward the end of the process, the linear regime extends until nearly 80% of the total expansion in area occurs, making it the relevant regime to study. This experimental result of the spreading velocity will be compared with the prediction of a simplified model on the basis of the viscous flow approximation. To calculate the spreading velocity of the substrate drop, we must first estimate the surface tension gradient established when the drop of surfactant is dosed on its surface. To estimate the surface tension gradient, we consider the results of the wave measurements on the large surface of substrate liquid as outlined in section 2.2. The difficulty in conducting the wave measurements on the much smaller drop of substrate used in the spreading experiments is that the dark fringes were not clear enough on the drop’s surface to permit the wavelength measurement. Although these measurements of wave propagation are on a large expanse of liquid surface compared to the area of the substrate drop used in the spreading experiments, the surface tension gradients in the two cases are expected to

2992

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007

Figure 6. Surface tension profile on flat water surface set up by dosage of a 5 µL 0.1 wt % Silwet L-77 drop.

be of the same order of magnitude. A plot of the surface tension as a function of the distance from the center is shown in Figure 6 for the large flat interface. The variation in the surface tension is nearly linear with the distance for this short time, implying that the surface tension gradient is a constant. Straight-line extrapolation of the curve to the center of symmetry yields a surface tension value at the center of 22 dyn/cm. At the center, the surfactant concentration is approximately equal to that in the dosage drop just before contact with the water surface (about 10 times cmc). Therefore, we expect that the surface tension at the center of the water surface will be close to the equilibrium value at cmc, which is 22 dyn/cm, consistent with the extrapolation in Figure 6. This finding underlines the reliability of the wave method to estimate the immediate surface tension gradient set up by dosing the surfactant onto the liquid surface. The surface tension gradient, estimated as the slope of the curve in Figure 7, is 25 dyn/cm2. When this value is substituted into the simplified lubrication approximation that retains only the surface tension gradient term,

Ud ≈

h dσ 2µ dx

(9)

we predict an average spreading velocity of 125 cm/s, using the known height of the substrate drop (0.1 cm, chosen to be 1/3rd the initial height of the drop to account for the decrease in height as spreading proceeds). The predicted value of the spreading velocity is more than 2 orders of magnitude higher than the observed experimental spreading velocity, as found from the slope of the curve plotted in Figure 5. Despite the fact that capillary pressures are neglected in eq 9, the discrepancy between the model and the experiment is too large to be reconciled. To further test the viscous flow regime approximation, we performed experiments with substrates of different viscosities. A blend of glycerol and water in varying proportions (20%, 30%, and 40% (v/v) glycerol) was used to change the viscosity up to 4 times that of water. An advantage of altering the viscosity is that the surface tension of the glycerol-water mixture changes only marginally from 72 dyn/cm (for water) to 68 dyn/cm for all the ratios of glycerol to water studied. The viscosity of each blend was measured using a Cannon Fenske viscometer (No. 150) obtained from Fisher Scientific, IL. For

this set of experiments, the dosage of the surfactant was increased to 0.2 wt % to observe a significant total change in the spreading diameter. Each of the substrate drops were 1 cm3, and due to their surface tensions (and by extension their contact angles with polystyrene) being nearly equal, the initial diameter in all cases was nearly identical, permitting easy comparison. The spreading rate for each of these cases is plotted against the inverse viscosity in Figure 7. The effect of the substrate viscosity on the spreading velocity is quite weaksfor a 4-fold increase in viscosity, there is only a 10% decrease in the spreading rate. The lubrication approximation, on the other hand, predicts a much stronger dependence (eq 9). The observations described in this section suggest that the motion of the substrate drop cannot be simply modeled as the flow of a thin film under the action of a sustained surface tension gradient. An alternative model of spreading based on the dominance of inertial forces over viscous forces is considered in the next section. 4.3. Inertial Spreading Regime. To understand the reasons for the failure of the commonly used viscous-regime-based lubrication approach in our experiments, we examined the time scale of the spreading process. The spreading process comes to a halt in less than 2 s for all sizes of the substrate drops consideredsin other words, spreading is a short-lived process. The characteristic time scale for momentum transfer from the free surface of the substrate drop to the solid surface is

τmom )

Ho2 (µ/F)

(10)

where Ho is the initial height of the substrate drop and µ and F are the viscosity and density of the substrate fluid. Using the values for the physical properties for water and the experimental value of 0.3 cm for the height of the water drop, we find that the time for the momentum transfer is about 10 s. However, it was experimentally observed that the spreading process ends in less than 2 s. Thus, the time scale of spreading is too short for viscous forces to dominate the inertial forces. This is the reason for the relatively minor effect of the substrate viscosity on the spreading velocity observed in the previous section. Another time scale relating to the duration of the imposed surface tension gradient is relevant. This is directly related to

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 2993

Figure 7. Spreading velocity of substrate drop as function of inverse viscosity.

Figure 8. Predicted surfactant concentration in dosed drop as a function of time due to convective flow transfer set up in substrate drop.

the time scale over which the dosed surfactant is transferred to the bulk of the substrate drop: the faster the “dissolution”, the quicker the imposed surface tension gradient dies out. Figure 8 plots the change in the surfactant concentration with time under the convective conditions imposed by the substrate flow as calculated by eq 5. The time scale for the decrease in the surfactant concentration in the dosed drop is much smaller than even the spreading time scale. Note than even a 20% decrease in the surfactant concentration translates into a significant increase in the surface tension, although this is not immediately obvious from Figure 8, because eq 5 has not incorporated a surface tension equation of state. However, the primary conclusion from Figure 8 is that the surface-imposed surface tension gradient dies out before the substrate drop stops spreading. This explains why the substrate drop’s motion cannot be explained by the viscous-regime-based approach as described by eq 9; that approach is valid only for spreading under a continuously acting surface tension gradient. In our case, the high surface tension gradient reported in wave measurements dies out almost

instantaneously. Increasing the concentration of the surfactants in the microdrop may increase the time span over which the

Figure 9. Schematic representation of different stages of substrate drop’s spreading process.

2994

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007

Figure 10. Spreading velocity of substrate drop as function of initial drop diameter.

imposed surface tension gradient continues to contribute directly to the spreading process. Thus, the physical picture that emerges is that of a sudden impulse in the surface energy delivered to the drop by the application of the surfactant (Figure 9). When the surfactant drop comes into contact with the substrate water drop, it creates a region of low surface tension. This sets up capillary waves, which only last for a very short time (∼1/30 s). Two events occur to attenuate the surface tension gradient: the spreading of the surfactant solution on the surface of the substrate drop and the convective flow of the surfactant solution into the bulk of the substrate drop. However, the effect of surfactant addition on the substrate drop persists via the pressure impulse that is set up in the substrate drop. This impulse causes the deformation of the free surface as a result of imbalance of the vertical forces and the “roof” of the drop “collapses”. Since the liquid substrate is incompressible, the resulting pressure gradient sends out shock waves and drives the drop to spread radially outward. As already mentioned in section 3.2, this spreading process can be approximated by an energy balance on the substrate drop. We apply eq 8 to calculate the velocity of a 1 cm3 water drop when a 5 µL drop of 0.1 wt % Silwet L77 is brought in contact with it. To estimate the area over which the surface tension of the substrate drop has been effectively reduced to that of the dosed surfactant drop, we assume that it is the surface area of a flat disk with radius (Rso) i.e., As ) BRso2. The contact radius, Rso, is given by the constraint that the spherical dosed drop has volume Vs ) 5 µL:

Rso )

( ) 3Vs 4π

1/3

(11)

If we insert the value of Rso ) 0.1 cm into eq 8, the spreading velocity, Ud, is 1.5 cm/s, which is of the same order of magnitude as the experimentally observed velocity of 1.4 cm/s (Figure 5). More importantly, the agreement between the inertial spreading model and the experiment is much better than the viscous-stress model that forms the basis of eq 9. Further, eq 8 predicts that as the drop’s mass (or volume) increases, the velocity decreases for the same impulse of energy imparted to it. Figure 10, which plots the experimental radial velocity as a

function of the initial drop diameter, shows that this prediction is borne out, qualitatively, at least, in the higher range of the drop volume. The reason that it is not met in the lower region of the drop volume is probably due to the restrictive assumption of a jump change in the surface energy of a fixed portion of the drop’s surface area, As. In reality, the dosed surfactant solution may spread radially as a monolayer on the surface of the substrate drop in a manner derived by Joos and Pinten (1977), who assumed a linear gradient of the surface tension was driving the spreading of the monolayer. The time required to cover the initial radius of the substrate drop (Rdo) as a monolayer is given by25

[

(

τmono ) Rdo(µF)1/4

3 4(σw - σs)

)]

1/2 4/3

(12)

where µ and F are the viscosity and density of the substrate and σw and σs are the surface tensions of the substrate and monolayer, respectively. Substituting the numerical values for this quantity, we see that a monolayer can spread over the initial radius of a 1 cm3 drop (Rdo ) 1.25 cm) in about 0.03 s. This means that more of the surface will be covered by surfactant molecules, resulting in a larger area over which the surface tension decreases. However, it must be noted that eq 12 is valid only when the monolayer is supplied by a much larger reservoir of that chemical species. Recall from Figure 8 that the dosed drop of surfactant (reservoir) dissolves very quickly into the substrate drop, over the same time scale as the spreading of the monolayer. Thus, although the radial spreading of the surfactant molecules and the extent of the area over which the surface tension gradient exists is oversimplified in eq 8, the approach that we take to model the inertial regime spreading appears to capture the essential features of the process. 5. Conclusions In this paper, we studied the spreading of a liquid drop on a hydrophobic surface under the action of an imposed surface tension gradient. The surface tension gradient was imposed by bringing a drop of surfactant (Silwet L77) solution into contact with a water drop on a polystyrene surface. Under our

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 2995

experimental conditions, it was observed that the motion of the water drop was quick (with a constant velocity of approximately 1 cm/s) and very short-lived (total time of spreading is less than 2 s). However, the spreading velocity was much too small to be explained by the classical approach based on the approximation of the viscous flow in a thin film driven by a surface tension gradient. This is because the imposed surface tension gradient dies out very quickly due to the convective transfer of the surfactant into the bulk of the substrate drop. Further, it was found that the velocity of spreading was very weakly dependent on the viscosity of the substrate drop. Both these findings pointed to the possibility that the spreading can be modeled as the response to a sudden (impulse) change in the surface energy of the substrate drop as a result of the surfactant dosage. A simple model based on the balance of the surface energy and kinetic energy of spreading is able to predict the experimentally observed spreading velocity. The results of this study can be used to prescribe a strategy for the dosage of a solute (like a pesticide or pulmonary surfactant) onto a wetted solid surface. Since the imposed surface tension gradient dies out rapidly despite the high dosage concentration of surfactant, it is not useful to increase the concentration in the spray. It is more useful to dose the solute in successive steps (or sprays) rather than in one shot. Each step (or spray) results in the expansion of the substrate fluid until the balance of the capillary and hydrostatic forces is approximately satisfied. Applying another impulse to this newly exposed substrate drop’s surface allows us to cause further expansion and cover a larger area of the solid surface. As the surface area covered by the substrate fluid increases, it permits the more rapid and efficient uptake of the solute to the solid surface. Acknowledgment We thank a reviewer for pointing out that pesticides are sprayed at dusk to avoid evaporation and, in cooler climates, dew drops form prior to the spraying. This research was supported by a grant from the National Science Foundation (Grant No. CTS-0553738). Literature Cited (1) Nikolov, A. D.; Wasan, D. T.; Chengara, A.; Koczo, K.; Policello, G. A.; Kolossvary, I. Superspreading Driven by Marangoni Flow. AdV. Colloid Interface Sci. 2002, 96, 325-338. (2) Shen, Y.; Couzis, A.; Koplik, J.; Maldarelli, C.; Tomassone, M. S. Molecular Dynamics Study of the Influence of Surfactant Structure on Surfactant-Facilitated Spreading of Droplets on Solid Surfaces. Langmuir 2005, 21, 12160-12170. (3) Kumar, N.; Maldarelli, C.; Couzis, A. An Infrared Spectroscopy Study of the Hydrogen Bonding and Water Restructuring as a Trisiloxane Superspreading Surfactant Adsorbs Onto an Aqueous-Hydrophobic Surface. Colloids Surf., A 2006, 277, 98-106. (4) Hill, R. M. Silicone Surfactants-New Developments. Curr. Opin. Colloid Interface Sci. 2002, 7, 255-261.

(5) Anderson, N. H.; Hall, D. J.; Wastern, N. H. The Role of Dynamic Surface Tension in Spray Droplet Retention. Proc. Int. Congr. Plant Prot., 10th 1983, 2, 576-581. (6) Knoche, M.; Tamura, H.; Bukovac, M. J. Performance and Stability of the Organosilicone Surfactant L-77; Effect of pH, Concentration and Temperature. J. Agric. Food Chem. 1991, 39, 202-206. (7) Zhang, Y.; Zhang, G.; Han, F. The Spreading and Superspreading Behavior of New Glucosamide Trisiloxane Surfactants on Hydrophobic Foliage. Colloids Surf., A 2006, 276, 100-106. (8) Conte, J. E.; Hollander, H.; Golden, J. A. Inhaled or Reduced-Dose Intravenous Pentamidine for Pneumocystis carinii Pneumonia. Ann. Intern. Med. 1987, 107, 495-498. (9) Gaver, D. P., III; Samsel, R. W.; Solway, J. The Effect of Surface Tension and Viscosity on Airway Opening. J. Appl. Physiol. 1990, 69, 7485. (10) Ludviksson, V.; Lightfoot, E. N. The Dynamics of Thin Liquid Films in the Presence of Surface Tension Gradients. AIChE J. 1971, 17, 1166-1173. (11) Cazabat, A. M.; Fournier, J. B.; Carles, P. Wetting Films Driven by Surface Tension Gradients. In Fluid Physics, Series on Nonlinear Science Series B; Velarde, M. G., Christov, C. I., Eds.; World Scientific: Singapore, 1995. (12) Bertozzi, A. L.; Munch, A.; Fanton, X.; Cazabat, A. M. Contact Line Stability and ‘Undercompressive Shocks’ in Driven Thin Film Flow. Phys. ReV. Lett. 1998, 81, 5169-5172. (13) Schneemilch, M.; Cazabat, A. M. Wetting Films in Thermal Gradients. Langmuir 2000, 16, 8796-8801. (14) Kataoka, D. E.; Troian, S. M. A Theoretical Study of Instabilities at the Advancing Front of Thermally Driven Coating Films. J. Colloid Interface Sci. 1997, 192, 350-362. (15) Munch, A.; Bertozzi, A. L. Rarefaction-Undercompressive Fronts in Driven Films. Phys. Fluids 1999, 11, 2812-2814. (16) Golovin, A. A.; Rubenstein, B. Y.; Pismen, L. M. Effects of van der Waals Interactions on the Fingering Instability of Thermally Driven Thin Wetting Films. Langmuir 2001, 17, 3930-3936. (17) Borgas, M. S.; Grotberg, J. B. Monolayer Flow on a Thin Film. J. Fluid Mech. 1988, 193, 151-170. (18) Gaver, D. P.; Grotberg, J. B. Droplet Spreading on Thin Viscous Film. J. Fluid Mech. 1992, 235, 399-414. (19) Halpern, D.; Grotberg, J. B. Dynamics and Transport of a Localized Soluble Surfactant on a Thin Film. J. Fluid Mech. 1992, 237, 1-11. (20) Jensen, O. E.; Grotberg, J. B. Insoluble Surfactant Spreading on a Thin Viscous Film: Shock Evolution and Film Rupture. J. Fluid Mech. 1992, 240, 259-288. (21) Starov, V. M.; de Ryck, A.; Velarde, M. G. On the Spreading of an Insoluble Surfactant over a Thin Viscous Liquid Layer. J. Colloid Interface Sci. 1997, 190, 104-113. (22) Ingard, K. U. Fundamentals of Oscillations and WaVes; Cambridge University Press: Cambridge, U.K., 1988. (23) Hartland, S.; Hartley, R. W. Axisymmetric Liquid-Liquid Interfaces; Elsevier: Amsterdam, 1976. (24) Rosen, M. J.; Wu, Y. Superspreading of Trisiloxane Surfactant Mixtures on Hydrophobic Surfaces 1. Interfacial Adsorption of Aqueous Trisiloxane Surfactant-N-Alkylpyrrolidinone Mixtures on Polyethylene. Langmuir 2001, 17, 7296-7305. (25) Joos, P.; Pintens, J. Spreading Kinetics of Liquids on Liquids. J. Colloid Interface Sci. 1977, 60, 507-513.

ReceiVed for reView May 31, 2006 ReVised manuscript receiVed August 14, 2006 Accepted August 16, 2006 IE060695Y