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Spreading Dynamics of Surfactant Solutions Maria von Bahr, Fredrik Tiberg,* and Boris V. Zhmud Institute for Surface Chemistry, Box 5607, SE-114 86 Stockholm, Sweden Received March 9, 1999. In Final Form: May 17, 1999 The spreading behavior of microdrops of surfactant solutions at solid surfaces has been studied. The influence on the spreading of different factors, such as the drop lifetime prior to surface contact, surface tension dynamics, surface energy, and surfactant properties, was systematically investigated. The results obtained suggest the existence of two spreading regimes exhibiting different spreading characteristics: In the first, nondiffusive regime, the spreading is very rapid and controlled to different extents by inertia, gravity, and capillarity, depending on the drop size, impact energy, and interfacial tension balance. It is shown in this study that the initial drop surface tension, which is set by the surface tension decay rate and the drop lifetime prior to the surface impact, strongly influences the maximum spreading distance in the nondiffusive spreading regime. The second, diffusion-controlled regime, is characterized by slower concentration-dependent spreading rates. The spreading rate is, here, mainly controlled by the diffusive transport of surfactant to the expanding liquid-vapor interface. In this regime, the drop base radius exhibits an approximate rb2 ∝ t dependence on time. The spreading kinetics at hydrophobic surfaces has been discussed in the framework of a simplified theory. Depending on the assumptions regarding the drop shape, rb5/2 ∝ t to rb2 ∝ t spreading laws are obtained. This agrees reasonably with the experimentally observed relationship.
Introduction It is often desirable to enhance the wetting ability of aqueous solutions by adding surfactants. Printing with water-borne inks, painting, manufacturing of photographic films, oil recovery, and spreading of agrochemicals are typical examples of processes where surfactants are used to promote spreading. Today’s industry is subject to a considerable pressure to adjust its processes and products to more environmentally friendly alternatives. Thus, there is a strong inducement for industry to increasingly use water-borne systems. Because most industrial processes rely on rapid spreading of the “active liquid” at the surface, dynamic aspects of the spreading process are of large interest. In contrast to simple liquids, the spreading rates of solutions are often controlled by the time evolution of the interfacial tensions or, more fundamentally, by solute adsorption rates at the different interfaces that join to form the three phase contact line (TPC).1 Since spreading of solutions is both practically important and scientifically challenging, many studies of the phenomenon have been carried out.2-16 * To whom correspondence should be directed. E-mail:
[email protected]. (1) Yaminsky, V.; Ninham, B.; Karaman, M. Langmuir 1997, 13, 5979. (2) Gau, C. S.; Zografi, G. J. Colloid Interface Sci. 1990, 140, 1. (3) Ananthapadmanabhan, K. P.; Goddard, E. D.; Chandar, P. Colloids Surf. 1990, 44, 281. (4) Hill, R. M.; He, M.; Davis, H. T.; Scriven, L. E. Langmuir 1994, 10, 1724. (5) Zhu, S.; Miller, W. G.; Scriven, L. E.; Davis, H. T. Colloids Surf. A: Physicochem. Eng. Aspects 1994, 90, 63. (6) Lin, Z.; Hill, R. M.; Davis, H. T.; Ward, M. D. Langmuir 1994, 10, 0. (7) Lin, Z.; Stoebe, T.; Hill, R. M.; Davis, H. T.; Ward, M. D. Langmuir 1996, 12, 345. (8) Rosen, M. J.; Song, L. D. Langmuir 1996, 12, 4945. (9) Frank, B.; Garoff, S. Colloids Surf. A: Physicochem. Eng. Aspects 1996, 116, 31. (10) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1996, 12, 337. (11) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7270.
Nevertheless, the understanding of the spreading mechanism of surfactant solutions is still far from complete. The aim of the present work was to elucidate the influence on the spreading phenomena of different factors, including drop lifetimes, surface tension dynamics, and surface energies. A study of the surface spreading of similar systems has been published earlier by Stoebe and co-workers.10 Some overlap exists between these two studies, but aside from some inconsistencies on certain issues, the two studies are overall complementary. A distinctive feature of the present work is also that it addresses, in a more broad sense, the entire spreading process, from the drop impact up to the wetting equilibrium. Experimental Section Instrumentation. Dynamic contact angle and spreading rates of the spreading drops were measured from side images of the drop profile which were monitored as a function of time with a DAT 1100 instrument (Fibro Systems AB, Sweden). This instrument, which is described in ref 17, gives dynamic measures of the drop base, volume, and height, as well as the contact angle. During the first seconds of spreading, the instrument collects 50 images of the spreading drop, one image every 20 ms, with the shutter open for 1 ms. Slower processes, i.e., occurring over minutes, can also be followed, collecting 5-10 images each second. The measurements were started when the drop had been formed on the tip of a syringe. Care was taken so that all interfaces in the syringe and the attached Teflon tubing had been saturated with surfactants before the actual solution used in the measurement was pumped to the syringe tip. The drop was applied on the surface by a short stroke from an electromagnet. The time between the drop formation and the stroke of the electromagnet was sometimes varied in order to study effects of surface tension (12) Stoebe, T.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7276. (13) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7282. (14) Hill, R. M. Curr. Opinion Colloid Interface Sci. 1998, 3, 247. (15) Chesters, A. K.; Elyousfi, A. B. A. J. Colloid Interface Sci. 1998, 207, 20. (16) Joanny, J. F. J.f Colloid Interface Sci. 1989, 128, 407. (17) Gerdes, S.; Cazabat, A.-M.; Stro¨m, G. Langmuir 1997, 13, 7258.
10.1021/la990276o CCC: $15.00 © 1999 American Chemical Society Published on Web 08/05/1999
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Table 1. Contact Angles for Water Drops on Thiolate-Modified Gold Surfaces with Different OH:CH3 End Group Ratios molar surface composition HS(CH2)16OH:HS(CH2)15CH3
contact angle θ rel to 18 MΩ water (deg)
0:100 15:85 25:75 40:60 60:40 75:25 100:0
108 ( 2 96 ( 4 92 ( 3 80 ( 4 70 ( 4 50 ( 3 22 ( 2
purchased from Nikko Chemicals and used without further purification. The values of the critical micelle concentration (cmc) were 1 × 10-5 mol/L and 9 × 10-4 mol/L for C14E6 and C10E6, respectively. The equilibrium surface tension values for concentrations well above the cmc, measured by the pendant drop technique, were 32 mN/m for both surfactants at 23 °C. Procedure. Experiments were performed at a relative humidity of 50 ( 3% and temperature of 23 ( 1 °C, if not otherwise stated. The drop volume varied from 2.8 to 4.0 µL. The radius of the drop base was normalized against the corresponding radius of a sphere with the same volume, i.e., rn ) rb(V)/rb(V)sphere, where V is the measured drop volume of the spreading drop. Using the normalized radius facilitated the comparison between different measurements. Note that we performed measurements with drops of different sizes (in the 1-6 µL range) to check possible volume effects. However, in this range, no effect of the volume on the initial spreading rate was observed. The surfaces were often used for more than one experiment. They were, between each measurement, cleaned for 5 min in pure ethanol in an ultrasonic bath, as well as rinsed several times with ethanol and Millipore water. The water contact angle was measured to ensure that the surface was not contaminated after this procedure. Spreading rates were calculated as the slope of the linear part of the drop base area versus time curve. A linear regression was used to get the best fit. The regression coefficient was generally better than 0.99, and the maximum deviation between spreading rates from different measurements varied no more than 5-20%, depending mainly on the rate of spreading (see below).
Results and Discussion
Figure 1. Contact angle for water drops on mixed thiol monolayer surfaces as a function of surface composition of selfassembled monolayers. relaxation. To minimize collateral inertia effects, the strength of the stroke was kept as low as possible. Dynamic surface tensions of surfactant solutions were measured with the pendant-drop technique (First Ten Angstrom 200, USA). Materials. Polished silicon surfaces were first coated with a 10-Å titanium layer followed by a 100-Å gold layer. Both layers were deposited using an electron-beam ultra-high-vacuum evaporation system (Balzers UMS 500P).18 Thiohexadecanol (HS(CH2)16OH) (g99.5%, courtesy of Prof. Liedberg, Linko¨ping University) and thiohexadecane (HS(CH2)15CH3) (>99%, Fluka) were used as received. Thiol adsorption onto the gold substrated was made from 1 mM solutions in ethanol immediately after evaporation, and incubation times were at least 15 h. This procedure results in well-ordered thiolate monolayers, in which the thiols previously have been shown to be oriented with the tail toward the fluid (air or water).19 In the case of the mixed thiol solutions, the two components have been shown to be evenly distributed over the surface.20,21 The surface energy at the surfaces, which is represented by the contact angle of Millipore water drops with a volume of 3.0-4.0 µL (see Table 1 and Figure 1), showed a linear dependence on the ratio of OH-to-CH3 termini. The roughness of all surfaces was a maximum 2 nm from peak to valley, as measured by atomic force microscopy. However, one rough unpolished surface was also used to investigate the effect of roughness. This showed peak-to-valley heights of more than 1 µm. The roughness was seen as discrete square craters (with areas of roughly 100 µm2) on the substrate surface. These were imaged with a noncontact profilometer, based on white light interferometry (New View 5010, Zygo Corporation). The surfactants used in this study were monodisperse polyoxyethylene glycol alkyl ethers (C14E6 and C10E6). These were (18) Ederth, T.; Claesson, P.; Liedberg, B. Langmuir 1998, 14, 4782. (19) Bain, C. D.; Troughton, E. B.; Tao, Y.-T.; Evall, J.; Whitesides, G. M.; Nuzzo, R. G. J. Am. Chem. Soc. 1989, 111, 321. (20) Bertilsson, L.; Liedberg, B. Langmuir 1993, 9, 141. (21) Ulman, A. Chem. Rev. 1996, 96, 1533.
Influence of Initial Concentration and Impact on Spreading. Surfactant adsorption at the drop surface inevitably depletes the drop interior of surfactant. Since spreading increases the surface-to-volume ratio, the depletion effect will increase as the spreading proceeds. This means that the measured contact angle does not necessarily have to be significantly smaller for a surfactant solution drop (with a start concentration equal to or above the cmc) than for a corresponding drop of pure water. Figure 2a shows such an example, where an initial surfactant (C14E6) concentration of 1 cmc in the drop interior is not sufficient to allow much spreading (relative to water) even though the reported surface tension of the solution is much smaller than that of water. Indeed, in the absence of surfactant excess in the bulk, the spreading and associated interfacial stretching will cause an unfavorable increase in the liquid-vapor (lv) and solid-liquid (sl) interfacial tensions. In contrast, extensive spreading is observed when the initial C14E6 concentration is increased to 10 cmc. The finial radius and contact angle at this concentration are identical with those obtained at higher surfactant concentrations. This indicates that plateau adsorption is reached at both the lv and sl interfaces. To achieve maximum spreading radius for a system with an initial surfactant bulk concentration, Cb, the following in equality has to be obeyed between the total amount of surfactant, )VCb, and the adsorbed amounts, Γ, at both the lv and sl interfaces:
AlvΓlv + AslΓsl , VCb
(1)
where Alv, Asl, and V represent the equilibrium interfacial areas and the volume of the drop, respectively. For a 4 µL droplet, the bulk depletion effect is manifested for concentration below 5 × 10-5 M, which is of the same order of magnitude as the cmc for C14E6. For surfactants with higher cmc values, such as C10E6 (cmc ) 9.8 × 10-4 M), the bulk depletion effect is not significant for microdrops (but would of course be for smaller inkjet drops). As can be seen in Figure 2b, the spreading radii are almost identical for the 1 and 10 cmc
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Figure 3. Contact angle and normalized radius versus square root of time dependencies for a C14E6 solution with a concentration of 10 cmc. The solution spreads on a 25:75 OH:CH3 substrate. The graph shows the effect of the drop lifetime.
Figure 2. Normalized radius versus time dependencies for spreading surfactant solution drops of (a) C14E6 (cmc ) 1.04 × 10-5 mol/L) and (b) C10E6 (cmc ) 9.0 × 10-4 mol/L). The concentrations used in the experiments were 1 cmc and 10 cmc, respectively. The substrate composition was 40:60 OH:CH3.
samples of C10E6. Note that the fact that no dynamic spreading is observed in the 10 cmc case only means that we have failed to access the short-time window (below 10 ms) during which the surfactant enhanced spreading actually takes place. This can be compared to the discussion about SDS and CTAB spreading on Parafilm in ref 10, in which solutions of these two surfactant were reported not to spread on hydrophobic surfaces. Because of the limited instrumental time resolution, the following study is focused on the C14E6/water system. As shown in Figure 2a, the adsorption-driven spreading of the 10 cmc C14E6/water system starts after about 1 s spreading delay. This delay is probably caused by the fact that, after the initial inertial or gravity-controlled spreading phase, the drop has reached a contact angle, which is smaller than that established by the balance of interfacial tensions at the TPC line. Therefore, further spreading driven by capillary forces does not commence until the diffusion-controlled adsorption of surfactant has produced a sufficiently large spreading force. Influence of Drop Lifetime. Figure 3 shows the spreading characteristics for surfactant solution drops that have been deposited on the surface at different times after the drop has been formed on the syringe tip. It is clear
that the spreading behavior depends strongly on the drop lifetime. The drop that was deposited “directly” after drop formation shows a clear spreading delay. The first measured contact angles are indeed approximately equal to those of pure water on the same surface. Diffusioncontrolled spreading is seen about 1 s after the drop has been placed on the surface. As is further seen in Figure 3, drops with longer prespreading lifetimes show no spreading delay. The initially measured contact angles are also significantly smaller for these drops. This can be attributed to the surfactant adsorption at the lv interface and its transfer to the sl interface during the initial nondiffusive spreading phase. It should further be noted that a drop surface with an initially high surface coverage can stretch quite extensively before the spreading process starts to manifest diffusion-controlled kinetic features. This is why it is so important to specify in detail the drop deposition procedure when studying complex fluid spreading. Effect of Surfactant Concentration and Surface Energy. Figure 4 shows typical examples of drop spreading curves measured for solutions with different concentrations. The spreading was in this case monitored on a 15:85 OH:CH3 surface. A close to linear increase of the drop with the square root of time is observed for most systems. Note again that a spreading delay is observed for the low concentration samples. The linear spreading law agrees well with the results published by Stoebe and co-workers.10 This behavior has also been observed in a number of other studies.14 The first measured rn values are nearly identical for all samples with concentration higher than 10 cmc. It can be inferred that the drop radius after the nondiffusive spreading regime is set by the amount of surfactant adsorbed at the lv interface at the moment the drop hits the substrate surface. Apart from the liquid-vapor adsorption, the initial spreading dynamics is also governed by inertia. The fact that the spreading radii start at approximately the same value for all surfactant drops with saturated lv interfaces implies that the surfactant adsorption and surface tension relaxation for the C14E6 solutions are much slower processes than the initial nondiffusive spreading process. This is not always the case. For instance, for rapidly spreading C10E6 drops, the starting values of the spreading radii differ significantly from those of C14E6 on the same surface.
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Figure 4. Normalized radius versus square root of time dependencies observed for spreading of C14E6 solutions with different concentration over a 15:85 OH:CH3 substrate.
Figure 5. Normalized radius versus square root of time dependencies observed for spreading of C14E6 solutions of different concentrations on a 75:25 OH:CH3 substrate.
Note that both systems have more or less the same equilibrium surface tension values. There is another interesting point to be stressed regarding the data in Figure 4. The drop spreading reveals the same kinetic features as the surface tension in pendant-drop measurements carried out with the same solution (see Figure 7). This shows that the long-term spreading behavior is controlled by diffusion and subsequent surfactant adsorption. As can be seen in Figures 4 and 7, the time needed for the surface tension to reach its equilibrium value exactly coincides with the time needed to reach the wetting equilibrium for the same surfactant system at the 15:85 OH:CH3 surface. One reason for such a good agreement is that the adsorption behavior at the sl and lv interfaces, having comparable interfacial tensions, is very similar. Moreover, the spreading is rather limited. Consequently, the corresponding surface reexposure effect, i.e., the effect of replenishment of the surface excess of surfactant at the expanding lv interface by diffusive transport from the interior of the drop resulting in a prolonged contact angle relaxation, is also limited. In Figure 5, the spreading behavior of C14E6 on a more hydrophilic 75:25 OH:CH3 surface is shown. On this
von Bahr et al.
Figure 6. Normalized radius versus square root of time dependencies observed for spreading of 10 cmc C14E6 solutions on substrates with different compositions.
Figure 7. Dynamic surface tension γ(t) against square root of time dependencies measured for C14E6 solutions with the pendant-drop technique.
surface, the spreading is clearly faster as compared to the 15:85 OH:CH3 surface discussed above, basically because a smaller adsorption is needed to produce the necessary surface pressure to drive the spreading. For the same reason, on more hydrophilic surfaces, the final radius appears to be larger and the corresponding contact angle smaller than on the more hydrophobic surfaces. The fact that drops spread faster and further at more hydrophilic surfaces highlights the importance of surface reexposure effects in the spreading kinetics. Despite the spreading being faster at hydrophilic surfaces, it often takes a longer time to reach a steady-state wetting (see below). The increased reexposure of surface area with increasing surface energy results in larger interfacial adsorption of surfactant and, therefore, a stronger bulk depletion at low concentrations. In support of the above conjecture regarding the surface reexposure effects is the fact that the time required to reach the steady-state wetting on more hydrophobic surfaces (0:100, 15:85, 25:75, 40:60, and 60:40 OH:CH3) increases slowly but steadily with increasing OH:CH3 ratio
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Figure 8. Spreading rates for C14E6 solution drops with different concentrations versus the substrate surface energy (expressed in terms of cos θ). Error bars show the 90% confidence intervals for the measured spreading rates.
Figure 9. Spreading rate versus concentration for C14E6 solutions on substrate with OH:CH3 ratio equal to 15:85 and 60:40, respectively. The error bars show the 90% confidence intervals.
(see Figure 6). The spreading time on, for instance, the 40:60 surface was only slightly longer than the time required to reach equilibrium surface tension in the pendant-drop experiments. For the more hydrophilic 60: 40 surface, the total spreading times of 10, 30, and 50 cmc drops were substantially longer. When the hydrophilicity of the surface increases further, the spreadings become extremely fast. Unfortunately, the equipment used in this study did not allow measurement of contact angles below about 10°, so only the first part of the spreading curve was accessible for measurement. Figure 6 shows a summary of the spreading behavior observed for the 10 cmc C14E6 solutions on surfaces with different OH:CH3 ratios. Again, we see the wetting delay phenomena on the more hydrophobic surfaces, but not on the more hydrophilic surfaces. As discussed above, we also observe that the spreading rates are generally faster on the more hydrophilic surfaces, but the total time of spreading is longer. The influence of the surface energy on the spreading rate in the adsorption controlled spreading regime is better shown in Figure 8. The spreading rate decreases at first with decreasing surface energy (given in terms of the water contact angle), and then it becomes more or less constant, or even increases slightly, for surfaces with contact angles larger than approximately 70°. We do not seesthe contrary of what was reported by Stoebe et al.10sa maximum in the spreading rate for intermediate surface energies. A possible explanation of this discrepancy could be the use of different surfactants in the two studies. However, considering the similarity of the surface tensions of the different surfactants, we expected to observe such a maximum at least in some region of the surface energies. Note also that the spreading rate of the most concentrated C14E6 solutions at the most hydrophilic surface was too rapid to be monitored by our technique (see below). In ref 10, much slower spreading was generally reported on the most hydrophilic surfaces. Other possible explanations for the differences in the spreading rate versus surface hydrophobicity curves relate to the differences in the measurement routine and possibly also in differences in substrate roughness and ambient humidity. To check whether substrate roughness affects the spreading, the spreading rates were measured for surfactant drops on surfaces with different surface rough-
nesses but the same chemical composition, 75:25 OH:CH3. Contrary to one’s expectations, no significant difference in the spreading rates was found. Apparently, as the roughness grows, the increase in the pulling force due to elongation of the triple contact line is canceled by increased viscous drag. The only exception might be if the average peak-to-valley distance turned out to be less than the thickness of the stagnant layer of liquid, in which case the flow hydrodynamics would be insensitive to the underlying surface relief. We did not observe any humidity effects on the spreading rate on hydrophobic surfaces either. The remaining insignificant difference to be checked lay in the experimental procedure. We monitored the spreading from the side, while Stoebe and co-workers did it from the top. Hence, we checked our data by tracking the spreading with a video camera positioned over the substrate surface. Except for the impossibility to monitor spreading for contact angles above 90°, no difference was detected. A final possible explanation for the differences in spreading rates reported in the two studies, in particular on the more hydrophilic surfaces, is the possibility that the spreading rates have been extracted from different parts of the spreading curve. This points to the importance of exactly stating the regime from which spreading rates are extracted. Finally, the effect of surfactant concentration on the spreading rate has been studied. The results obtained for C14E6 surfactant drops on substrates with different surface composition are summarized in Figure 9. In the measured concentration range, the spreading rate increases proportionally with the surfactant concentration. Again, this indicates a diffusion-controlled spreading mechanism in this regime. This relation should not be extrapolated to higher surfactant concentrations, when micellar growth may slow the surfactant transport. Some Remarks on the Contact Angle and Surface Tension Dynamics. When an equilibrated drop hits the surface, its kinetic energy is redistributed in a number of ways. A part of the energy is dissipated as heat, a part is transformed into the surface energy of expanding solidliquid and liquid-vapor interfaces, and, finally, a part is preserved in the form of kinetic energy of liquid flow. If the action of gravity is important, the potential energy of the drop should also be taken into consideration. Furthermore, unless the surface has been prewetted, heat
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evolution due to solvation and heat consumption due to evaporation may take place at the triple contact line and at the lv interfaces, respectively. Thus, even the spreading of pure liquids appears to be an extremely complex phenomenon, which is fundamentally governed by the global laws of conservation of mass, momentum, and energy. Mathematically, this is expressed as a set of coupled, nonlinear, partial-differential equations in terms of the velocity components, pressure, and temperature. In isothermal conditions, the energy equation is uncoupled from the momentum, and it is only the latter, known as the Navier-Stokes equations, that needs to be solved. In general, this is still a formidable task. Thus, further simplifications are normally introduced. The spreading of the so-called thick thin films of incompressible Newtonian liquids is usually described in the lubrication approximation. As applied to a drop spreading on a smooth surface, this gives22
∂h 1 ∂ + (rQ) ) 0 ∂t r ∂r
Q)
(
)
∂h ∂H 1 ∂γsl rh3 + - Fg 2γlv 3η ∂r h ∂r ∂r
(2)
where h is the film thickness, H is the mean curvature of the (hemispherical) cap, F is the density of the liquid, η is the viscosity, and g is the acceleration of gravity. On the physical side, this shows that the flow can be driven by gradients in the interfacial tensions, γsl and γlv, and film curvature, H. Surface tension gradients can most likely be produced by evaporation of volatile impurities and adsorption-desorption processes. In a different variation, the above equation has been applied to study spreading of liquid drops by a number of authors.23-25 However, although the subject has been a subject of numerous studies, both experimental and theoretical,26-30 there remain a number of points open for discussion. Therefore, given the complexity of the spreading phenomena in pure liquids, one might expect an even more hopeless situation with the spreading of surfactant solutions. The major complication comes from the fact that surfactant diffusion and adsorption at the interfaces can affect the spreading dynamics. In fact, it is the adsorption of surfactant to the sl interface that makes possible the spreading of microdrops on hydrophobic surfaces. Fortunately, the situation is not so hopeless as it seems in a special case where (i) the gravity effects can be neglected (microdrops), (ii) the surfactant concentration is quickly leveled over the lv interface (mobile adsorption layers), so that there exists no gradient in the surface tension, and (iii) the adsorption at the sl interface is essentially irreversible. Here, some specific features of the capillarity-driven spreading of surfactant solution in this specific case are highlighted, including the contact angle and surface tension dynamics for low-energy surfaces. (22) Teletzke, G. F.; Davis, H. T.; Scriven, L. E. Chem. Eng. Commun. 1987, 55, 41. (23) Lopez, J.; Miller, C. A.; Ruckenstein, E. J. J. Colloid Interface Sci. 1976, 56, 460. (24) Hocking, L. M. Q. J. Mech. Appl. Math. 1981, 5, 129. (25) Greenspan, H. P.; McCay, B. M. Studies Appl. Math. 1981, 64, 95. (26) Cazabat, A. M. Adv. Colloid Interface Sci. 1992, 42, 65. (27) Cazabat, A.-M.; Gerdes, S.; Valignat, M.-P.; Villete, S. Int. Sci. 1997, 5, 129. (28) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (29) Chen, J. D. J. Colloid Interface Sci. 1988, 122, 60. (30) Lelah, M. D.; Marmur, A. J. Colloid Interface Sci. 1981, 82, 518.
Let us consider the force vector at the contact line. This includes three components, viz., the tensions at the solidvapor (γsv), liquid-vapor (γlv), and solid-liquid (γsl) interfaces. It seems logical to expect that γsv and γsl are essentially constant, as should be the case if there is no transport through the vapor phase and the surfactant adsorbs irreversibly at the solid substrate. However, γlv varies with time owing to the time-dependent character of surfactant adsorption at the lv interface and the relaxation of the adsorbed layer. In fact, the spreading over hydrophobic substrates is enabled by preadsorption of surfactant from the surfactant-rich lv interface to the hydrophobic surface. Therefore, to restore the thermodynamic excess of surfactant at the lv interface which is continuously depleted by the carryover of surfactant from the droplet surface to the solid substrate, a diffusion flux will develop that feeds the lv interface with surfactant from the drop interior. At later stages of spreading, an axisymmetrical segment like droplet geometry can be assumed during spreading. Since the volume, V, of the droplet remains constant (no evaporation), the droplet height is approximately given by
h=
xπRV (1 - 6R1 xπRV ),
(V , R3)
(3)
R is here the segment radius. The droplet base radius is then simply rb ) R sin θ. In the case of diffusion-controlled kinetics, the time needed to restore the interfacial excess by diffusive transport of surfactants from the bulk to the surface should be of the same order of magnitude as the time needed for surfactant to diffuse through a layer of thickness h, i.e., τ ) h2/D. D is here the diffusion coefficient for the surfactant. This suggests the following relaxation kinetics for the surface tension, γlv
( )
0 eq γlv(t) ) γeq lv + (γlv - γlv ) exp -
tD h2
(4)
and the contact angle, θ
cos θ )
γsv - γsl γlv(t)
(5)
Thus, for h ) 0.1 mm and D ) 10-9 m2 s-1, the relaxation takes 10 s. Notice that the product γlv cos θ is independent of time and equal to its equilibrium value, γeq lv cos θ. To get a qualitative idea about the time dependence of the droplet base size, let us analyze the force balance for a narrow sector of spreading liquid. Assuming a steady spreading process and neglecting the gravity force, one can write
1 drb 1 sγlv cos θ ) (const)η srb 2 h dt
(6)
where s is the arc length, const is a geometrical factor of the order of unity, η is the viscosity, (1/2)srb is the area under the segment, and h-1 drb/dt is the velocity gradient. If the droplet height, h, and the contact angle, θ, were constant, the base area would change linearly with time, i.e., rb2 ∝ t. However, h is a descending function of time: h decreases as spreading progresses. Therefore, the droplet geometry needs to be known to relate h to rb. So, for a disklike geometry, h ∝ rb-2, and for a more realistic segment like geometry
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h=
x
V sin θ πrb
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(7)
If the contact angle changes only slightly, one gets rb5/2∝ t. Note that the dependence rb2 ∝ t can be deduced by from quite different positions. We assume that only surfactant monomers are capable of adsorbing at the lv interface. If monomers are generated by micellar dissociation, then the amount of monomers produced by dissociation must be equal to the amount of surfactant consumed by adsorption, i.e., kdNt ) πΓrb2. Here, kd is the micelle dissociation rate constant, N is the number density of micelles, and Γ is the adsorption capacity of the solid substrate. In this case, the micellar decay is assumed to be the rate-determining step of the spreading process. Conclusions When discussing spreading rates it is very important to specify the conditions of drop formation and the drop history prior to the actual contact with the substrate surface. The drop lifetime is important for the initial spreading, since it affects the starting value of the surface tension. Even if the driving force for spreading initially is inertia, the initial surface tension balance may act after longer times, to both promote and restrict spreading. In the diffusion-controlled regime, the spreading rate depends on the diffusive transport of surfactant to the lv
interface during spreading. On hydrophobic surfaces, where the drop area increase is limited, the contact angle relaxation time is found to be equal to the surface tension relaxation time measured for pendant drops. In the case of hydrophilic surfaces, the surface reexposure effect slows down the contact angle relaxation. However, in the diffusion-controlled regime, the measured initial spreading rates are always much faster on the more hydrophilic surfaces. The spreading rates also increase with increasing surfactant concentration, as well as the cmc of surfactant. Thus, the spreading of C10E6 solution (cmc ≈ 1 mM) appears to be too fast to be measured with our technique, whereas for C14E6 (cmc ≈ 0.01 mM), the spreading can be conveniently monitored and increases linearly with time. The effect of cmc on spreading demonstrates that the presence of micelles significantly slows the surfactant transport. A heuristic theoretical treatment of our experimental findings has also been put forward, which can be used as a starting point for a more sophisticated analysis in the future. The short-time spreading is only briefly mentioned in this work. More studies need to be carried out in the millisecond time range. Such work is currently under way. Acknowledgment. This work was supported by the Swedish Foundation for Strategic Research (SSF) and the Swedish Pulp and Paper Research Foundation. Thomas Ederth is thanked for providing the thiol surfaces. LA990276O
