Unstable Spreading of Aqueous Anionic Surfactant Solutions on

Jan 8, 2003 - instability developing behind the spreading front. In this paper, the role of solubility on this instability is investigated by conducti...
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Langmuir 2003, 19, 703-708

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Unstable Spreading of Aqueous Anionic Surfactant Solutions on Liquid Films. 2. Highly Soluble Surfactant Abia B. Afsar-Siddiqui, Paul F. Luckham, and Omar K. Matar* Department of Chemical Engineering & Chemical Technology, Imperial College of Science, Technology & Medicine, London, SW7 2BY, UK Received April 20, 2002. In Final Form: September 20, 2002 The spreading of a surfactant solution across a thin water film may be accompanied by a fingering instability developing behind the spreading front. In this paper, the role of solubility on this instability is investigated by conducting spreading experiments using highly soluble surfactant solutions of sodium dodecyl sulfate over a range of concentrations on water films ranging from 25 to 100 µm in thickness. It is found that at surfactant concentrations up to and around the critical micelle concentration, spreading is largely accompanied by fingers upstream of the spreading front. In comparison with sparingly soluble AOT solutions (sodium di-2-ethylhexyl sulfosuccinate) studied in part 1 of this series, the instability becomes apparent sooner and the fingers are more pronounced and branched. Above the cmc, the instability takes a different form on thinner films, which was not noted in the sparingly soluble case, while fingering develops on thicker films.

1. Introduction There has been a significant amount of both theoretical and experimental work carried out in the field of spreading surfactant solutions on thin liquid films, which is motivated because of applications including drug delivery1 and coating flows.2 When surfactant solution comes into contact with a thin homogeneous liquid film, the surfactant adsorbs at the air-liquid interface, leaving a surfactantrich region surrounded by relatively uncontaminated liquid. The Marangoni stresses induced at the surfactantliquid junction advance both the surfactant and the underlying liquid toward regions of higher surface tension, thus causing deformations in the liquid layer. The strength of the Marangoni stresses is proportional to the surface tension gradient and the thickness of the underlying film. In part 1 of this study,3 the case of insoluble/sparingly soluble surfactant was considered, where a shocklike structure develops at the leading edge of the spreading front of height almost twice the undisturbed film thickness together with corresponding thinning upstream.4,5 If the surfactant is soluble and the solid wall beneath the liquid layer absorbs surfactant, then the continual absorption of surfactant at the wall reduces surfactant concentration gradients at the interface, thus reducing spreading rates and decreasing deformations. At long times there may be a region in which backflow arises even with negligible gravitational effects.6 In the case of a soluble surfactant where the solid wall is impermeable to surfactant (the case considered in this paper), the surfactant will desorb from the surface to the bulk until both bulk and surface concentrations are in local equilibrium. The behavior of the front is dependent on the sorption kinetics. If desorption is rapid, then an advancing pulse of fluid develops instead of the shocklike structure of the * Corresponding author. E-mail: [email protected]. (1) Shapiro, D. L. In Surfactant Replacement Therapy; AR Liss: New York, 1989. (2) La Due, J.; Muller, M. R.; Swangler, M. J. Aircraft 1996, 33, 131. (3) Afsar-Siddiqui, A. B.; Luckham, P. F.; Matar, O. K. Langmuir 2003, 19, 696. (4) Borgas, M. E.; Grotberg, J. B. J. Fluid Mech. 1988, 193, 151. (5) Gaver, D. P.; Grotberg, J. B. J. Fluid Mech. 1990, 213, 127. (6) Halpern, D.; Grotberg, J. B. J. Fluid Mech. 1992, 237, 1.

insoluble case. The height of this pulse can be in excess of 3 or 4 times the undisturbed film thickness.7 As the solubility of the surfactant increases, the upstream slope of the pulse becomes increasingly steep. The downstream slope is less affected by solubility and so is similar to the insoluble case. If desorption is slow, initially the surfactant will spread as in the insoluble case. Then as desorption begins to occur preferentially from regions of high surface surfactant concentration to the bulk, spreading rates are reduced. Once surface and bulk concentrations are in local equilibrium, the pulse of fluid develops.7 Surface deformations in the soluble case are therefore more severe than in the insoluble case, as shown in Figure 1. While solubility affects the flow patterns, it does not have a significant effect on the spreading exponent and the t1/4 prediction made by Grotberg and co-workers from the insoluble study remains valid.7,8 Marmur and Lelah9 first reported seeing the fingering patterns accompanying the spreading of various aqueous surfactant solutions on what they believed to be dry glass. Their observations on the anionic surfactant SDBS showed uniform circular spreading at concentrations below the critical micelle concentration (cmc). However, spreading at concentrations above the cmc was accompanied by “fingers” of surfactant originating near the point of original deposition, which appeared to branch as they developed. Frank and Garoff10 observed the development of fingers when a negatively charged silicon oxide substrate was brought into contact with a reservoir of sodium dodecyl sulfate (SDS) solution in a vertical geometry. The fingers were seen to propagate several millimeters up the sample before stopping. When spreading the same solution on a positively charged sapphire substrate, however, autophobing occurred. Cazabat and co-workers11-14 studied the nonionic CnEm surfactants in ethylene and diethylene (7) Jensen, O. E.; Grotberg, J. B. Phys. Fluids 1993, 5, 58. (8) Jensen, O. E.; Grotberg, J. B. J. Fluid Mech. 1992, 240, 259. (9) Marmur, A.; Lelah, M. D. Chem. Eng. Commun. 1981, 13, 133. (10) Frank, B.; Garoff, S. Langmuir 1995, 11, 87. (11) Bardon, S.; Cachile, M.; Cazabat, A. M.; Fanton, X.; Valignat, M. P.; Villette, S. Faraday Discuss. 1996, 104, 307. (12) Cachile, M.; Cazabat, A. M. Langmuir 1999, 15, 1515. (13) Cachile, M.; Cazabat, A. M.; Bardon, S.; Valignat, M. P.; Vadenbrouck, F. Colloids Surf. A 1999, 159, 47.

10.1021/la025851u CCC: $25.00 © 2003 American Chemical Society Published on Web 01/08/2003

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Figure 1. Schematic diagram of the liquid height profile, H, for (a) an insoluble spreading surfactant solution and (b) a soluble spreading surfactant solution where Ho is the undisturbed film thickness and R is the instantaneous extent of surfactant spreading. Note that in the soluble case, the upstream slope is steeper and the leading edge resembles a pulse.

glycol over a range of relative humidity. They observed profusely branching fingers at the edge of the surfactant drop. Other groups who have reported seeing fingering with surfactant solutions include Nikolov et al.15 and Stoebe et al.,16 who both used aqueous trisiloxane surfactant solutions on various substrates. There have been several studies modeling the fingering instability in an attempt to understand the driving mechanism. Most recently, a transient growth analysis of insoluble surfactant spreading was conducted to probe early time dynamics in the presence of Marangoni, capillary, and diffusion forces.17,18 This showed that an explosive growth of disturbances in the film thickness occurred on a time scale of fractions of a second. Disturbances of all wavelengths decay eventually. In the presence of van der Waals forces, the amplification of initially small transverse disturbances was enhanced, leading to sustained growth consistent with experimental patterns.19 Having investigated the effect of varying surfactant concentration and initial film thickness on the behavior of a spreading drop of a sparingly soluble anionic surfactant, AOT,3 this study reports on experimental results using a highly soluble anionic surfactant solution, SDS, to understand the role of solubility. As before, surfactant concentrations above and below the cmc are deposited on water films up to 100 µm in thickness, and the effect of this on the instability onset time and pattern wavelength is examined. 2. Experimental Details 2.1. Materials. The liquid substrate was ultrapure water obtained from a Barnstead NANOpure II filter system with a resistivity of 18 MΩ cm and a surface tension of 72.2 ( 0.5mN/m at 25 °C. This was used to clean the glassware and syringe as well as to prepare the surfactant solutions. The surfactant used was SDS (sodium dodecyl sulfate, MW 288.4, 99+%, Aldrich), which is a highly soluble anionic surfactant with a cmc of 8 × 10-3 M.20 The solubility parameter, β ) ka/ kdHo,7 where Ka and Kd are adsorption and desorption coefficients of the surfactant respectively, is on the order 10-2 at a range of concentrations below the cmc.21 This indicates the high bulk solubility of SDS. The surface tension of the surfactant solutions was determined using a platinum Wilhelmy plate suspended (14) Cachile, M.; Schneemilch, M.; Hamraoui, A.; Cazabat, A. M. Adv. Colloid Interface Sci. 2002, 96, 59. (15) Nikolov, A. D.; Wasan, D. T.; Chengara, A.; Koczo, K.; Policello, G. A.; Kolossvary, I. Adv. Colloid Interface Sci. 2002, 96, 59. (16) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7270. (17) Matar, O. K.; Troian, S. M. Phys. Fluids 1998, 10, 1234. (18) Matar, O. K.; Troian, S. M. Phys. Fluids 1999, 11, 3232. (19) Matar, O. K.; Troian, S. M. Chaos 1999, 9, 141. (20) Mukerjee, P.; Mysels, K. J. National Bureau of Standards, US Dept. of Commerce, Washington, DC, 1970. (21) Chang, C. H.; Franses, E. I. Colloids Surf. A 1995, 100, 1.

Figure 2. Schematic diagram of the experimental setup. Table 1. Variation of Spreading Behavior with Initial Surfactant Concentration and Water Film Thickness initial surfactant concn (mM) 0.32 (0.04 cmc) 3.2 (0.4 cmc) 6.4 (0.8 cmc) 9.6 (1.2 cmc) 12.8 (1.6 cmc) 22.4 (2.8 cmc) 32 (4 cmc)

spreading behavior 25 µm

50 µm

100 µm

stable spreading fingering stable fingering fingering “disk” instability fingering “disk” instability fingering “disk” instability fingering

from a Kruss microbalance at a constant temperature of 25 °C. The values obtained were in agreement with published data.20 2.2. Visualization Technique. All the spreading experiments were carried out in a glass Petri dish fitted with an optically flat bottom. This was illuminated from above using a fiber optic lamp, and the resulting image was projected onto a tracing paper screen placed beneath the Petri dish. The images were recorded using a 120 Hz CCD progressive scan camera (Pulnix TM6710) via a mirror. The setup has been described in greater detail previously3 and is illustrated in Figure 2. 2.3. Experimental Procedure. After cleaning the Petri dish with RBS50 detergent (Chemical Concentrates Ltd.) and thorough rinsing with ultrapure water, the water film thickness was evaluated gravimetrically as previously described.3 A 9-µL drop of aqueous surfactant was contacted with the water film using a clean 20-µL precision Hamilton syringe. The spreading was followed for about 4 s after deposition, and the images were analyzed using commercially available software. Each spreading run was repeated three times to ensure good reproducibility with complete cleaning prior to each run.

3. Results The spreading of aqueous SDS solutions over a wide range of concentrations on water films gives rise to three distinct types of behavior, which are presented in Table 1. At very low surfactant concentrations, spreading is stable. Increasing the surfactant concentration up to and around the cmc results in fingered spreading. At still

Unstable Spreading of Aqueous Anionic Surfactant

Figure 3. Variation in the spreading exponent with initial surfactant concentration and water film thickness: (b) 25 µm, (9) 50 µm, (2) 100 µm. These points represent the average of the three runs, and the error bars arise from the uncertainty in the measurement of the radius of the spread drop. Table 2. Variation in Fingering Instability Onset Time with Surfactant Concentration and Initial Film Thickness initial surfactant concn (mM) 0.32 (0.04 cmc) 3.2 (0.4 cmc) 6.4 (0.8 cmc) 9.6 (1.2 cmc) 12.8 (1.6 cmc) 22.4 (2.8 cmc) 32 (4 cmc)

onset time (s) 25 µm

50 µm

0.125 0.10 0.075

0.17 0.14 0.12

100 µm

0.15 0.13 0.12 0.11 0.11

higher concentrations, above the cmc, two different types of behavior are observed, depending on the initial thickness of the underlying film. On the thinner films, a “disk” of liquid remains in the center at the point of original surfactant deposition from which protrusions extend. Increasing the film thickness results in the fingering instability. Typically the speed of spreading is about 0.5 cm/s. Spreading rates increase with increasing concentration up to 1.2 cmc and decrease thereafter. There is no great variation in spreading rates with film thickness, despite reduced viscous dissipation effects at the higher thicknesses. Given that the radius of the spreading front, R, advances with time, t, raised to an exponent, R, the exponents for each of the spreading runs were determined from a logarithmic plot of the radius of the spreading front against time for the same time duration in each case. These results are shown in Figure 3. The spreading exponents are broadly in agreement with the t1/4 prediction made by Grotberg and co-workers.5,8 3.1. Evolution and Onset of Fingering Instability. Figure 4 shows the evolution of the spreading drop and the onset and development of the fingers with time. Almost immediately after deposition, the thickened front can be seen as a dark ring, which initially spreads uniformly (Figure 4a). At a time which depends on the initial surfactant concentration and film thickness, a lighter front becomes visible behind the thickened rim and it is from this thinned region that small protrusions grow and develop (Figure 4b). They appear to be pushing the dark front, which becomes distorted. The fingers grow and branch (Figure 4c), and at later times, the thin region is barely discernible (Figure 4d). The bright, white region beyond the spreading front appears to be an optical effect. The time taken for the fingers to become visible as a function of the surfactant concentration and initial film thickness is presented in Table 2. The fingering instability onset time decreases with increasing surfactant concentration and with decreasing film thickness. Note that the onset times for SDS are an order of magnitude faster than those observed for AOT.3

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In addition to the instability onset time, the average width of the fingers has been measured using image analysis software on developed profiles. The fingers on the thinner films were found to be broader at the base and narrower at the tips. The measurement taken is that at the tip. The variation in finger width with initial film thickness and surfactant concentration is presented in Figure 5. There is an increase in finger width with decreasing surfactant concentration and increasing film thickness. Figure 6 shows qualitatively the difference in spreading patterns that arise at different film thickness when a surfactant solution of 1.2 cmc SDS is deposited on a water film. With increasing film thickness, the fingers become increasingly round-tipped, shorter, straighter, and wider. The distance between the tips of the fingers and the thickened front is smaller and the front itself is more circular and less corrugated. A similar effect is seen with decreasing surfactant concentration, as shown in Figure 7. 3.2. Evolution of “Disk” Instability. Surfactant deposition at concentrations above the cmc (1.6, 2.8, and 4 cmc) on 25 and 50 µm films gives a different type of spreading behavior. Figures 8 and 9 show the evolution of this instability for 1.6 cmc on a 25 µm water film and 2.8 cmc on a 50 µm water film, respectively. Figure 10 shows side views for the same parameters as those used to generate Figure 9. In the fingering case described in section 3.1, there remains no remnant of the original surfactant deposition. For the cases depicted in Figures 8 and 9, however, a central “disk” or cap remains in the center, which is clearly shown by the side views in Figure 10. Protrusions then appear to extend from the edge of this disk, the shape and onset time of which seem to be determined by the surfactant concentration and film thickness. These protrusions are long and straight on a thinner film, while they appear sooner and much shorter on the thicker film. In this respect, the protrusions behave in a fashion similar to the fingers. However, growth of fingers leads to corrugation of the thickened front (see Figure 4b), whereas the protrusions do not affect the spreading front, which remains circular (see Figures 9 and 10). The disk itself does not grow appreciably with time; the spreading front, however, does grow. The early time pattern in Figure 8a bears some resemblance to patterns obtained by Fournier and Cazabat,22 who observed forklike dendrites at the bulk edge of a water-ethanol mixture, which gave rise to a solutal Marangoni effect. There is a darker zone at the bulk edge that is interpreted as a depression in the liquid surface. The side views in Figure 10 do not readily indicate that there is a depression; it is possible that the darker zone results from the curvature at the edge of the lens. 4. Discussion The spreading exponents shown in Figure 3 are in good agreement with the t1/4 prediction.5,8 This indicates that spreading is driven primarily by Marangoni convection at all film thicknesses and surfactant concentrations. However, the qualitative difference in behavior over the range of concentrations and thickness is due to the strength of Marangoni stresses in relation to other forces. The experimental results show that at very low surfactant concentrations spreading is stable. This is because the surface tension gradient is too low to generate sufficient Marangoni flows to initiate the fingering instability. At higher concentrations up to the cmc, Marangoni stresses (22) Fournier, J. B.; Cazabat, A. M. Europhys. Lett. 1992, 20, 517.

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Figure 4. The development of the fingering pattern when a 9-µL drop of 1.2 cmc (9.6 mM) SDS is deposited on a 25 µm water film: (a) formation of the thickened front at 0.067 s, (b) onset of fingering from the thinned region at 0.14 s, (c) development of the fingers at 0.31 s, and (d) fully developed fingers at 2 s. Corresponding side views are shown in parts e, f, g, and h, respectively.

Figure 5. Variation in finger width with variation in surfactant concentration and water film thickness: ([) 1.2 cmc, (9) 1.6 cmc. Each value is the average of 30 measurements over three runs. The solid lines are a power law fit to the data. The exponents are ([) 0.61 and (9) 0.63. The regression coefficients are 0.99 for both sets of data.

Figure 6. Fingering patterns produced after 0.5 s when a 9-µL drop of 1.2 cmc (9.6 mM) SDS is deposited on (a) a 25 µm water film and (b) a 100 µm water film.

are strong enough to cause fingering, which is reflected by the shorter instability onset times and narrower, more branched fingers. With increasing film thickness, the fingers are broader and straighter and have a longer onset time. This can be explained by considering the effect of sorption kinetics.3 Although sorption kinetics constants do not change with concentration up to the cmc (both ka and kd increase at the same rate21), the solubility parameter, β ) ka/kdHo (see section 2.1), decreases with increasing film thickness, implying that the time scale for adsorption is larger than that for desorption. The

increased rate of desorption, therefore, diminishes the magnitude of the Marangoni driving force on thicker films and exerts a stabilizing influence on the spreading process. This may counteract the decrease in viscous drag contributing to the observed weak dependence of the spreading rates on the film thickness. At concentrations above the cmc, the reverse trend occurs, with fingering occurring on the thicker films and a different instability developing on the thinner films. At these concentrations the sorption kinetics are different, because of the large number of micelles in the bulk. This

Unstable Spreading of Aqueous Anionic Surfactant

Figure 7. Fingering patterns produced after 0.5 s when a 9-µL drop of (a) 0.4 cmc (3.2 mM) and (b) 1.2 cmc (9.6 mM) SDS is deposited on a 25 µm water film.

Figure 8. Spreading pattern produced after (a) 0.08 s and (b) 3.5 s when a 9-µL drop of 1.6 cmc (12.8 mM) SDS is deposited on a 25 µm water film.

Figure 9. Spreading pattern produced after (a) 0.1 s and (b) 3.5 s when a 9-µL drop of 2.8 cmc (22.4 mM) SDS is deposited on a 50 µm water film.

results in a higher rate of surfactant adsorption from the bulk to the interface. This reduces the magnitude of the surface tension gradients, which is reflected in the lower spreading rates observed above the cmc. However, on thicker films the surface cannot be replenished rapidly enough to suppress Marangoni flow, even at these high concentrations, so fingering occurs at this thickness. The reasons for the instability at the edge of the “disk” are not yet understood. The surface and bulk Peclet numbers, showing the relative strength of Marangoni stresses to surface or bulk diffusion, are on the order of 106 and 103 respectively.

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While confirming the dominance of Marangoni forces, these also indicate that bulk diffusive transport is relatively more significant than surface diffusion (more so on thinner films). Gravitational forces are characterized by the Bond number, which relates hydrostatic pressure to spreading pressure. This is on the order of 10-3 for the thinner films, rising to ∼0.2 on the 100 µm film at the lower surfactant concentrations, accounting for the elevations in film thickness and the relaxation of the concentration gradients. The fact that significant variation in spreading rates was not found with film thickness, despite reduced viscous retardation on thicker films, could also be brought about by gravitational forces, which reinforce the effects of surfactant desorption in counterbalancing the reduced viscous effects on thicker films. Moreover, spreading exponents for surfactant concentrations up to the cmc on the 100 µm film were found to be closer to 0.2. The exponent takes this value when gravitational effects become significant (derived in part 1 of this study3), thus further indicating that these effects may be significant on the thicker films, particularly at the lower concentrations. The dimensionless Hamaker constant, which gives the relative strength of van der Waals forces to Marangoni convection, achieves values on the order of 10-11. While this does not appear to be significant, it does not account for the substantial degree of thinning that can occur in the film. It was previously demonstrated3 that in the case of dominant Marangoni forces, the finger width, λ, is related to the initial film thickness, Ho, by the following simple relation derived using scaling analysis: λ ∼ Ho2/3. In the case of significant van der Waals forces, this relation becomes λ ∼ Ho2. The results shown in Figure 5 show good agreement with the 2/3 scaling law, indicating that the spreading is Marangoni-driven. However, this does not rule out the possibility of van der Waals interactions. Comparison of Spreading Behavior of SDS with AOT3 on Water Films. The spreading behavior of both the highly soluble SDS and the sparingly soluble AOT solutions on water films shows many similarities in trends with some notable differences. Upon deposition, a central cap is visible at the point of deposition at early times for AOT, whereas there is no remnant of surfactant deposition in the case of SDS. Spreading rates are on the same order for both surfactants and exhibit a maximum around the cmc. However, while spreading rates increase with increasing film thickness due to reduced viscous effects for AOT, no such variation is seen with SDS. This is attributed to rapid surfactant desorption and relatively significant gravitational effects at larger film thicknesses for SDS, which counterbalance the reduced viscous dissipation. Fingering occurs over a broader range of concentrations in the case of SDS, with fingering patterns apparent at a surfactant concentration of 0.4 cmc, while in the case of AOT fingering is not seen until 0.8 cmc for all values of the film thickness considered. Fingers developing as a result of SDS deposition are more pronounced, with earlier

Figure 10. Side view of a 9-µL drop of 2.8 cmc (22.4 mM) SDS on a 50 µm water film after (a) 0.07 s and (b) 2 s.

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onset times (