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Drop Retention Force as a Function of Drop Size Preeti S. Yadav, Prashant Bahadur, Rafael Tadmor,* Kumud Chaurasia, and Aisha Leh Department of Chemical Engineering, Lamar UniVersity, Beaumont, Texas 77710 ReceiVed August 9, 2007. In Final Form: December 12, 2007 The force, f, required to slide a drop past a surface is often considered in the literature as linear with the drop width, w, so that f/w ) const. Furthermore, according to the Dussan equation for the case that the advancing and receding contact angles are constant with drop size, one can further simplify the above proportionality to f/V1/3 ) const where V is the drop volume. We show, however, that experimentally f/V1/3 is usually a decaying function of V (rather than constant). The retention force increases with the time the drop rested on the surface prior to sliding. We show that this rested-time effect is similar for different drop sizes, and thus the change of f/V1/3 with V occurs irrespective of the rested-time effect which suggests that the two effects are induced by different physical phenomena. The time effect is induced by the unsatisfied normal component of the Young equation which slowly deforms the surface with time, while the size effect is induced by time independent properties. According to the Dussan equation, the change of f/V1/3 with V is also expressed in contact angle variation. Our results, however, show that contact angle variation that is within the scatter suffices to explain the significant force variation. Thus, it is easier to predict contact angle variation based on force variation rather than predicting force variation based on contact angle variation. A decrease of f/V1/3 with V appears more common in the system studied compared to an increase.
Introduction The Young equation1 describes the flat surface thermodynamic equilibrium2-4 contact angle, θY, as a function of the interfacial tensions between the liquid and the solid, the liquid and the vapor, and the solid and the vapor: γSL, γ, and γSV, respectively:
γSL + γ cos θY ) γSV
f ) γ(cos θR - cos θA) w
(1)
θY describes the equilibrium contact angle on a flat homogeneous surface and should be a unique contact angle. Since surfaces are not completely flat or homogeneous, deviations from θY toward another global equilibrium5 contact angle θ0 have been described in the literature (e.g., the Wenzel and Cassie equations6,7). An extension of this approach can explain the variation of contact angle with drop size via nonextensive thermodynamics argumentations.8 Yet, the common case of contact angles that are constant with drop size is expected to result in retention force that is constant with drop size. As we show here experimentally, this is not the case. Contact angle hysteresis can arise from a number of factors including surface roughness and chemical heterogeneity. As we show below, by comparing eqs 2 and 3, the line tension can also be related to contact angle hysteresis. The line tension, k, is defined by the modified Young equation9,10 which describes the energy associated with the three phase contact line (termed either line energy or line tension):11-19
k ) γ(cos θ - cos θ0)r
where θ is the apparent contact angle, θ0 is the equilibrium contact angle, and r is the drop’s radius. Dussan,20 and before her Furmidge,21 showed that the retention force, f, obeys
(3a)
where θA and θR are the tilt-advancing and tilt-receding contact angles and w is the drop’s width, that is, for spherical drops, w ) 2r. Note that while eq 2 assumes a circular contact line, eq 3 does not. By tilt-advancing and tilt-receding, we refer to the front edge maximal and rear edge minimal contact angles that a drop obtains due to tilting of the surface on which it rests. (A pioneering study by Krasovitski and Marmur22 showing that these differ from the planar-advancing and planar-receding contact angles was supported recently theoretically and experimentally.23) Dussan showed that for small contact angle hysteresis one can write eq 3a as20
()
f ) γV1/3
[
96 π
1/3
(cos θR - cos θA)
]
(1 + cos θA)3/4(1 - 3/2 cos θA + 1/2 cos3 θA) (cos θA + 2)3/2(1 - cos θA)9/4
2/3
(3b)
(2)
(1) Young, T. Phil. Trans. R. Soc. London 1805, 95, 65. (2) Safran, S. A. Statistical thermodynamics of surfaces, interfaces, and membranes; Addison-Wesley Publishing Company: Reading, MA, 1994. (3) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; John Wiley & Sons: New York, 1997. (4) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: London, 1991. (5) Marmur, A. Soft Matter 2006, 2, 12. (6) Wenzel, R. Ind. Eng. Chem. 1936, 28, 988. (7) Cassie, A. Discuss. Faraday Soc. 1952, 75, 5041. (8) Letellier, P.; Mayaffre, A.; Turmine, M. J. Colloid Interface Sci. 2007, 314, 604. (9) Vesselovsky, V. S.; Pertzov, V. N. Zh. Fiz. Khim. 1936, 8. (10) Toshev, B. V.; Platikanov, D.; Scheludko, A. Langmuir 1988, 4, 489.
(11) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464. (12) Gretz, R. D. J. Chem. Phys. 1966, 45, 3160. (13) Gretz, R. D. Surf. Sci. 1966, 5, 239. (14) Milchev, A. I.; Milchev, A. A. Europhys. Lett. 2001, 56, 695. (15) Pethica, B. A. J. Colloid Interface Sci. 1977, 62. (16) Sheludko, A. Colloid J. 1986, 48, 917. (17) Tadmor, R. Langmuir 2004, 20, 7659. (18) Marmur, A. J. Colloid Interface Sci. 1997, 186, 462. (19) Shapiro, B.; Moon, H.; Garrell, R. L.; Kim, C. J. J. Appl. Phys. 2003, 93, 5794. (20) Dussan, E. B. J. Fluid Mech. 1985, 151, 1. (21) Furmidge, C. G. L. J. Colloid Sci. 1962, 17, 309. (22) Krasovitski, B.; Marmur, A. Langmuir 2005, 21, 3881. (23) Tadmor, R.; Yadav, P. S. J. Colloid Interface Sci. 2008, 317, 241.
10.1021/la702473y CCC: $40.75 © 2008 American Chemical Society Published on Web 02/29/2008
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Figure 1. Various dimensions and parameters used in the measurements.
Thus, for an axisymmetric drop, one can relate k to f:
kR kA f f ) ) )w 2r r r
(4)
From eq 3b, we learn that for constant θA and θR also f/V1/3 ) const. This is a bit surprising because there are cases which report f/V1/3 * const for constant θA and θR. In this paper, we show how the Dussan equation can still be correct at least for the small contact angle hysteresis conditions for which it was derived (the systems we use in this paper are in the lower contact angle hysteresis range with respect to other papers using the Dussan equation) and the reported θA and θR values may look experimentally constant but are probably not. Apparently, the case of constant θA and θR is nonexistent but hard to resolve experimentally. Thus, the forces can explain the variations of contact angles better than the direct measurements of the contact angles themselves; that is, deducing the force variation based on the angle variation is more difficult than vice versa. We show that f/V1/3 can either decrease or increase with V depending on the system, though it seems that decreasing is more common.
Figure 2. θA (O) and θR (0) vs the time, t, that a HD drop rested on an OTA covered mica surface prior to motion. Both angles increase with t. θA and θR correspond to the steepest tilt just before the drop slides, when θR reaches its minimum. The linear fits are θR ) 31.894 + 0.285t and θA ) 42.564 + 0.184t (where t is in minutes). detail are only of hexadecane (HD) on octadecyl trimethylammonium (OTA) covered mica. However, the size effect was measured in less detail also for other systems and was found to be general as we describe later. Another liquid used was water; other surfaces used were OTA covered glass and Teflon. Covering mica or glass with OTA was done by self-assembly24 as described in other places.25-30 Briefly, 0.4 mg/mL C18H37N[CH3]3Br (OTA-Br; 99% pure from Aldrich) was dissolved in water by heating to 60 °C, thereby dissociating to OTA+ and Br- ions. Freshly cleaved mica surfaces or cleaned glass surfaces were immersed in the aqueous solutions of the surfactants. The surfaces were withdrawn dry following rinsing in pure water. OTA is known to form a rigid smooth monolayer of 10 ( 3 Å on mica.25 Note that in ref 25 OTA is named STAI where the “S” stands for “stearic” (the acid derivative of OTA) and is not related to the OTAI there. In this paper, we adopt the more modern nomenclature, hence OTA. In each experiment, the surface was loaded into a pocket goniometer (model PG2 from Gardco Co.). A built-in charge-coupled device (CCD) camera in the PG2 goniometer captured a video sequence of the liquid droplet on the surface. The contact angles were measured in two ways: (1) by passing a tangent at the dropsurface contact point and (2) by measuring the contact angle using a curve fit to the drop picture and calculating the contact angle from the geometrical relations. Both cases agreed within an error of (1° in precision.
Experimental Section
Results and Discussion
Measurements. In each experiment, we placed a drop of a given volume on a solid surface using microsyringe prescribed reading (errors of 0.1 µL). The drop looked perfectly symmetric when placed on the surface, and the picture of the “as placed”23 drop was used to calculate and verify the drop’s volume, V. The error of our measured volume using the as placed drop picture ranged from (0.05 µL for the smallest drop volumes (i.e., 0.5 µL) to (1 µL for the biggest drop volumes (i.e., 35 µL), matching the accuracy of our microsyringe ((0.1 µL) at a volume of 2 µL. We then tilted the surface until the drop began to slide, at which point the lengths of “a” and “b” (see Figure 1) were measured and the tilt angle, R, was deduced. The lateral retention force, f, was calculated as
Change of Angle with Drop Size. We start by describing the nature of the θA and θR dependence on the time, t, at which the drop was resting on the surface prior to sliding. Figure 2 shows an increase of both θA and θR with time. We attributed this to the unsatisfied normal component of the Young equation that is pulling on the surface and deforming it with time. Note that Figure 2 addresses the motion of the whole drop, which is characterized by the detachment of the receding edge. The rate of tilt did not seem to affect the change of the angle with time (within the scatter). Since in this paper we are primarily interested in the dependency of the retention force on drop size, Figure 2 is required only for the purpose of normalizing the contact angle to zero rest time. Moreover, one may question its need, since one can attempt to keep a constant (and short) rest time, thereby bypassing the rest time complication. Indeed, we too studied the θA and θR
f ) FVg sin R
(5)
where F is the density of the drop’s liquid and g is the gravitational acceleration. The drops were videotaped during the tilting, and the values of θA and θR were measured from the video frame just prior to commencement of drop motion. To explain the exact point considered as “drop motion”, we describe here the stages associated with drop motion: (1) The first movement is that of the drop’s leading edge. This movement is followed by a spontaneous stop. (2) We then further increase the tilt of the supporting surface until a subsequent movement of the leading edge is followed by the trailing edge, that is, a slide of the whole drop (we observed the same qualitative description also for drops sliding centrifugally on a perfectly horizontal surface). This slide of the drop is often short, yet the data shown here correspond to this situation. All experiments were performed in a dust free laminar flow hood (Terra Universal, ULPA filters). The experiments reported here in
(24) Ulman, A. An Introduction to Ultrathin Organic Films; Academic Press Inc.: New York, 1991. (25) Tadmor, R.; Rosensweig, R. E.; Frey, J.; Klein, J. Langmuir 2000, 16, 9117. (26) Raviv, U.; Giasson, S.; Kampf, N.; Gohy, J. F.; Jerome, R.; Klein, J. Nature 2003, 425, 163. (27) Raviv, U.; Giasson, S.; Frey, J.; Klein, J. J. Phys.: Condens. Matter 2002, 14, 9275. (28) Tadmor, R.; Klein, J. J. Colloid Interface Sci. 2002, 247, 321. (29) Kampf, N.; Gohy, J. F.; Jerome, R.; Klein, J. J. Polym. Sci., Part B: Polym. Phys. 2005, 43, 193. (30) Tadmor, R.; Chen, N. H.; Israelachvili, J. Macromolecules 2003, 36, 9519. (31) Carre, A.; Gastel, J. C.; Shanahan, M. E. R. Nature 1996, 379, 432.
Drop Retention Force as a Function of Drop Size
Figure 3. θA (O) and θR (0) extrapolated to t ) 0 vs V1/3 (V ) drop’s volume) for HD drops about to slide on an OTA covered mica surface. The solid lines are linear fits to the data.
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Figure 5. Slopes obtained from linear fits to curves like those shown in Figure 2 normalized by their corresponding drop size, namely 1/V1/3 ∂f/∂t vs the drop size V1/3. The high scatter is caused in part by dividing two measured properties (slopes and sizes), which results in the amplification of the error in the quotient; in part by the fact that the slope itself is a quotient; in part by the fact that a slope (∆f/∆t) requires subtraction (“∆”) of two measured properties and subtraction again amplifies the error, and in part to the difficulty of this tedious experiment.
Figure 4. Lateral force per unit length, f/V1/3, required to slide a hexadecane drop past an OTA covered mica surface as a function of the time, t, at which the drop was resting on the surface prior to sliding. Cube roots of the drop volumes, V1/3, are noted near their corresponding plots.
dependencies on drop volume in another series of experiments in which the surface was tilted very shortly after the drop was placed (as opposed to waiting several minutes as was done in the Figure 2 experiments). However, since heavier drops begin sliding down at a smaller tilt, which takes less time to reach, and smaller drops slide at higher tilts, which takes more time to reach, neglecting the time effect affects different drop sizes differently, thereby reducing both the accuracy and the precision of the angle variation with drop size. Thus, the results need to be rest time corrected. To obtain rest time independence, we extrapolated the experimental values of θA and θR to t ) 0 using the slopes of Figure 2. Though the rest time makes a small modification to the results, it is noted here especially in view of the eventual slopes (or lack thereof) obtained in Figure 3. The t normalized values of θA and θR are plotted in Figure 3 versus V1/3. After normalization, both advancing and receding angles seem constant with drop volume. Within the experimental scatter, linear fits to these correspond to θA decreasing and θR slightly increasing with V, and though one would not consider these meaningful based on this figure only, we mention them because we support these slopes via the Dussan equation and our force measurements later on. Lateral Retention Force versus Drop Size. To possibly obtain a size dependence of the lateral force, the corresponding time correction needs to be size specific. Indeed, Figure 4 shows the retention force for three drop sizes, and we see that the bigger the drops, the smaller is their corresponding f/V1/3. Another feature suggested from Figure 4 is that the slopes d(f/V1/3)/dt are similar. To quantify this similarity, we show in Figure 5 all those slopes for a variety of drop sizes. They seem to scatter around 2 nN/(s‚mm). Thus, we see a rather peculiar phenomenon: while the initial (t ) 0) adhesion force per unit length, f0/V1/3, differs for different drop sizes, still its variation with time, d(f/V1/3)/dt, is similar for different drop sizes. This suggests that f0 does not represent the same physical phenomenon as df/dt. It was suggested in the previous section that df/dt is induced by the unsatisfied normal component of the Young equation which deforms the surface with time. At t ) 0, however, the surface is not deformed yet, and any f0 would correspond to other phenomena such as surface inhomogeneities of either
Figure 6. f0/V1/3 vs V1/3 for hexadecane drops on OTA covered mica (O ) data; embedded line ) linear fit). The horizontal upper line shows the prediction of eq 3b for constant (average) θA and θR. The slanted upper line shows the prediction of eq 3b for θA and θR that vary linearly according to the fits in Figure 3.
topographical or chemical nature. A recent study by Amirfazli and co-workers shows that a drop placed on a tilted surface slides at a smaller tilt angle than a drop placed on a horizontal surface and then tilted.32 The Amirfazli experiment is therefore the first evidence of a force difference that is associated with the drop-surface contact history! The time effect extrapolation noted here recalls the Amirfazli experiment, since the extrapolation is to a situation of a drop on a surface that reaches the critical inclination at no time. Before we continue to explore the origin of f0, we need to address more specifically the force law that we obtain. Figure 6 shows the normalized intersections, namely the value of f/V1/3 when t f 0 (noted as f0/V1/3). According to Dussan equation (eq 3b), one does not expect any variation of the force with the drop size so long as the contact angles do not change with drop size. The blue line in Figure 6 describes the prediction of eq 3b for average constant values of θA and θR taken from Figure 3. The value of the calculated f0/V1/3 is much higher than the measured one. To realize why this is so, we compare eqs 3a and b and note that except (cos θR - cos θA) all other terms in eq 3b should be a geometrical representation of the drop’s width. Yet, they are only a function of θA (and drop volume). One reason for the discrepancy between the measured and calculated values is related to the fact that the width is also a function of θR and indeed eq 3b was originally written as a proportionality. Moreover, other parameters such as physical or chemical defects on the solid surface which eq 3b does not take into account also influence the retention force. Additionally, we see that f0/V1/3 is clearly not constant. To account for this, we plot in a green line f0/V1/3 calculated by (32) Pierce, E.; Carmona, F. J.; Amirfazli, A. Colloids Surf., A, published online Sept 25, http://dx.doi.org/doi:10.1016/j.colsurfa.2007.09.032.
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Figure 7. f0/V1/3 vs V1/3 for water drops on (a) OTA covered glass (for which θA ) 72 ( 4° and θR ) 59 ( 8°) and (b) Teflon (for which θA ) 147 ( 5° and θR ) 121 ( 7°).
substituting the fits of Figure 3 to eq 3b (Figure 6). It appears that though the slopes in Figure 3 are mild and within the scatter, they do account for the significant experimental change of f0/V1/3 with V1/3. We thus conclude that the forces are much more sensitive than the contact angles, and it is easier to predict the latter from the former than vice versa. Other systems were studied with less detail, mainly to see if f0/V1/3 increases or decreases with drop size. It was found that both water and hexadecane drops had decreasing f0/V1/3 with V when placed on OTA covered glass. The only system which showed an increase of f0/V1/3 with V was water on Teflon. The two water systems are shown in Figure 7. Reasons for the variation of f0/V1/3 with V1/3 can (among others) relate to the different contributions of the three phase contact line (which is proportional to the drop’s circumference) and the two phase contact area (drop-surface) to the total force. Additionally, the topography and chemistry of the surface can have different contributions to the circumference and the contact area. The two phase interaction can be either attractive or repulsive, resulting in increasing or decreasing total force (cf. Figure 7). Both OTA covered glass and Teflon (Figure 7) are hydrophobic surfaces, yet hydrophobic interaction is attractive. For example, oil and water molecules attract. The reason oil still does not mix with water is because oil-oil self-attraction, O2, plus water-water self-attraction, W2, are much higher than the water-oil mixed-attraction, 2OW, as explained by Israelachvili4
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(thus, the total energy is lower for the case of phase separation). Yet, simple hydrophobic interaction is attractive. For repulsive interaction, there needs to be a thin layer33,34 between the liquid and the substrate such that its refractive index is in between that of the substrate and the drop.2,4 The OTA surfactant itself may be that thin layer, or even functional groups within the surfactant35 may be that thin layer, but a generally homogeneous substrate such as Teflon is unlikely to have the unique functional groups or the ordered arrangement required for the repulsive interaction. The contact angles of the two systems are given in the caption to Figure 7. Note, however, that the contact angles are also functions of the air-surface interaction as is evident by eq 1 (Young’s). Two other parameters that can induce f0/V1/3 variation with 1/3 V are the hydrostatic and Laplace pressures, which vary with drop size. Yet another parameter is the tilt stage experiment itself which varies the normal force, f⊥, at the same time it varies the lateral force, f||. Thus, the ratio f⊥/ f|| differs for different drop sizes. Yet, regardless of the evident complexity of the problem and regardless of the exact details of the various parameters that compose the total drop-surface retention force, the drop will also need to form a vapor-liquid interface enclosure that sustains it on the substrate. That enclosure will need to adapt contact angles that obey the Dussan proportionality. The sensitivity of that proportionality to contact angle variations is very high, and thus, it is easier to deduce such small contact angle variations from force measurements than from contact angle measurements themselves. In summary, we show experimentally that the retention force normalized by drop size f/V1/3 is a function of drop size and of the time the drop was resting on the surface prior to motion (rested time). The variation of f/V1/3 with rested time, t, does not seem to be a function of drop size, but its extrapolation to t ) 0 is (f0/V1/3). For three out of four systems, f0/V1/3 was decreasing, and for one system it was increasing with drop size. Though the variation in f0/V1/3 is significant, its expression in contact angle variation is very small. Thus, it is easier to establish the variation in contact angle from force measurements than from contact angle measurements. Acknowledgment. We acknowledge Alexandra Azocar, Carl Jenkins, and Sudeep Barsin for helping with some experiments. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, and to the National Science Foundation through Grant No. DMR0619458 for support of this research. LA702473Y (33) Tadmor, R. J. Phys.: Condens. Matter 2001, 13, L195. (34) Tadmor, R.; Pepper, K. G. Langmuir 2008, 24, 3185. (35) Yadav, P. S.; Dupre, D.; Tadmor, R.; Park, J. S.; Katoshevski, D. Surf. Sci. 2007, 601, 4582.
