Spreading of Aqueous Dimethyldidodecylammonium Bromide

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Spreading of Aqueous Dimethyldidodecylammonium Bromide Surfactant Droplets over Liquid Hydrocarbon Substrates Tatiana F. Svitova,†,‡ Randal M. Hill,§ and Clayton J. Radke*,† Chemical Engineering Department, University of California, Berkeley, California 94720-1462, and Dow Corning Corporation, Midland, Michigan 48686 Received December 4, 1998. In Final Form: June 2, 1999 The dynamics of dimethyldidodecylammonim bromide (DDAB) aqueous surfactant solutions spreading over a deep layer of liquid hydrocarbons is studied by video-enhanced microscopy at 23 °C and at different relative humidities ranging from 35 to 100%. Aqueous DDAB droplets do not spread over simple liquid hydrocarbons until the DDAB concentration exceeds 0.005 wt % where the initial spreading coefficient is positive and the aqueous DDAB solutions consist of dispersed small vesicles and large liposome-like aggregates. The rates of spreading strongly depend on surfactant concentration, even though the initial spreading coefficient is essentially constant with concentration. Neither relative humidity nor drop volume significantly influences spreading rates, although larger drop volumes do spread to larger areas. Classical tension-gradient-driven spreading theory, developed for pure, nonvolatile, and immiscible liquid spreading on a second liquid predicts lens expansion rates that are an order of magnitude higher than those experimentally observed for DDAB solutions. With the aqueous DDAB surfactant solutions, spreading eventually ceases in the form of equilibrium lenses whose areas increase linearly with both surfactant concentration and drop volume. A surprising and important observation is that substrate viscosity has only a minor effect on the rate of surfactant solution spreading. Fascinatingly, DDAB solution droplets actually spread faster on mineral oil than on dodecane, which is 18 times less viscous, even though the initial spreading coefficients of these two substrates are essentially identical. We argue that the rate of surfactant arrival at the stretching air-water and oil/water interfaces determines the droplet spreading kinetics.

Introduction Spreading of a liquid 1 over the surface of another liquid 2, according to Harkins and Feldman,1 happens when the initial spreading coefficient, S, is positive, where S is defined as

S ) σ2 - (σ1 + σ12) > 0

(1)

Here σ1 is the surface tension of liquid 1, σ2 is the surface tension of liquid 2, and σ12 is the interfacial tension between these two liquids. The generality of this rule is now well accepted.2-15 The dynamics of lens spreading on a substrate liquid is a complicated fluid mechanic process.7,9-12 For pure liquid-on-liquid spreading, a thin * To whom correspondence should be addressed. † University of California. ‡ Currently on leave from Institute of Physical Chemistry, Russian Academy of Sciences, Moscow, Leninsky pr., 31, 117915, Russia. § Dow Corning Corp. (1) Harkins, W. W.; Feldman, A. J. Am. Chem. Soc. 1922, 44 (12), 2665. (2) Davies, J. T.; Rideal, E. K. In Interfacial Phenomena; Academic Press: New York and London, 1963; pp 25-28. (3) Ellison, A. H.; Zisman, W. A. J. Phys. Chem. 1956, 60, 416. (4) Suciu, D. G.; Smigelschi, O.; Ruckenstein, E. J. Colloid Interface Sci. 1970, 33 (4), 520. (5) Ahmad, J.; Hansen, R. S. J. Colloid Interface Sci. 1972, 38 (3), 601. (6) Joos, P.; Pintens, J. J. Colloid Interface Sci. 1977, 60 (3), 507. (7) Foda, M.; Cox, R. G. J. Fluid Mech. 1980, 101 (1), 33. (8) Joos, P.; van Hunsel, J. J. Colloid Interface Sci. 1985, 106 (1), 161. (9) Camp, D. W.; Berg, J. C. J. Fluid Mech. 1987, 184, 445. (10) Joanny, J.-F. PCH, PhysicoChem. Hydrodyn. 1987, 9 (12), 183. (11) Fraaije, J. G. E. M.; Cazabat, A. M. J. Colloid Interface Sci. 1989, 133 (2), 452. (12) Jensen, O. E. J. Fluid Mech. 1995, 293, 349.

(often referred to as a monolayer7) precursor film extends beyond the main body of the lens, driven by tension gradients. The spreading radius of such precursor films depends on the initial spreading coefficient and on the supporting fluid viscosity and density, but not on the lens viscosity. When the substrate liquid is deep, the transient precursor film radius R(t) well obeys a simple 3/4 power scaling law in time5-14

R(t) ) K

S1/2 3/4 t (µF)1/4

(2)

where µ and F are the viscosity and density of the substrate liquid, respectively, and K is an experimental constant that can vary between 0.665 and 1.52.14,15 Camp and Berg9 demonstrate that this classical spreading law can be obeyed for surfactant-driven spreading of oils on water, while Bergeron and Langevin13 establish that it also applies to pure PDMS (poly(dimethylsiloxane)) lenses spreading over aqueous surfactant solutions. Although aqueous surfactant solution spreading dynamics over hydrophobic fluids is of practical importance, for instance, for fluid fuel fire extinction, experimental data for these systems are restricted to only a few publications.6,8,16,17 As noted in refs 6 and 8, the rate of aqueous surfactant solution spreading over a benzene surface is less than that theoretically predicted. The (13) Bergeron, V.; Langevin, D. Phys. Rev. Lett. 1996, 76 (17), 3152. (14) Dussaud, A. D.; Troian, S. M. Phys. Fluids, 1998, 10 (1), 23. (15) Camp, D. W. Ph.D. Thesis, University of Washington, 1985. (16) Svitova, T. F.; Hoffmann, H.; Hill, R. M. Langmuir 1996, 12, 1712. (17) Stoebe, T.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7282.

10.1021/la981683n CCC: $18.00 © 1999 American Chemical Society Published on Web 08/12/1999

Spreading of Aqueous DDAB Surfactant Droplets

authors attribute this fact due to “the smallness of the surfactant concentration and, thus, dynamic effects operate”. Svitova et al.16 establish that for nonionic surfactant solution/hydrocarbon systems the rate of spreading correlates with surface/interfacial tension dynamics and the dynamic spreading coefficient. Recently, Stoebe et al.17 report that the rate of spreading of trisiloxane nonionic surfactant solutions over mineral oil strongly depends on surfactant concentration and ethoxy chain length and that the area of the spreading lens is approximately linearly proportional to time. They also find that DDAB solutions do not spread over mineral oil surfaces. Unfortunately, the values of surface tension and viscosity of the particular mineral oil used are not recorded. An advantage of studying liquid rather than solid substrates is that of providing a molecularly smooth and homogeneous substrate surface and of allowing measurement of the interfacial tensions for all interfaces. Accordingly, a direct calculation can be made of the initial spreading coefficient, S, from eq 1. Since the surface tension of water, σ1, is ∼70 mN/m, while the interfacial tension between oil and water, σ12, can be as high as 50 mN/m, only very effective surfactants can lower these two tensions enough to make S positive and permit spreading over hydrocarbon liquids. DDAB provides such tension lowering at concentrations greater than 0.005 wt %. Also, DDAB exhibits minimal solubility in the hydrocarbon liquids studied and thus avoids the complication of surfactant loss into the substrate. Another feature of aqueous surfactant solutions spreading over hydrophobic surfaces is that a precursor surfactant monolayer spreading on the low-energy surface appears unlikely. Neither do we expect a precursor bilayer film of DDAB in the tailto-air nor tail-to-oil molecular orientation, as has been suggested for nonionic trisiloxane surfactant solutions spreading on hydrophobic solids.18-20 Double-tailed surfactants are known to form tail-to-tail bilayer structures in water. However, the presence of such tail-to-tail bilayer microstructures appears not to be strongly related to structuring in a precursor film but rather to the efficiency of these solutions in lowering the interfacial tensions at the air/water and oil/water interfaces,21 thereby providing positive spreading coefficients. The purpose of this paper is a detailed investigation of aqueous DDAB droplets spreading over simple hydrocarbon surfaces in the 0.1-1.0 wt % surfactant concentration range. New experimental spreading data are compared with the tension-gradient-driven spreading law of eq 2 and found to be in contradiction. We argue that surfactant mass transport from the bulk aqueous solution toward the spreading lens interfaces and/or surfactant adsorption dynamics are the major factors determining the rates of aqueous-DDAB-solution spreading over hydrocarbon liquids. Experiment Section Materials. Didodecyldimethylammonium bromide (DDAB), 99% purity from Eastman Kodak, is used as received. Water, distilled and deionized by a Millipore system, is used for solution preparation. DDAB suppresses the surface tension of water to low values of 22.6 mN/m at the critical vesicle concentration (CVC) and above about 0.5 wt % reduces the interfacial tension against hydrocarbons to ultralow values of less than 0.1 mN/ m.21 (18) Tiberg, F.; Cazabat, A. M. Langmuir 1994, 10, 2310. (19) Tiberg, F.; Cazabat, A. M. Europhys. Lett. 1994, 25, 205. (20) Ruckenstein, E. J. Colloid Interface Sci. 1996, 179, 136. (21) Svitova, T. F.; Smirnova, Yu. P.; Pisarev, S. A.; Berezina, N. A. Colloids Surf., A: Physico-Chem. Eng. Aspects 1995, 98, 107.

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Figure 1. Schematic of experimental apparatus for spreading dynamics. Hexadecane (F ) 0.7733 kg/m3, µ ) 3.03 mPa s, surface tension σ2 ) 31.7 mN/m, and interfacial tension against water σ12 ) 52.1 mN/m), tetradecane (F ) 0.7628 kg/m3, µ ) 2.128 mPa s, σ2 ) 30.4 mN/m, σ12 ) 51.3 mN/m), dodecane (F ) 0.7486 kg/m3, µ ) 1.383 mPa s, σ2 ) 29.3 mN/m, σ12 ) 51.1 mN/m), and mineral oil (F ) 0.838 kg/m3, µ ) 24.3 mPa s, σ2 ) 28.6 mN/m, σ12 ) 52.4 mN/m), spectroscopic grade of purity, and all obtained from Fluka, are used as the subphases for the spreading-dynamics studies. Poly(dimethylsiloxane) oil (PDMS), SF99 from GE Silicones (F ) 0.964 kg/m3, µ ) 1 Pa s, σ1) 20.9 mN/m, σ12 ) 36.2 mN/m against water) is used for pure liquid spreading experiments on water. Apparatus and Procedures. A schematic of the apparatus for measuring spreading rates is presented in Figure 1. To avoid any influence of the vessel wall we employ crystallizing dishes of 95-185 mm inner diameters depending on the final area of spreading. In most cases, the dishes are filled with a water layer 95 mm deep to eliminate any influence of the container bottom on the spreading process, which otherwise may be quite substantial.11 The substrate hydrocarbon liquid is placed atop the thick water layer and is 10 mm thick, unless otherwise noted. Droplets of DDAB solution, 0.4-1.5 mm3 in volume, are formed at the end of a 0.5 mm diameter stainless steel needle, flat cut, and attached to 2-mm3 Hamilton precision syringe for HLPC. Twenty to thirty seconds after formation, the drop is carefully placed on the substrate surface and left in contact for 0.3-0.5 s. Then the needle is retracted quickly but smoothly. We also perform several spreading experiments using a constant surfactant solution supply by an ISCO syringe pump (model 1) at a flow rate of 0.83 mm3/s, similar to the procedure outlined by Suciu et al.4 In these experiments, a constant flow of aqueous solution is provided through a 0.5-mm thin glass capillary whose tip is located about 1 mm above the oil surface. The flow rate is carefully chosen both to establish a steady jet without formation of separate drops and to keep the spreading aqueous surfactant solution directly on the surface of the oil substrate without submerging down into the bulk of the oil. Spreading events are recorded at 30 frames per second with 6-25× magnifications. A concave aluminum-reflecting sheet is positioned relative to the light source to establish reflection back onto the oil surface, thus providing both incident and reflected illumination for clear visualization of the transparent spreading lens. The video system consists of a Panasonic wv-GL-320 color video camera, with a Toyo Optics TV Zoom Lens, connected to a Panasonic AG-1980 VCR and a Trinitron, Sony color monitor. NIH version 1-59 image processing and analysis programs for

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Figure 3. Radius versus time for 1 mm3 volume drops of PDMS spreading on water. The solid line corresponds to eq 2. Figure 2. The initial spreading coefficient of aqueous DDAB solutions on mineral oil as a function of concentration. The critical spreading concentration is identified at S ) 0. the Macintosh platform define and quantify the spreading-lens diameter. We use the average of three values for each captured lens image. As illustrated in Figure 1, the entire apparatus, except for the VCR and TV monitor, is placed into an environmental chamber at room temperature. Measurements are performed at ambient (∼30%) and saturated (>95%) relative humidity. Spreading experiments are repeated at least three times for each solution/ substrate combination. The reproducibility of a transient lens diameter is (5%. In several experiments we employed small, 0.05-0.1 mm in the large dimension, Teflon particles to trace the spreading-lens edge and to check whether a thin precursor film moves ahead of the visible lens edge or not.11 Sometimes, the Teflon particles attach to the spreading-lens perimeter causing shape distortion. Hence, we utilize these particles infrequently. Interfacial Tension Measurement. Surface and interfacial tension measurements against mineral oil are from the pendantdrop method using NIH image processing software and numerical solution to the Young-Laplace equation.22,23 Detailed surface and interfacial tension isotherms for DDAB aqueous solutions against liquid alkanes are available in the literature.21 The initial spreading coefficient on mineral oil as a function of surfactant concentration is presented in Figure 2. In the DDAB solution/ mineral oil system, S becomes positive at DDAB concentrations above 0.005 wt % (1.1 × 10-4 kmol/m3), that is about three times higher than the CVC of this surfactant (3.5 × 10-5 kmol/m3 21). Only above this concentration of surfactants do solution droplets spread on the hydrocarbons. The critical spreading concentration corresponds roughly to the transition from a single phase of small vesicle-containing solutions to a biphasic region of the DDAB/ water phase diagram where small vesicles coexist with larger liposome-like aggregates.21 The presence of the large aggregates and their thermodynamic instability are the reasons that DDAB solutions are cloudy above 0.02 wt % and tend to separate into two phases when left standing for more than 1 week. For all of our experiments, we use fresh, vigorously mixed solutions, which do not exhibit any phase separation within 2 days. Complete quantitative initial spreading coefficient isotherms for the alkane (22) Padday, J. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley-Interscience: New York, 1969; Vol. 1. (23) Beverung, C. J. M.S. thesis, University of California at Berkeley, 1996.

hydrocarbons were not measured in the present work, although, according to ref 21 they appear qualitatively quite similar to those shown in Figure 2. The drop weight/volume method is used for dynamic surface and interfacial tension measurements. This method is described elsewhere.16,24,25 We adopt the procedures proposed in refs 26 and 27 for the hydrodynamic correction for drop volume. PDMS Spreading over Water. To establish that our apparatus provides a correct quantitative picture of spreading, we performed several experiments with polydimethysiloxane (PDMS) as a spreading fluid on water. PDMS is insoluble in water, and its initial spreading coefficient on water is 15.4 mN/ m. Radius histories, R(t), for 1-mm3 drops of PDMS spreading on water are presented in Figure 3 on logarithmic scales. We use the Teflon particles to trace the spreading liquid edge in the silicone-oil spreading experiments. Similar to the results in refs 11, 13, and 14, formation of a microscopically thin precursor film ahead of the macroscopic lens edge is confirmed for silicon-oil lenses spreading on water. The precursor film manifests itself in that the Teflon particles are rapidly pushed to the sides of the dish before the main lens reaches that area. Thus, open symbols in Figure 3, including a repetitive run, correspond to the transient precursor-film radius. The solid line in this figure reflects the classical spreading theory of eq 2 using the known physical properties of the fluids and using a fit value of K ) 0.75. After an initial startup phase of about 0.5 s,11 the spreading kinetics well obeys theory. Spreading of PDMS on water sustains the 3/ -power scaling with the time out to several seconds. Thus, our 4 results are in agreement with those in the literature for the spreading of the precursor film ahead of the expanding lens.11,13,14 We conclude that our apparatus provides reliable results and that eq 2, although surprisingly simple, works well for pure immiscible liquid/liquid systems.

Spreading Results for Aqueous DDAB Droplets The role of the oil-layer thickness on spreading kinetics is reported on logarithmic scales in Figure 4 illustrating how the radii of 1-mm3 aqueous lenses of 0.2 wt. % DDAB on mineral oil increase as a function of time in a saturated (24) Jho, C.; Burke, R. J. Colloid Interface Sci. 1983, 95, 61. (25) Svitova, T. F.; Smirnova, Yu. P.; Yakubov, G. Colloids Surf. A: Physico-Chem. Eng. Aspects 1995, 101, 251 (26) Miller, R.; Hofman, A.; Hartman, R.; Shano, K. H.; Halbig, A. Adv. Mater. 1992, 5 (4), 370. (27) Miller, R.; Shano, K. H.; Hofman, A. Colloids Surf., A: PhysicoChem. Eng. Aspects 1994, 92, 189.


Figure 4. Radius versus time for 1 mm3 volume drops of 0.2 wt % DDAB solutions spreading on mineral-oil layers of different thickness at saturated humidity: (1, 2) 35 mm; (3, 4) 7 mm; (5, 6) 2 mm.

humidity environment. This figure has the features that are common to many of those shown later. Similar to Figure 3 and ref 11, at early times less than about a second, there is a lag in the spreading rate as the drop first contacts the oil surface. As a rule, we deposit the drop on the oil surface 20-30 s after the drop is formed at the needle tip to allow the surface tension at solution/air interface to equilibrate. Thus, the spreading induction period is partly due to the originally high interfacial tension at the freshly formed oil/solution interface. Physical placement of the drop onto the oil surface and hydrodynamic perturbations related to this placement may also play a role. After this induction period, the spreading rate (i.e., dR/dt) increases with time, but unlike the pure liquid PDMS/water system, it does not attain a single-exponent power-law form. Indeed, the spreading rate eventually falls to zero as the lens reaches an equilibrium radius. Three different oil-layer thicknesses are studied in Figure 4 over the range from 2 to 35 mm while keeping the total fluid layer thickness constant at 105 mm by simultaneously changing the underlying water-layer thickness. Various lines simply connect the experimental points. Spreading behavior over the 7- and 35-mm oil layers is in excellent agreement. There is a slight increase in spreading rate for 2-mm oil-layer thickness. This is due to motion in the water sublayer apparently reducing the viscous drag that the higher viscosity mineral oil exerts on the expanding droplet. When the oil layer is thinner than 5 mm, Teflon particles placed at the interface between the oil and the underlying water phase are observed to move slowly backward (i.e., toward the lens center) as the lens spreads outward. For oil layers thicker than 5 mm, we do not observe any motion of the lower oil-water interface, indicating that the viscous boundary layer growing beneath the spreading lens does not reach the cushioning water subphase. Accordingly, in all the experiments to follow, we adopt an oil-layer thickness of 10 mm. We also do not observe a clearly pronounced rim surrounding the leading lens edge. Lenses appear quite flat at the late stages of spreading.

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To ascertain whether a precursor film also forms with spreading surfactant-solution drops, we sprinkled the small Teflon particles onto the hydrocarbon surface. In the case of DDAB aqueous solutions spreading over hydrocarbons, unlike for the PDMS spreading events on water, we do not find a thin precursor film forming ahead of spreading-lens edge. The Teflon particles remain immobile until the outer edge of the spreading lens reaches them, and then they move in concert with the spreadinglens front. Lack of a precursor film is consistent with our earlier arguments made in the Introduction. Accordingly, the radii in this figure, R(t), and in those to follow, correspond to the edge of the main body of the lens, as determined visually. Videographs of the lenses, formed by spreading a 1-mm3, 0.3 wt % DDAB solution drop on the mineral-oil surface, in 30% ambient humidity, are presented in Figure 5. As seen from these pictures, our videotechnique reveals clearly the edge of the spreading lenses, which is perfectly circular until the later stages of the spreading process. The small black spot seen in the center of each videograph reflects a small air bubble that was inadvertently trapped between the solution droplet and the mineral-oil surface. In videographs h and i of Figure 5 the lens spreads thin enough to permit visible interference patterns. In Figure 5i the lens apparently reduces its area. However, the Teflon particles, after moving with the spreading edge, remain fixed at the maximum distance from the lens center they originally attained and do not retract when the lens apparently shrinks. It is known that asymmetric pseudoemulsion films, unlike free foam films, of thicknesses less than 100 nm appear silver gray and become white when they are less than 50 nm in thickness.28 Thus, under the chosen lighting conditions we simply may not be seeing the outermost part of lens in videographs h and i in Figure 5. Possibly, due to water evaporation the outermost part of the lens thins into the invisible thickness range giving the apparent radius reduction seen at later times. The role of humidity is further pursued in Figure 6. Spreading-film radius histories for 0.3 wt % DDAB solutions on mineral oil are shown at saturated relative humidity in comparison with ambient (∼30%) relative humidity. Note that the radius of the spreading lens grows only somewhat faster at high humidity. More importantly, the visible final radius of a lens at 100% relative humidity is larger and stays at a constant equilibrium value. DDABsolution lenses are very stable and do not change their shape for more than 10 min when high humidity is carefully maintained. This confirms the comments made above that the apparent decrease of lens radius seen in Figure 5 is due to evaporative loss of water at ambient humidity. Thus, to prevent evaporation during droplet spreading, we pay special attention to maintain high humidity. Figure 7 reports as open and closed symbols the time dependence of the main lens radii for 1-mm3 drops of DDAB solution on mineral oil at saturated humidity for four different surfactant concentrations. Three repeat runs are recorded for the 0.5 wt % solution revealing the excellent reproducibility of our results. As noted earlier, the spreading of DDAB solutions does not commence immediately after the drop is brought into contact with the oil surface but rather after an induction period. From Figure 7 we discover that the lower the surfactant concentration is, the longer this induction period lasts. Also from the experimental data, we see that the rate of spreading increases with surfactant concentration, even though the initial spreading coefficients remain quite (28) Bergeron, V.; Radke, C. J. Colloid Poym. Sci. 1995, 273, 165.

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Figure 5. Videographs of 1 mm3 volume drops of 0.3 wt % DDAB solutions spreading on mineral oil at ambient relative humidity (∼30%).

constant for these concentrations (cf. Figure 2). Concomitantly, higher DDAB concentrations lead to larger equilibrium lens radii. A maximum occurs in the rate of spreading that both increases in magnitude and shifts to earlier time with increasing surfactant concentration. No theories to date for lens spreading predict a maximum rate, but rather a constant decrease in the rate of spreading with increasing time.

The solid line in Figure 7 reflects eq 2 for tensiongradient-driven spreading of a precursor film in the monolayer regime7,9 with a spreading coefficient of 6.6 mN/m (which corresponds to the initial spreading coefficient of DDAB solutions from Figure 2 at concentrations greater than 0.2 wt %). Lack of an observed precursor film ahead of the spreading lenses raises an important question about the applicability of eq 2 to describe the


Figure 6. Radius versus time for 1 mm3 volume drops of 0.3 wt % DDAB solution spreading on mineral oil: (1) ambient relative humidity (∼30%); (2) saturated relative humidity (>95%).

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Figure 8. Radius versus time for 0.3 wt % DDAB solutions spreading on mineral oil at saturated relative humidity and for drop volumes of (1, 2) 0.6 mm3, (3, 4) 0.8 mm3, (5, 6) 1.0 mm3, and (7, 8) 1.2 mm3.

kinematic viscosity of the substrate. Third, the tension gradients driving the spreading must vary as

d(σ1 + σ12)/dr ∼ S/R

Figure 7. Radius versus time for 1 mm3 volume drops of DDAB solutions spreading on mineral oil at saturated relative humidity and for four concentrations: (1) 0.1 wt %; (2) 0.2 wt %; (3) 0.3 wt %; (4) 0.4 wt %; (4-6) 0.5 wt %. The solid line corresponds to eq 2.

spreading behavior of the main drop radii measured in our experiments. Interestingly, we show in Appendix A that tension-gradient-driven spreading of bulk lenses may in principle obey eq 2, provided three criteria are met. First, the lens must be thin enough to be in plug flow. Equivalently, the flow resistance parameter κ ) µ1δ/µh11 must be much greater than unity, where µ1 and h are the viscosity and characteristic thickness of the lens, respectively, and δ is the characteristic distance over which viscous dissipation occurs in the substrate liquid. This criterion is well met in our experiments (cf. Appendix A). Second, δ must scale as (νt)1/2 7,9 where ν ) µ/F is the

(3)

where here the initial spreading coefficient corresponds to that measured in Figure 2. If these three criteria are met, then eq 2 provides a useful benchmark theory for surfactant solution spreading of main lenses, even when those lenses are thick enough to exhibit bulk tensions at the air/water and oil/water interfaces. Nevertheless, comparison of the R(t) data here for the DDAB-solution droplets on mineral oil with eq 2 dramatically shows that DDAB surfactant-driven spreading of aqueous lenses is much slower than theory predicts. Even for the most concentrated 0.5 wt % DDAB solutions, the rate of spreading is more than 1 order of magnitude slower than that predicted. Apparently, the mechanism(s) controlling spreading of DDAB-solution lenses on mineral oil must differ from those underlying eq 2. Figure 8 portrays the radius-time histories for 0.3 wt % DDAB solutions, but now for the drops of different volume with two runs for each drop volume. Excellent reproducibility is again seen in the repeat experiments. Changing the drop volume from 0.8 to 1.2 mm3 does not produce a noticeable effect on the rate of spreading outside the induction-period region. But the maximum equilibrium radius of the spreading lens clearly increases. Smaller drops of 0.6-mm3 volume do spread slightly slower. For a more concentrated 0.5 wt % DDAB solution, the decrease of spreading rate is only noticeable when the drop volume is reduced to 0.4 mm3. Figure 9 compares the radius histories for 1-mm3 drops of 0.2 wt % DDAB solution in 100% relative humidity spreading on mineral oil, hexadecane, tetradecane, and dodecane. The initial spreading coefficients for these systems are 6.6, 9.9, 8.4, and 7.2 mN/m, respectively. All initial spreading coefficients for the various substrates show similar behavior with DDAB concentration as that

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Figure 9. Radius versus time for 1 mm3 volume drops of 0.2 wt % DDAB solutions spreading at saturated relative humidity on four substrates: (1) dodecane; (2) tetradecane; (3) hexadecane; (4) mineral oil.

seen in Figure 2 for mineral oil. We find in Figure 9 that DDAB-laden droplets spread most quickly on mineral oil, yet this substrate exhibits the smallest initial spreading coefficient. It is especially surprising to find that higher spreading rates occur for the higher viscosity liquid substrates. In fact, the rates of spreading are just in the opposite trend for viscosity from that predicted by classical tension-gradient-driven spreading theory in eq 2. The viscosity trend in Figure 9 is remarkable. It suggests that viscous resistance in the substrate fluid is not the ratecontrolling mechanism for aqueous DDAB surfactantdriven spreading over hydrophobic liquids. Finally, in Figure 9 we observe that the aliphatic hydrocarbons yield smaller final lenses and that these lenses have a tendency to shrink at later times even at 100% relative humidity. This is especially true for the shorter chain alkanes. Figure 10 explores the role of constant surfactant supply in aqueous surfactant spreading following the technique utilized by Suciu et al.4 Here the radius history on logarithmic scales is shown for 0.2 wt % DDAB solution spreading on hexadecane and mineral oil when the volumetric supply is fixed at 0.83 mm3/s. Two different runs are shown for each substrate. We see first that no equilibrium lens size is attained when the surfactantsolution supply is continuous. Rather, the radii continue to increase in time linearly. The best fit power-law exponent is 1.0 ((0.05). Importantly, the rate of spreading of the DDAB aqueous solution is again over an order of magnitude slower compared to that predicted by eq 2. Moreover, the rate of spreading on hexadecane is essentially the same as that for mineral oil, even though the viscosity of mineral oil is much higher. Thus, in harmony with the findings in Figures 7-9, aqueous-surfactantsolution spreading over hydrophobic liquids, even at constant surfactant supply, does not follow the theory obeyed by pure, immiscible liquid droplet spreading. Discussion The case of aqueous DDAB surfactant-driven droplet spreading over simple hydrocarbon liquids differs sub-

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Figure 10. Radius versus time for 0.2 wt % DDAB solutions spreading on hexadecane and mineral oil with constant solution supply of 0.833 mm3/s. The solid line corresponds to spreading on mineral oil according to eq 2.

stantially from that of surfactant-free droplet spreading, which rather universally follows the classic tensiongradient-driven theory in eq 2. These differences are not superficial. About the only commonalty observed is that for spreading to occur at all the initial spreading coefficient must be positive. No precursor film is observed ahead of the DDAB-laden droplet, as is seen for surfactant-free lenses. DDAB-driven droplet spreading is much slower, by several orders of magnitude at the lowest concentrations studied, than the corresponding spreading rates of surfactant-free droplets, even with identical initial spreading coefficients for the two cases. Equation 2 predicts that spreading continues, whereas DDAB droplets eventually cease to spread and attain an equilibrium lens area. R(t) does not simply scale with the time raised to the 3/4 power for the surfactantladen solutions. Also, increasing the concentration of DDAB does not materially change the initial spreading coefficient, but it does substantially increase both the rate of spreading and the final equilibrium lens area. Neither effect follows from eq 2. Increasing the drop volume weakly alters the spreading rate but does increase the final lens radius, again an effect not explained by the classical tension-gradient-driven spreading theory. Surprisingly, an increase in viscosity of the substrate liquid by a factor of 11 between mineral oil and tetradecane actually increases the relative spreading rate of DDAB droplets on mineral oil by 20%. When a constant surfactant solution supply is provided to the spreading lens, spreading continues unabated, but again at a rate much smaller than that predicted by eq 2. Moreover, in this case the rate of spreading is practically independent of substrate viscosity and initial spreading coefficient, as is seen in Figure 10. Taken together, these observations suggest that the mechanism of slow surfactant-driven spreading is not that directly underlying eq 2 of a tension-gradient driving force opposed by a viscous drag in the substrate liquid. Dynamic Spreading Coefficient. Perhaps the simplest correction to make in eq 2 is that of a constant initial


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Figure 11. Variations of dynamic surface and interfacial tension with surface dilatation rate for 0.2 wt % DDAB solutions against (1) air, (2) mineral oil, (3) tetradecane, and (4) hexadecane.

spreading coefficient. As the droplet spreads, the oil/water and air/water interfaces expand. Surfactant must be supplied to these interfaces to maintain the spreading process. There may be mass transfer and/or adsorption/ desorption kinetic limitations to prevent the tension at these interfaces from achieving instantaneous equilibrium, as is tacitly assumed in adopting the initial-spreadingcoefficient values in Figure 2. One way to test this idea is to introduce a dynamic spreading coefficient into eq 2. Under dynamic conditions, the tensions at the oil/water and air/water interfaces are higher than those at the equilibrium. Following the earlier study of Svitova et al.,21 we measure here the dynamic surface tension and the dynamic interfacial tensions (against mineral oil, hexadecane, and tetradecane) of 0.2 wt % DDAB aqueous solutions using the drop-weight method. The results are plotted in Figure 11 in terms of the dynamic tension as a function of the dilatation rate θ ) d ln A/dt, where A is the surface area of the expanding drop. We obtain the dilatation rate from the approximate formula for a growing sphere

θ)

2 3

(

Q

V0 1 +

)

Qt V0

(4)

where Q is the volumetric flow rate, V0 is initial drop volume taken as 2/3πRt3,29,30 and Rt is the radius of the capillary tip. Figure 11 indicates that each of the interfaces studied obeys a power law in the dilatation rate but that the exponent is not universal. From the experimental results in Figure 11 a dynamic spreading coefficient may be calculated assuming a constant air/oil surface tension. Figure 12 shows such dynamic spreading coefficients as a function of the surface dilatation rate. At small dilatation rates, the dynamic spreading coefficient is positive, and spreading is allowed. (29) Joos, P.; Van Uffelen, M. J. Colloid Interface Sci. 1995, 171, 297. (30) Van Uffelen, M.; Joos, P. Colloids Surf. A: Physico-Chem. Eng. Aspects 1994, 85, 107.

Figure 12. Dynamic spreading coefficients versus surface dilatation rate of 0.2 wt % DDAB solutions for three substrates: (1) mineral oil; (2) tetradecane; (3) hexadecane.

However, if the interface stretches too quickly at higher surface dilatation rates, the dynamic spreading coefficient becomes negative, and spreading is no longer supported. At high dilatation rates, the surfactant cannot adsorb fast enough onto the oil/water and air/water interfaces to sustain spreading. Unfortunately, we see in Figure 12 that hexadecane has relatively the highest positive dynamic spreading coefficient and mineral oil the lowest. This is in the opposite order to the spreading velocities in Figure 9. In addition, the kinematics and mass transfer resistances in the dropweight dynamic tension measurement are unlikely to mimic those occurring during lens spreading. Thus, simple correction of eq 2 with a dynamic spreading coefficient is quantitatively in error, which again calls into question the underlying mechanisms of that model for DDAB-laden aqueous droplets spreading on hydrophobic liquids. Surfactant Adsorption Rate-Limited Spreading. Without surfactant present, water droplets do not spread on hydrophobic liquids since the spreading coefficient is quite negative. Here we offer a simple qualitative explanation of surfactant-driven spontaneous spreading of aqueous surfactant solutions over the surface of hydrophobic liquids. As the air/water and oil/water interfaces expand during lens spreading, surfactant must be supplied to maintain a positive spreading coefficient near the droplet center. At the three-phase contact line, the tension forces balance yielding a zero local spreading coefficient (cf. Appendix A). This difference in spreading coefficients, or equivalently the difference in the corresponding tensions, drives spreading by Marangoni forces. However, a limited amount of surfactant supply and surfactant mass transfer and/or sorption kinetic resistances prevent the establishment of a constant and large spreading coefficient even near the droplet center. Hence, the large tension gradients presumed in eq 3 are not maintained. For DDAB aqueous lenses spreading on hydrophobic oils, tension gradients are much smaller than those obtained in eq 3; accordingly, spreading rates are much smaller than those predicted by eq 2. The net rate of surfactant adsorption to the dilating air/water and oil/water interfaces depends on a balance among the falling net rate of surfactant arrival due to a falling concentration in the bulk solution, the

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rate of surfactant depletion due to interface expansion, and the rate of surfactant desorption and dissolution into the liquid substrate. The basic premise is that finite surfactant mass transfer and/or sorption kinetics and finite surfactant inventory limit the rate of droplet spreading. Spreading commences only when adsorption of surfactant at the air/water interface, Γ1, and at the oil/water interface, Γ12, is sufficient to lower the tensions at these two interfaces enough for the spreading coefficient to become positive (see Figure 2). Once enough surfactant adsorbs to make S just positive, the droplet begins to spread, increasing the interfacial area. Because surfactant does not instantaneously fill the newly available area, the air/water and oil/water interfacial tensions tend to rise, and, correspondingly, the transient spreading coefficient diminishes toward zero. Meanwhile, surfactant continues to arrive at the air/water and oil/water interfaces competing with the area expansion to maintain S positive. Limited rates of surfactant arrival to the expanding interfaces thus evolve a barely positive spreading coefficient and a concomitant small tensiongradient driving force (see eq 3). Hence, we propose that Γ1 and Γ12 at the lens interfaces remain essentially constant during spreading at the values reflecting interfacial tensions corresponding to S ∼ 0. It is the small values of the transient S and tension gradients attained in surfactant-driven spreading that give rise to the slow spreading rates experimentally observed. These much smaller spreading rates, compared to pure liquid drops, explain why viscous resistance to spreading is apparently not controlling the process. As more total surfactant transfers the interface during spreading, the bulk surfactant concentration in the droplet falls. Eventually, the bulk surfactant concentration drops to the point where it is now in equilibrium with the adsorption at the interfaces for the spreading coefficient near zero. This value for aqueous DDAB solutions is about c∞ ) 0.005 wt %, the critical spreading concentration labeled in Figure 2. Once the bulk droplet concentration falls to this value, spreading stops, and an equilibrium lens emerges provided evaporation is negligible (see Figures 6-9). A simple test of this idea is available from conservation of surfactant mass. If we approximate the final lens as a flat disk, then the final equilibrium lens area Amax ) πRmax2 is given by

Amax ) (c0 - c∞)V/(Γ1 + Γ12)

(5)

where V is the (initial) drop volume and c0 is the initial surfactant concentration. Since in our cases c∞ , c0 , a plot of Amax versus the amount of surfactant in the droplet N ) c0V should yield a straight line whose slope gives the adsorption at the interfaces. Figure 13 presents this plot obtained from the final lens radii in Figures 7 and 8 for the effects of DDAB concentration and aqueous drop volume on spreading over mineral oil. Both sets of data independently confirm the suggested linear behavior. If we do not discriminate between the adsorption at the air/ water and oil/water interfaces, then the slope of the Figure 13 gives an average area occupancy at each interface of 1/Γ∞ ) 49 ( 2 Å2/molecule. The literature value for saturation adsorption of DDAB at the air/water interface is 68 Å2/molecule, based on the inflection point observed for a spread monolayer surface pressure-area curve during a Langmuir-balance compression cycle25 and on small-angle X-ray scattering.31 This result suggests a less (31) Dubois, M.; Zemb, T. Langmuir 1991, 7, 1352.

Figure 13. Maximum or final area, πRmax2, of the spreading lenses as a function of DDAB amount in the spreading lens. The closed squares reflect the concentration data from Figure 7 while the open circles correspond to the drop-volume data in Figure 8. The solid line obeys eq 5.

Figure 14. Average final lens thicknesses versus DDAB concentration.

compact surface coverage than we find. However, within the accuracy of our data and within the approximations in our model, this discrepancy is acceptable. Final thicknesses of the equilibrium lenses are also available from solvent mass conservation. Figure 14 gives the calculated results from the data for spreading on mineral oil in Figure 7. At concentrations slightly above the critical spreading concentration, the final lens thickness is several microns. As the concentration of DDAB increases to an order of magnitude higher than the critical


spreading concentration, the equilibrium lens thickness falls to about 0.35 µm. Such thicknesses are consistent with the iridescent colors observed during the later stages of spreading for high concentration droplets. Figure 14 also indicates that thin-film forces need not be invoked to explain the spreading dynamics. The proposed surfactant adsorption rate-limited spreading mechanism accounts only for surfactant accumulation at the expanding interfaces. If the surfactant is soluble in the subphase, then by extension spreading rates should be even slower than when there is no surfactant loss to the substrate. Also, when surfactant can dissolve into the subphase, no stable equilibrium lens can be formed, and the lens must shrink in size at later times, even in a saturated humidity environment. These arguments are confirmed by the data for the aliphatic hydrocarbons seen in Figure 9. DDAB is quite insoluble in mineral oil. Although we do not have quantitative solubility data, our observations indicate that DDAB does dissolve in the alkane hydrocarbons and that this solubility increases with a decrease in alkane carbon number. Thus, our explanation for the apparent inverse role of viscosity on the spreading rates seen in Figure 9 is that of surfactant loss to the substrate and subsequent loss of the driving force for spreading. When surfactant loss due to dissolution in the substrate is minimized because of constant surfactant supply, the rates of spreading are essentially the same for substrates of substantially different viscosity (see Figure 10). In our simplified mechanism for slow surfactant-driven spreading, substrate viscosity plays no important role. Of course, if it is possible to raise the concentration of surfactant high enough so that spreading rates begin to approach those characteristic of pure, immiscible liquid drops, then surfactant-supply rate limitations diminish and viscous losses likely become important again. Finally, our proposed slow surfactant-driven spreading mechanism is consistent with the constant-surfactantsupply results in Figure 10. Here there is minimal surfactant depletion from the bulk lens and hence no slowing to an equilibrium lens. Nevertheless, the rate of spreading is still controlled by mass transfer and/or kinetic sorption rate limitations. Accordingly, the spreading rates in Figure 10 remain an order of magnitude smaller than those for pure liquids. Conclusions A systematic study of aqueous DDAB solutions spreading over liquid hydrophobic surfaces is performed over a wide surfactant concentration range and at different humidities. Under saturated humidity conditions, the formation of a thin precursor film, moving ahead of the main thick film edge, is not observed. Humidity control is important because of solvent (water) evaporation from the leading film edge making it thinner than about 50 nm and invisible to observation. Thus, the spreading lens appears to shrink at later times. The rate of aqueous DDAB solution spreading strongly increases with increasing surfactant concentration, regardless of an almost constant initial spreading coefficient in the concentration range studied. Drop volume changes in the range of 0.8-1.2 mm3 do not produce a noticeable effect on the rates of spreading, whereas the final radius of the spreading lens increases linearly with drop volume. A similar linear increase of equilibrium lens radius is found for increasing surfactant concentration. Variations in the subphase viscosity do not produce a major effect on the rate of DDAB-solution spreading, but

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the observed effect is surprisingly opposite to that predicted by current tension-gradient-driven spreading theory. Surfactant solutions spread faster on mineral oil, having a higher viscosity, than on dodecane, tetradecane, and hexadecane, which are much less viscous. We hypothesize that surfactant dissolution into the shorter chain hydrocarbons slows droplet spreading in comparison to mineral oil. Spreading of DDAB solutions with constant surfactantsolution supply do not slow to an equilibrium lens, but nevertheless the spreading rate is still ∼15 times smaller than that theoretically calculated using the initial spreading coefficient. Comparison with pure liquid spreading on immiscible liquid substrates reveals that surfactant solutions spread drastically slower than pure liquid droplets even when the initial spreading coefficients are comparable. Correction of the classical tension-gradient-driven theory with a dynamic, rather than an initial, spreading coefficient proves inadequate. We propose a simple, surfactant rate-limited adsorption mechanism to explain our measured spreading results. Surfactant supply and supply rate limitations to the expanding interfaces compete with the dilatation of the spreading interfaces to maintain an effective spreading coefficient that is positive but close to zero. It is the small value of the transient spreading coefficient and of the corresponding small tension gradients that give rise to the slow expansion rates observed in surfactant-driven droplet spreading. Acknowledgment. This work was partially supported by the U.S. Department of Energy under Contract DEAC03-76SF00098 to the Lawrence Berkeley National Laboratory of the University of California. Financial support to Dr. T. F. Svitova was supplied by the Dow Corning Corporation, Midland, MI. Appendix A: Tension-Gradient Spreading of a Thin Surfactant Laden Lens on an Immiscible Liquid Substrate in the Monolayer Regime Consider a nonwetting liquid lens containing enough surfactant to permit spreading of the lens over an immiscible liquid substrate of infinite depth. We focus on tension-gradient-driven spreading when the thickness of the spreading lens is thin, but still macroscopic. By macroscopic we mean that, for example, continuum mechanics holds in the lens, interfacial tensions are their bulk values, and thin-film forces (i.e., disjoining pressures) are negligible. Under these circumstances, we establish that that the lens may still be thin enough to be in the “monolayer” spreading regime of Foda and Cox.7 Let z be the linear coordinate directed vertically from the lens/substrate interface into the lens, and let r be the radial coordinate originating at the center of the expanding lens whose terminus radius is R(t). Since only tension gradients drive the flow, we have in the lubrication approximation that ∂2νr(r,z,t)/∂2z ) 0, where νr is the radial velocity in the lens. Hence, at pseudo steady state the radial velocity profile is locally a straight line. At the lens/ air interface where z ) h, with h defining the thickness of the lens, the stress continuity boundary condition reads

µ1(U - Us)/h ) dσ1/dr

(A1)

at z ) h where µ1 and σ1 are the viscosity and surface tension of the lens. In eq A1, U is the surface velocity of the lens at

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z ) h, and Us is the surface velocity at the lens/substrate interface where z ) 0. We demand that σ1 be higher near the periphery of the expanding lens than near the center, thus driving the lens spreading. A similar tension-gradient driving force may exist also at the lens/substrate interface. Thus, the corresponding stress boundary condition at z ) 0 reads

µUs/δ - µ1(U - Us)/h ) dσ12/dr

(A2)

at z ) 0 Again, we anticipate that σ12 increases in value toward the drop periphery thus driving spreading. In this expression, δ reflects the characteristic distance over which the viscous resistance of the substrate is evident. Equations A1 and A2, along with the identification that dR/dt ) (U + Us)/2, relate the spreading rate to the tensiongradient driving forces

µdR/dt (1 + 2κ) ) dσ1/dr + dσ12/dr δ 2κ

(A3)

where the important parameter κ ) µ1δ/µh arises. κ gauges the hydrodynamic resistance of the lens relative to that of the substrate.11 Thus, when κ . 1 the lens viscous resistance is high enough that it essentially slips in plug flow over the substrate. In this case, all viscous dissipation occurs in the substrate. Conversely, when κ , 1 the substrate behaves as a solid with all viscous resistance occurring in the lens. This regime is that of a droplet spreading over a solid surface. We define the so-called monolayer-spreading regime to be that when the lens flows as a plug or when κ . 1. For the monolayer-spreading regime, Foda and Cox establish that a Blasius hydrodynamic boundary layer grows underneath the spreading film and that, except near the leading edge, δ ∼ (νt)1/2, where ν is the kinematic viscosity of the substrate liquid.7,12 It is important to establish which regime corresponds to our spreading data. If we take water for the lens and mineral oil for the substrate, then µ1/µ ) 0.05. The hydrodynamic boundary layer thickness after about 1 s of spreading is approximately 1 cm. The largest value for h is that initially or about 0.1 mm while near the final stages of spreading h can be less than 1 µm (see Figure 14). Hence, we find that 5 e κ e 500. Accordingly, except for the very early stages of spreading, our lens-expansion data fall into the monolayer-spreading regime.

In this limiting regime, eq A3 reduces to

µdR/dt ∼ d(σ1 + σ12)/dr xνt

(A4)

Equation A4 is the traditional stress boundary condition for the monolayer-spreading model.7,9,11 Here it is assumed that although the monolayer-spreading regime holds, the tensions at the air/lens and lens/substrate interfaces correspond to their bulk values. They can vary along the droplet only when the surfactant adsorption densities also vary along the droplet. It now remains to discuss the physical behavior of the tension gradients appearing in eq A4. To arrive at eq 2 of the text it is necessary to assume that the tensions in the spreading lens change over the scale of the droplet radius

d(σ1 + σ12)/dr ∼

[(σ1 + σ12)r)R - (σ1 + σ12)r)0] (A5) R

But by the definition of the spreading coefficient, S ) σ2 - σ1 - σ12, we recover d(σ1 + σ12)/dr ) [Sr)0 - Sr)R]/R. We next approximate Sr)0 as S, the initial spreading coefficient. At the leading edge of the spreading lens, tension forces balance so that Sr)R ) 0 yielding finally

d(σ1 + σ12)/dr ∼ S/R

(A6)

Substitution of eq A6 into eq A4 followed by integration in time gives eq 2 of the text. Thus, even though our spreading lenses exhibit bulk films several micrometers in thickness, they, nevertheless, may exhibit the tension-driven mechanism of a strict molecular mono- or multilayer. Three criteria are necessary for this result. First, the viscous resistance in the lens must be larger than that in the substrate, κ . 1, so that plug flow exists in the spreading lens. Second the substrate hydrodynamic boundary layer is of the Blasius form.7,9,12 Third, there must be a radial surfactant adsorption density variation in the expanding lens such that eq A6 is obeyed. That such bulk variations in tensions are indeed possible in micrometer-thickness spreading lenses with surfactants present is established in the work of Camp and Berg when there is ample supply of surfactant and local equilibrium prevails between the surfactant and the expanding interfaces.9,15 LA981683N

Spreading of Aqueous Dimethyldidodecylammonium Bromide

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